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# Is there more than one “median” formula?

In my work, when individuals refer to the "mean" value of a data set, they're typically referring to the arithmetic mean (i.e. "average", or "expected value"). If I provided the geometric mean, people would likely think I'm being snide or non-helpful, as the definition of "mean" is known in advance.

I'm trying to determine if there are multiple definitions of the "median" of a data set. For example, one of the definitions provided by a colleague for finding the median of a data set with an even number of elements would be:

### Algorithm 'A'

• Divide the number of elements by two, round down.
• That value is the index of the median.
• i.e. For the following set, the median would be 5.
• [4, 5, 6, 7]

This seems to make sense, though the rounding-down aspect seems a bit arbitrary.

### Algorithm 'B'

In any case, another colleague has proposed a separate algorithm, which was in a stats textbook of his (need to get the name and author):

• Divide the number of elements by 2, and keep a copy of the rounded-up and rounded-down integers. Name them n_lo and n_hi.
• Take the arithmetic mean of the elements at n_lo and n_hi.
• i.e. For the following set, the median would be (5+6)/2 = 5.5.
• [4, 5, 6, 7]

This seems wrong though, as the median value, 5.5 in this case, isn't actually in the original data set. When we swapped out algorithm 'A' for 'B' in some test code, it broke horribly (as we expected).

# Question

Is there a formal "name" for these two approaches to calculating the median of a data set? i.e. "lesser-of-the-two median" versus "average-the-middle-elements-and-make-new-data median"?

• I have never seen algorithm "A" considered a median. It shouldn't be a problem that a descriptive statistic of the central tendency of data is not among the data themselves: after all, most means are not in the data, either. A more fundamental property we would like the median to have is that it does not change when the sequence of data is reversed, since ordering data from smallest to largest or largest to smallest is an arbitrary matter of taste. For this reason most authors define the median as in algorithm "B," because that is by far the simplest possible order-invariant procedure. – whuber Sep 18 '18 at 16:15
• @whuber Algorithm 'A' is sometimes called the low-median. There is also of course a corresponding High-median. Typically the median is the average of the two (which may or may not be one element from the set the median is computed over). – user603 Sep 18 '18 at 17:24
• A good time and place to repeat the comment that the two central values in an ordered sample with an even number of observations -- like 3 and 4 in 1, 2, 3, 4, 5, 6 -- are to be regarded as comedians (independently quipped by S.M. Stigler, R. Koenker, and no doubt others). – Nick Cox Sep 18 '18 at 17:38
• Aren’t both algorithms missing the crucial step of sorting the data? – Emil Sep 18 '18 at 19:53
• If you need your "median" to always be an element of the data set, you might actually be looking for a medoid. – Ilmari Karonen Sep 18 '18 at 21:59

TL;DR - I'm not aware of specific names being given to different estimators of sample medians. Methods to estimate sample statistics from some data are rather fussy and different resources give different definitions.

In Hogg, McKean and Craig's Introduction to Mathematical Statistics, the authors provide a definition of medians of random samples, but only in the case that there are an odd number of samples! The authors write

Certain functions of the order statistics are important statistics themselves... if $$n$$ is odd, $$Y_{(n+1)/2}$$ ... is called the median of the random sample.

The authors provide no guidance on what to do if you have an even number of samples. (Note that $$Y_i$$ is the $$i$$th smallest datum.)

But this seems unnecessarily restrictive; I would prefer to be able to define a median of a random sample for even or odd $$n$$. Moreover, I would like the median to be unique. Given these two requirements, I have to make some decisions about how to best find a unique sample median. Both Algorithm A and Algorithm B satisfy these requirements. Imposing additional requirements could eliminate either or both from consideration.

Algorithm B has the property that half the data fall above the value, and half the data fall below the value. In light of the definition of the median of a random variable, this seems nice.

Whether or not a particular estimator breaks unit tests is a property of the unit tests -- unit tests written against a specific estimator won't necessarily hold when you substitute another estimator. In the ideal case, the unit tests were chosen because they reflect the critical needs of your organization, not because of a doctrinaire argument over definitions.

• (+1) We can add too that (1) When values come with weights then the definition of medians in principle and in practice must cover that too. (Implicitly in answers so far, all weights are equal, therefore immaterial.) While linear interpolation in the cumulative sum of weights is simplest, there are situations where other kinds of interpolation might make sense. (2) More rigorous definitions of median are usually intended to cover discrete and continuous and hybrid distributions alike, including those with spikes of probability somewhere. – Nick Cox Sep 18 '18 at 17:43

What @Sycorax says.

As a matter of fact, there are surprisingly many definitions of general quantiles, so in particular also of medians. Hyndman & Fan (1996, The American Statistician) give an overview that is, AFAIK, still comprehensive. The different types do not have formal names. You may simply need to be clear on which type you are using. (It often does not make a big difference with data sets of realistic sizes.)

Note that it is commonly accepted to have a value that is not present in the data set as the median, e.g., 5.5 as a median for (4, 5, 6, 7). This is the default behavior for R:

> median(4:7)
[1] 5.5

R's median() by default uses type 7 of Hyndman & Fan's classification.

• +1 for "It often does not make a big difference with data sets of realistic sizes." I'll steal that, instead of my usual "if it makes material a difference, you probably need more data." :) – Jason Sep 19 '18 at 4:49
• If you have a binary variable with values 0, 1 (say) and with about equally many 0s and 1s (mean close to 0.5) then large sample size will not necessarily stop the reported median flipping back and forth between 0, 0.5 and 1. Mosteller and Tukey (Data Analysis and Regression 1977) cite strongly bimodal and nearly symmetric distributions as cases where the median might not behave especially well. – Nick Cox Sep 20 '18 at 8:31

In R's mad function, it uses the terms "lo-median" to describe your algorithm A, "hi-median" to describe rounding up instead, and just "median" to describe your algorithm B (which, as others have noted is by far the most common definition).

Curiously, there is no such option on R's median() function! (But R's quantile() has type for fine control.)


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# Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then 1. $Cl(A) = Cl(Int(A))$ 2. $Int(A) = Int(Cl(A))$

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then

1. $Cl(A) = Cl(Int(A))$
2. $Int(A) = Int(Cl(A))$

I believe both of these statements are false and I think I have two counterexamples and I just want to double check that I am correct. Both these counterexamples are in the usual topology.

1. Let $A= [0,1] \cup \{2\}$ then $Cl(A) = [0,1] \cup \{2\}$ and the $Int(A) = (0,1)$ therefore the $Cl(Int(A))= [0,1]$ which does not equal the Cl(A).

2. Let $A= (0,1) \cup (1,2)$ then the $Cl(A) = [0,2]$ and the $Int(A)= (0,1) \cup (1,2)$ therefore the $Int(Cl(A))= (0,2$) which does not equal the interior of A.

My definition of closure is:
Let $(X,\mathfrak T)$ be a topological space and let $A \subseteq X$ . The closure of $A$ is $Cl(A) = \bigcap \{U \subseteq X: U$ is a closed set and $A \subseteq U\}$ Based on this I know $A \subseteq Cl(A)$

Am I correct? After typing this I am starting to doubt my calculations for the closure of both sets.

Your work looks correct to me! You could have gone with an even easier example for $(1)$ like letting $A = \{0\}$, but what you did is certainly fine. For your second problem I think you picked the perfect set to disprove the claim.
Your examples are correct. Something that I personally like to use for questions like these are the rational numbers $\mathbb{Q}$. Note that it has empty interior and its closure equals the whole real line. So it serves as a counter example for both $1.$ and $2.$

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# Chapter 5 Slutsky’s Theorem

## 5.1 Theorem in plain English

Slutsky’s Theorem allows us to make claims about the convergence of random variables. It states that a random variable converging to some distribution $$X$$, when multiplied by a variable converging in probability on some constant $$a$$, converges in distribution to $$a \times X$$. Similarly, if you add the two random variables, they converge in distribution to $$a$$ plus $$X$$. More formally, the theorem states that if $$X_n \xrightarrow{d} X$$ and $$A_n \xrightarrow{p}$$, where $$a$$ is a constant, then:

1. $$X_n + A_n \xrightarrow{d} X + a$$
2. $$A_nX_n \xrightarrow{d} aX$$

Note that if $$A_n$$ or $$B_n$$ do not converge in probability to constants, and instead converge towards some distribution, then Slutsky’s Theorem does not hold. More trivially, if all variables converge in probability to constants, then $$A_nX_n + B_n \xrightarrow{p} aX+B$$.

## 5.2 Coded demonstration

This theorem is reasonably intuitive. Suppose that the random variable $$X_n$$ converges in distribution to a standard normal distribution $$N(0,1)$$. For part 1) of the Theorem, note that when we multiply a standard normal by a constant we “stretch” the distribution (assuming $$|a|>1$$, else we “compress” it). Recall from the discussion of the standard normal in Chapter 5 that $$aN(0,1) = N(0,a^2)$$. As $$n$$ approaches infinity, therefore, by definition $$A_n \xrightarrow{p} a$$, and so the degree to which the standard normal is stretched will converge to that constant too. To demonstrate this feature visually, consider the following simulation:

library(ggplot2)
set.seed(89)
N <- c(10,20,500)

results <- data.frame(n = as.factor(levels(N)),
X_n = as.numeric(),
A_n = as.numeric(),
ax = as.numeric())

for (n in N) {
X_n <- rnorm(n)
A_n <- 2 + exp(-n)
aX <- A_n * X_n

results <- rbind(results, cbind(n, X_n, A_n, aX))

}

ggplot(results, aes(x = aX)) +
facet_wrap(n~., ncol = 3, labeller = "label_both") +
geom_density() +
labs(y = "p(aX)")

Here we have defined two random variables: X_n is a standard normal, and A_n converges in value to 2. Varying the value of n, I take $$n$$ draws from a standard normal distribution and calculate the value the converging constant $$A_n$$. I then generate the product of these two variables. The figure plots the resulting distribution aX. We can see that as n increases, the distribution becomes increasingly normal, remains centred around 0 and the variance approaches 4 (since 95% of the curve is approximately bounded between $$0 \pm 2\times\sqrt{var(aX)} = 0 \pm 2\times2 = 0 \pm 4$$).

Similarly, if we add the constant $$a$$ to a standard distribution, the effect is to shift the distribution in its entirety (since a constant has no variance, it does not ‘’stretch’’ the distribution). As $$A_n$$ converges in probability, therefore, the shift converges on the constant $$a$$. Again, we can demonstrate this result in R:

library(ggplot2)
set.seed(89)
N <- c(10,20,500)

results <- data.frame(n = as.factor(levels(N)),
X_n = as.numeric(),
A_n = as.numeric(),
a_plus_X= as.numeric())

for (n in N) {
X_n <- rnorm(n)
A_n <- 2 + exp(-n)
a_plus_X <- A_n + X_n

results <- rbind(results, cbind(n, X_n, A_n, a_plus_X))

}

ggplot(results, aes(x = a_plus_X)) +
facet_wrap(n~., ncol = 3, labeller = "label_both") +
geom_density() +
geom_vline(xintercept = 2, linetype = "dashed") +
labs(y = "p(a+X)", x="a+X")

As n becomes larger, the resulting distribution becomes approximately normal, with variance of 1 and a mean value centred around $$0 + a = 2$$.

Slutsky’s Theorem is so useful precisely because it allows us to combine multiple random variables with known asymptotics, and retain this knowledge i.e. we know what the resultant distribution will converge to assuming $$n \to \infty$$.

## 5.3 Proof of Slutsky’s Theorem

Despite the intuitive appeal of Slutsky’s Theorem, the proof is less straightforward. It relies on the continuous mapping theorem (CMT), which in turns rests on several other theorems such as the Portmanteau Theorem. To avoid the rabbit hole of proving all necessary antecedent theorems, I simply introduce and state the continuous mapping theorem (CMT) here, and then show how this can be used to prove Slutsky’s Theorem.

### 5.3.1 CMT

The continuous mapping theorem states that if there is some random variable such that $$X_n \xrightarrow{d} X$$, then $$g(X_n) \xrightarrow{d} g(X)$$, so long as $$g$$ is a continuous function. In approximate terms (which are adequate for our purpose), a continuous function is one in which for a given domain the function can be represented as an single unbroken curve (or hyperplane in many dimensions). For example, consider the graph of $$f(x) = x^{-1}$$. For the domain $$D_{+}: \mathbb{R} > 0$$, this function is continuous. But for the domain $$D_{\infty}: \mathbb{R}$$, the function is discontinuous because the function is undefined when $$x = 0$$.

In short, CMT states that a continuous function preserves the asymptotic limits of a random variable. More broadly (and again, I do not prove this here), CMT entails that $$g(P_n,Q_n,...,Z_n) \xrightarrow{d} g(P,Q,...,Z)$$ if all $$P_n, Q_n,...$$ etc. converge in distribution to $$P,Q,...$$ respectively.

### 5.3.2 Proof using CMT

How does this help prove Slutsky’s Theorem? We know by the definitions in Slutsky’s Theorem that $$X_n \xrightarrow{d} X$$ and, by a similar logic, we know that $$A_n \xrightarrow{d} a$$ (since $$A_n \xrightarrow{p} a$$, and converging in probability entails converging in distribution). So we can note that the joint vector $$(X_n, A_n) \xrightarrow{d} (X,a)$$. By CMT, therefore, $$g(X_n, A_n) \xrightarrow{d} g(X,a)$$. Hence, any continuous function $$g$$ will preserve the limits of the respective distributions.

Given this result, it is sufficient to note that both addition and multiplication are continuous functions. Again, I do not show this here but the continuity of addition and multiplication (both scalar and vector) can be proved mathematically (for example see one such proof here For an intuitive explanation, think about the diagonal line $$y=X$$ – any multiplication of that line is still a single, uninterrupted line ($$y = aX$$) assuming $$a$$ is a constant. Similarly, adding a constant to the function of a line also yields an uninterrupted line (e.g. $$y= X + a$$).

Hence, CMT guarantees both parts 1 and 2 of the Theorem. $$\square$$

## 5.4 Applications

Slutsky’s Theorem is a workhorse theorem that allows researchers to make claims about the limiting distributions of multiple random variables. Instead of being used in applied settings, it typically underpins the modelling strategies used in applied research. For example, Aronow and Samii (2016) consider the problem of weighting multiple regression when the data sample is unrepresentative of the population of interest. In their proofs, they apply Slutsky’s Theorem at two different points to prove that their weighted regression estimates converge in probability on the weighted expectation of individual treatment effects, and subsequently, that the same coefficient converges in probability to the true average treatment effect in the population.

### 5.4.1 Proving the consistency of sample variance, and the normality of the t-statistic

In the remainder of this chapter, I consider applications of both Central Mapping Theorem and Slutsky’s Theorem in fundamental statistical proofs. I first show how CMT can be used to prove the consistency of the variance of a random variable, and subsequently how in combination with Slutsky’s Theorem this helps prove the normality of a t-statistic. These examples are developed from David Hunter’s notes on asymptotic theory that accompany his Penn State course in large-sample theory.

#### 5.4.1.1 Consistency of the sample variance estimator

First, let us define the sample variance ($$s^2_n$$) of a sequence of i.i.d random variables drawn from a distribution $$X$$ with $$\mathbb{E}[X] = \mu$$ and $$var(X) = \sigma^2$$ as:

$s^2_n = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X_n})^2.$

We can show that the sample variance formula above is a consistent estimator of the true variance $$\sigma^2$$. That is, as the sequence of i.i.d. random variables $$X_1, X_2,...$$ increases in length, the sample variance estimator of that sequence converges in probability to the true variance value $$\sigma^2$$.

We can prove this by redefining $$s^2$$ as follows:

$s^2_n = \frac{n}{n-1} \left[\frac{1}{n}\sum_{i=1}^n(X_i - \mu)^2 - (\bar{X_n}-\mu)^2 \right],$ which clearly simplifies to the conventional definition of $$s^2$$ as first introduced.

From here, we can note using WLLN, that $$(\bar{X_n} - \mu) \xrightarrow{p} 0$$, and hence that $$(\bar{X_n}-\mu)^2 \xrightarrow{p} 0$$. Note that this term converges in probability to a constant. Moreover, $$\frac{1}{n}\sum_{i=1}^n(X_i - \mu)^2 \xrightarrow{p} \mathbb{E}[X_i - \mu] = var(X) = \sigma^2$$, by definition.

Now let us define an arbitrary continuous function $$g(A_n,B_n)$$. We know by CMT that if $$A_n \xrightarrow{p} A, B_n \xrightarrow{p} B$$ then $$g(A_n, B_n) \xrightarrow{p} g(A,B)$$. And hence, using the implications above we know that for any continuous function $$g$$ that $$g(\frac{1}{n}\sum_{i=1}^n(X_i - \mu)^2, (\bar{X_n} - \mu)) \xrightarrow{p} g(\sigma^2,0)$$.

Since subtraction is a continuous function, we therefore know that:

$\left[\frac{1}{n}\sum_{i=1}^n(X_i - \mu)^2 - (\bar{X_n}-\mu)^2\right] \xrightarrow{p} \left[\sigma^2 - 0\right] = \sigma^2.$

Separately we can intuitively see that $$\frac{n}{n-1} \xrightarrow{p} 1.$$ Hence, by applying CMT again to this converging variable multiplied by the converging limit of the above (since multiplication is a continuous function), we can see that:

$s^2_n \xrightarrow{p} 1 \times \sigma^2 = \sigma^2 \;\;\; \square$

#### 5.4.1.2 Normality of the t-statistic

Let’s define a t-statistic as:

$t_n = \frac{\sqrt{n}(\bar{X}_n-\mu)}{\sqrt{\hat{\sigma^2}}}$

By the Central Limit Theorem (CLT, Chapter 5), we know that for a random variable $$X$$ with mean $$\mu$$ and variance $$\sigma^2$$ that $$\sqrt{n}(\bar{X}_n - \mu) \xrightarrow{d} N(0,\sigma^2)$$.

We also know from the proof above that if $$\hat{\sigma}^2 = s^2$$ then $$\hat{\sigma}^2 \xrightarrow{p} \sigma^2$$ – a constant. Given this, we can also note that $$\frac{1}{\hat{\sigma^2}} \xrightarrow{p} \frac{1}{\sigma^2}$$.

Hence, by Slutsky’s Theorem:

\begin{align} \sqrt{n}(\bar{X}_n-\mu) \times \frac{1}{\sqrt{\sigma^2}} &\xrightarrow{d} N(0,\sigma^2) \times \frac{1}{\sqrt{\sigma^2}} \\ &= \sigma N(0,1) \times \frac{1}{\sigma}\\ &= N(0,1) \;\;\; \square \end{align}

One brief aspect of this proof that is noteworthy is that since Slutsky’s Theorem rests on the CMT, the application of Slutsky’s Theorem requires that the function of the variables $$g$$ (in this case multiplication) is continuous and defined for the specified domain. Note that $$\frac{1}{0}$$ is undefined and therefore that the above proof only holds when we assume $$\sigma^2 > 0$$. Hence why in many statistics textbooks and discussions of model asymptotics, authors note that they must assume a positive, non-zero variance.

### References

Aronow, Peter M., and Cyrus Samii. 2016. “Does Regression Produce Representative Estimates of Causal Effects?” American Journal of Political Science 60 (1): 250–67. https://doi.org/10.1111/ajps.12185.


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anonymous 5 years ago If the rotation matrix in the rotation equation is matrix[x_y]=matrix[-√(2)/2,-√(2)/2,_√(2)/2,-√(2)/2,]*matrix[x'_y']. What angle of rotation is being used?

1. anonymous

I just wrote a huge piece and this site kiboshed what I wrote. Did you get anything for this question?

2. anonymous

Basically, identify the entries in your matrix with each of the trig. functions that describe a rotation matrix in two dimensions. You should have then,$\sin \theta = -\frac{1}{\sqrt{2}}$and $\cos \theta =-\frac{1}{\sqrt{2}}$

3. anonymous

In one rotation, $\sin \theta = -\frac{1}{\sqrt{2}} \rightarrow \theta =225^o, 315^o$

4. anonymous

$\cos \theta = -\frac{1}{\sqrt{2}} \rightarrow \theta = 135^o, 225^o$

5. anonymous

The only angle BOTH trig. functions have in common is 225 degrees. Unless I've misread your matrix entries, this is the angle of rotation.


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date: 27 June 2022

# Goal-Setting Theory: Causal Relationships, Mediators, and Moderators

• Gary P. LathamGary P. LathamRotman School of Management, University of Toronto

### Summary

Consciously setting a specific, difficult, challenging goal leads to high performance for four reasons. Specificity results in (1) the choice to focus on goal-relevant activities and to ignore those that are irrelevant. Challenge leads to an increase in (2) effort and (3) persistence to attain the goal. The combination of specificity and difficulty cue (4) the search for strategies to attain the goal. However, for this to occur, an individual or team must have the ability and the situational resources to attain the goal. In addition, the goal must be important; there must be commitment to goal attainment. Finally, feedback must be provided on goal progress so that adjustments can be made, if necessary, regarding effort or strategy for attaining the goal.

### Subjects

• Organizational and Institutional Psychology

Theories in psychology enable predicting, explaining, and influencing behavior. To qualify as a theory, causal relationships must be specified, mediators that explain the causal relationships must be identified, and the boundary conditions within which the theory is applicable must be known. Locke and Latham’s (1990, 2002, 2013; Latham & Locke, 2007, in press) goal setting theory of motivation satisfies these three criteria. This article explains the theory, describing the inductive method used to develop that theory, examples of experiments supporting the theory, the perils in ignoring the theory’s moderator variables, the various ways of setting goals, and the economic benefits of setting specific, high goals. It concludes with a discussion of goals that are primed in the subconscious.

### Goal Setting Theory

With regard to causal relationships, goal setting theory makes three assertions. First, specific, high goals lead to higher performance than setting no goals or even a vague goal such as the exhortation to “do your best.” Second, the higher the goal, the higher an individual’s performance. Third, such variables as feedback or knowledge of one’s results, participation in the making of decisions, or competition with others have little or no effect on a person’s behavior unless they lead to the setting of a goal that is both specific and difficult.

The mediators that explain why specific, high goals increase an individual’s performance are four-fold. First, consistent with the definition of motivation, a specific goal involves the choice to take action to pursue X to the exclusion of other factors. Thus a goal that is specific enables people to focus, to have a purpose in what they do rather than to meander relatively aimlessly. Second, a goal that is difficult as well as specific engenders effort, a second cornerstone of motivation. Hence, the higher a specific goal, the more effort that is expended. The third mediator is persistence. When a goal that is chosen is specific rather than vague, and difficult rather than easy, people persist in their pursuit of the goal until it is attained. The problem with a vague goal is that it allows multiple interpretations as to whether the goal is attained (e.g., my goal is to lose weight). Thus people may pat themselves on the back undeservedly. A specific goal makes explicit the desired level of performance and hence whether it has been attained (e.g., my goal is to lose 15 pounds). Persistence is a third cornerstone of motivation (Latham, 2012). A fourth mediator is relatively cognitive in nature. Setting a specific, high goal cues an individual’s extant strategies necessary to attain it.

Moderator variables specify the boundary conditions within which the above assertions are applicable. A specific, high goal leads to higher performance than an easier goal, a vague goal, or no goal only under four conditions. First, the person must have the ability to attain a high goal or the person is unlikely to commit to attaining it. Goal commitment is a second, and arguably the most important, moderator variable. If an individual is not committed to goal attainment, by definition that individual does not have a goal. Third, people must receive feedback on their performance in relation to the goal they are striving to attain. In the absence of feedback, people lack the information necessary to ascertain whether they should adhere to or change their strategy, a mediator in goal setting theory, for goal attainment.1 Finally, the requisite resources must be available for goal attainment. Situational constraints can mitigate ability for and commitment to goal attainment, no matter how difficult or easy the goal may be.

### Induction

Many, if not most, theories in psychology are developed through deduction. The authors of a deductive theory typically begin with plausible statements based on their observations. They then make predictions regarding the relationships between variables, they offer an explanation for the alleged relationships (i.e., mediators), and they state the conditions (e.g., boundary/moderators) under which the relationships should occur. Finally, empirical experiments are conducted to test the predicted causal relationships, mediators, and moderators that the theorist expects to observe. An example of a deductively derived theory in organizational psychology is Vroom’s (1964) expectancy theory.2

A major limitation of developing a theory through deduction is that it can lead to what Kahneman (2011, p. 211), the Nobel Prize–winning psychologist, labeled as theory-induced blindness: “Once you have scripted a theory and used it as a tool in your thinking, it is extraordinarily difficult to notice its flaws. If you come upon an observation that does not seem to fit the model, you assume that it must be a perfectly good explanation that you are somehow missing. You give the thought the benefit of the doubt, trusting the community of experts who have accepted it.”

The alternative to deductively developing a theory is to do so inductively. The primary difference between deduction and induction is the time-period or sequential order for conducting empirical experiments. As noted above, the deductive method involves the specification of causal variables, its mediators and moderators, before rather than after conducting empirical experiments to determine whether the deductions can be supported by scientifically obtained evidence. The inductive method requires conducting experiments, accumulating knowledge from these experiments, and then and only then developing a theory.

Empirical research was conducted inductively from the 1960s through the 1980s in both laboratory and field settings before the theory of goal setting was formally developed (Locke & Latham, 1990). The theory is based on nearly 400 studies involving close to 40,000 participants from eight different countries who performed one or more of 88 different tasks. The time span of these tasks ranged from 1 minute to three years. A decade later more than 1,000 studies had been conducted (Mitchell & Daniels, 2003). These studies show that goal setting theory is not only applicable to the motivation of an individual, it is applicable to groups/teams (Kramer, Thayer, & Salas, 2013), departments (Porter & Latham, 2013), and organizations as well (Pritchard et al., 2013; Saari, 2013).

The time span showing the beneficial relationship of goal setting to performance has been shown to be considerably longer than three years. A goal for job promotion at AT&T correlated positively with actual job promotion 25 years later (Howard, 2013).

### Empirical Research

Field experiments have been conducted with pulpwood crews in the southeastern United States. They were matched on such variables as crew size, type of terrain where they were cutting trees, and the level of mechanization they owned to cut the trees. Then they were randomly assigned to the experimental or control condition. The crews that were assigned a specific, high weekly goal to attain had significantly higher productivity the very first week of the three-month experiment than the crews in the control condition who were urged to do their best. “To do one’s best” was not a meaningless exhortation because all the crews were paid on a piece-rate basis. Yet those with a specific, high goal not only had higher productivity (cords per employee hour) throughout the experiment, they also had higher job attendance than those who were urged to do their best. People were now eagerly coming to work because their job, previously viewed by them as tedious, was now viewed as meaningful. The performance goal became a self-evaluative standard for comparing their current performance with their previous performance. The goal also allowed the employees to assess their personal effectiveness relative to others. Thus, the goal setting initiative engendered competition among the crews, competition that could just as easily have occurred, but did not occur, among the crews in the control condition who had been urged to do their best (Latham & Kinne, 1974).

The wood supply of pulp and paper companies sometimes exceeds their processing capacity. Consequently, the companies impose a wood quota where they restrict the number of days they will buy wood to three rather than five. Independently owned logging crews were found to perceive this restriction as a challenging goal. That is, they made the choice to exert the effort necessary and to persist in doing so until they harvested as many trees in three days as they normally did in five (Latham & Locke, 1975).

The beneficial effect of goal setting on job performance has also been shown with high level employees, namely, engineers and scientists with masters and doctoral degrees in an R&D department (Latham, Mitchell, & Dossett, 1978). There were 10 conditions. Employees (1) were assigned a goal, (2) participated in setting the goal, or (3) were urged to do their best. This latter condition was highly relevant for these employees because an ad hoc task force of line managers was examining ways for the company to reduce costs. Rumors abounded that R&D was likely to be reduced in both funding and manpower. The employees in these three goal conditions received feedback in the form of (1) praise, (2) public recognition, or (3) a monetary bonus. This 3 × 3 experimental design yielded nine conditions. A 10th condition was a true control group consisting of engineers/scientists who were not aware that they were involved in this field experiment.

Consistent with the predictions of goal setting theory, the employees in the do-your-best conditions performed no better than the employees in the control condition, even though those in the control condition did not receive performance feedback in the form of praise, public recognition, or a monetary bonus. Remember, the theory states and research shows that feedback only affects performance positively if it leads to the setting of a specific, high goal.

Consistent with goal setting theory, those engineers/scientists whose goals had been assigned to them performed better than those in the do-best and control conditions. But, the highest performing employees were those who had participated in the goal setting process even though their goal commitment was not significantly higher than their peers with assigned goals. The reason for the higher performance is explained by the theory—they set higher goals. The theory states that the higher the goal, the higher the performance, given the presence of the four moderators.

In all of these studies, money was not necessary for goal commitment. When money is tied to goal setting, it should not be tied to only goal attainment but rather to the attainment of subgoals as well as the final goal.

### Perils in Ignoring Moderators

Ignoring the moderators in goal setting theory is done at one’s peril.

#### Feedback

In a dynamic environment where what is true in one time period is no longer true at a later point in time, blindly adhering to a strategy for goal attainment will likely prove to be costly. In such circumstances, proximal goals (i.e., subgoals) should be set. Proximal goals are advantageous for two reasons. First, they are motivational for maintaining focus, effort, and persistence until the distal goal is attained. Second, and arguably more important in a dynamic setting, is the informative nature of proximal goals. They provide feedback as to whether the strategy for attaining the distal goal requires modification (Seijts & Latham, 2001).

#### Ability

A series of laboratory experiments show that setting performance goals that only 10% of participants can attain sometimes leads to unethical behavior, as defined by overstating one’s performance. What is fascinating is that the exaggeration is only done by people who are close to attaining the goal, especially people who in addition receive a monetary bonus for goal attainment. An even more fascinating finding is that people who exaggerated their performance regarding goal attainment did not take the money even though the experiment was designed to allow them to do so (Schweitzer, Ordonez, & Douma, 2004).

Ordonez and colleagues (2009) concluded from their laboratory findings that performance goals should not be set in the workplace. There are at least two problems with their conclusion. First, by setting goals that only 10% of the participants could attain, they failed to take into account one of the theory’s moderator variables, ability. They did so based on Appendix C of Locke and Latham’s (1990) book, where this level of difficulty is recommended solely for laboratory settings to ensure variance in the participants’ performance.3 Second, Ordonez and colleagues failed to realize that the goals set in field settings reflect the values of the leaders and an organization’s culture. Goal setting is both a theory and a technique for increasing performance. As is any technique, goal setting is subject to misuse. Nevertheless, few if any historians blame goal setting for Hitler’s egregious behavior leading to the Holocaust. Rather, they blame his values and the Nazi culture. Few if any management scholars blame goal setting for market penetration into legitimate businesses by the Mafia. Instead, they blame the values of the Mafia leaders and the Mafia culture. Similarly, few if any people in the judiciary blame goal setting for the illegal behavior that took place in the undoing of Enron. The fault has been attributed to the company’s leaders and the values they inculcated throughout the organization.

In the field of education, there are students who set a specific high goal for the grade they want to attain and then cheat to ensure that they attain it. This is particularly true for some students who aspire to attain high grades in order to get into medical school, law school, or any of the graduate departments (e.g., psychology) where the number of acceptances relative to the number of applicants is small. Yet few people argue for the abolishment of grades because they sometimes lead to unethical behavior on the part of students whose values allow them license to engage in it.

As is the case in any endeavor where there are standards, there will likely be people who will lie about or cheat on ways to ensure their attainment. To paraphrase Shakespeare, the fault is not in the goals but in ourselves, that is, the values we hold while pursuing them (Latham & Locke, 2009). In short, goal setting theory provides an excellent framework for managers and employees to increase their performance. Yet as is the case with any scientific theory and/or technique, there is no foolproof way of ensuring that it will not be misused.

#### Situational Resources

During a downturn in the economy, senior management may set performance goals that are perceived by supervisors as too high for them to attain. A correlational study revealed that when supervisors see the goal as exceeding their ability to attain it, when in addition they believe they lack the resources to attain it, they experience “hindrance stress.” This in turn has been shown to correlate with their subsequent abuse of their subordinates (Mawritz, Folger, & Latham, 2014). This correlational finding is consistent with a seminal experiment conducted years ago that showed that frustration leads to aggression (Dollard, Doob, Miller, Mowrer, & Sears, 1939). Hence the importance of taking into account a person or teams resources when setting a goal.

#### Learning Goals

There are occasions when an individual lacks the ability to attain a performance goal (e.g., generate X new revenue streams; develop Y products that will not be easily copied by competitors) regardless of its specificity or level of difficulty. People may simply lack knowledge of the strategies necessary for goal attainment. When this is the case, urging people to do their best typically leads to higher performance than a specific, high performance goal. This is because the latter often increases anxiety that in turn yields to a mindless scramble to find solutions (Mone & Shalley, 1995). The solution is to set a specific, challenging learning goal. A learning goal shifts attention from the desired performance level to be attained to the discovery of X processes, procedures, systems, or strategies. Laboratory experiments show consistently that a specific, high learning goal leads to higher performance than urging people to do their best (e.g., Latham, Seijts, & Crim, 2008). A field study revealed that during a turbulent economic cycle, only learning goals correlated positively with a department’s performance (Porter & Latham, 2013).

### Goal Setting Method

Goal setting theory is silent about the optimum method for setting the goal. Programmatic research involving at least 11 experiments revealed that an assigned goal is as effective a method for increasing task performance as a goal that is set participatively between the employee and the supervisor. However, there are a number of caveats regarding this conclusion. First, the statement is correct as long as there are no significant differences in the level of goal difficulty between the two goal setting methods (Latham & Saari, 1979a). If the assigned goal is significantly higher than the goal the employee was involved in choosing, consistent with goal setting theory, the higher goal leads to higher performance (Latham, Steele, & Saari, 1982). Second, a rationale must accompany an assigned goal; a goal that is assigned curtly is unlikely to increase an individual’s performance (Latham, Erez, & Locke, 1988).

Latham and Steele (1983) manipulated independently participative decision-making (PDM) on task strategy versus assigned, PDM, and do-best goals. The results revealed that only setting a specific high goal increased performance. PDM had no effect. However, another caveat is in order. In all the field and laboratory experiments conducted by Latham, the supervisor/experimenter who assigned the goal did so in a supportive manner in interacting with the employee/participant. In the one laboratory experiment where the experimenter deliberately behaved in a nonsupportive manner, the participants set significantly lower goals than those who had been randomly assigned to the supportive condition (Latham & Saari, 1979b).

These findings are consistent with Dember (1974), who, after reviewing the literature on motivation, concluded that in the right setting, being told to do something is tantamount to being motivated to do it. It seems that instructions that are deemed appropriate by an individual take on the formation of powerful internally generated drives. Similarly, Salancik and Pfeffer (1977) concluded that the assignment of a goal implies to an individual that she or he is capable of attaining it. A meta-analysis by Wagner and Gooding (1987) of the research on this topic revealed no noteworthy relationship between participation in decision-making and either job performance or job satisfaction.

Subsequent research has shown that researchers were going down the wrong path in their attempts to show the beneficial effect of an individual’s participation in deciding on the goal that should be set. When there is a beneficial effect for PDM, the effect is primarily cognitive rather than motivational. When the task is complex, participants ask more questions than do those who are assigned goals. The information gleaned from these questions improve their performance (Latham & Saari, 1979b). Moreover, participation in decision-making can lead to the development of an effective strategy for attaining the goal; this in turn increases a participant’s self-efficacy that the goal is attainable. Strategy and self-efficacy have been shown to have a reciprocal effect on one another, and both have been shown to mediate the PDM–performance relationship (Latham, Winters, & Locke, 1994). Self-efficacy is defined as one’s belief or confidence that the goal is attainable. Self-efficacy influences goal choice and goal commitment (Bandura, 2013). That is, people with high self-efficacy choose and commit to high goals. They are resilient in the face of goal setbacks. People with low self-efficacy quickly abandon the goal when they experience difficulties in goal pursuit.

With regard to self-set goals, an experiment conducted in a government agency showed that goal difficulty level, goal acceptance, goal attainment, and task performance were as effective as goals that were assigned or set in a participatory manner (Latham & Marshall, 1982). In summary, from a motivational standpoint, one method of goal setting is not necessarily more effective than another. From a motivational standpoint, the critical factor for increasing performance is the level of difficulty of the goal that is set.

There are contexts where only self-set goals are appropriate, especially off-the-job settings. Frayne and Latham (1987) successfully taught a self-management program to unionized state government employees whose job attendance was low. The core of the program was goal setting and its moderator, feedback, regarding weekly/monthly job attendance. Millman and Latham (2001) taught displaced managers who had been out of work for 13 months to set a goal for re-employment and to use verbal self-guidance to increase self-efficacy for goal attainment, namely, re-employment. Similarly, Latham and Budworth (2006) used this approach for enabling Native North Americans to obtain employment, as did Yanar, Budworth, and Latham (2009) for enabling women in Turkey over the age of 40 to attain their goal of becoming re-employed.

### Economic Benefits

Latham and Baldes (1975) reported that goal setting regarding loading trucks to their maximum legal weight saved a forest-product company a quarter of a million dollars over 9 months. Schmidt (2013) used utility analysis procedures to estimate the economic value of goal setting to employers in today’s dollars. Specifically, he examined the difference between do-your-best or no-goal conditions and specific, difficult goals. The dollar value figures indicate the increase in revenue from improved performance. His sample is based on four meta-analyses of goal setting experiments. These four meta-analyses included 19,839 data points. The results revealed that the average increase in employee output per year as a result of a goal setting intervention is $9,200. A goal setting intervention that lasts for 5 years, involving only 35 employees, costing$200 per employee yields an organization \$1,603,000 due to the increase in production. The percentage increase in employee output is 9.2%.

### Priming Goals in the Subconscious

In suggesting directions for future research on motivation, Locke and Latham (2004) pointed to a limitation of goal setting theory, namely that it is a cognitive theory that ignores the subconscious. This is a limitation because cognitive resources are limited (Miller, 1956). In contrast to consciousness, the subconscious is a vast reservoir of information (Vorhauser-Smith, 2011).

Bargh’s (1990) automaticity model focuses on goals that are primed in the subconscious. As is the case with goal setting theory, the model asserts that a goal is a mental representation of a desired state that is pursued through action. The goal can be primed in the subconscious in one of two ways, subliminally or supraliminally. The model further asserts that an external stimulus in the environment can passively, subtly, and unobtrusively activate a goal. If the priming is done subliminally, the stimulus is presented below focal awareness; if the priming is done supraliminally, the individual is aware of the stimulus yet is unaware of its influence on subsequent behavior. In short, the model states that a primed goal guides behavior in the absence of conscious intention. In agreement with goal setting theory, this occurs only if the goal is important to the person.

After reviewing the literature in social psychology on primed goals, Latham, Stajkovic, and Locke (2010) concluded that this methodology should be examined with regard to organizational behavior. A laboratory experiment involving brainstorming had revealed that making sentences from scrambled achievement-related words (e.g., win) led to higher performance than making sentences from scrambled neutral words (e.g., tree) (Stajkovic, Locke & Blair, 2006).

Most social psychology experiments on primed goals involve the presentation of the prime and the measurement of the dependent variable seconds/minutes later (e.g., length of time to walk from a laboratory to an elevator). Thus a field experiment was conducted in a call center to determine whether laboratory findings on primed goals generalize to work settings.4 As was the case in the preceding laboratory experiment, a supraliminal prime was used to prime the goal for achievement, namely a photograph of a woman winning a race. At the end of the work shift, the employees with the primed goal raised significantly more money from donors than did those in the control group (Shantz & Latham, 2009). These results were replicated in two additional call centers (Shantz & Latham, 2011). Of further practical significance was the finding that the two goals, consciously set and primed, led to higher productivity than either goal alone (Shantz & Latham, 2009).

Consistent with goal setting theory, Latham and Piccolo (2012) hypothesized that a context-specific goal that is primed leads to higher performance than a more general goal. The results from a fourth call center supported this hypothesis. The employees who were primed with a photograph of call center employees in the work-place raised 16% more money than those who were primed with the photograph of the racer, and 85% more money than the employees in the control group. Those employees who were primed with the racer raised 60% more money than those in the control condition. Of further practical significance is the finding that these results were obtained on the first day and lasted throughout the four-day work week.

What remains to be explored is the time length with which a primed goal influences behavior and the frequency with which the prime should be changed to maintain high performance. The effect of a consciously set specific, high goal on an employee’s behavior has been shown to last for months, if not years (e.g., Latham & Baldes, 1975; Howard, 2013).

### Conclusion

Goal setting theory is among the most valid and useful theories of motivation of organizational behavior (Lee & Earley, 1992; Miner, 2003; Pinder, 1998). The theory is straightforward: set a specific, high goal and the result will be high performance. This is because the specific goal that is chosen focuses an individual’s attention on goal-relevant activities. Individuals exert far more effort for a higher goal than they do for an easier one, and they persist in doing so until the goal is attained. However, the beneficial effect of the goal–performance relationship only occurs if the person has the ability to attain the goal, is committed to goal attainment, receives feedback on goal pursuit, and has the requisite resources to pursue and attain the goal. Dysfunctional behavior is likely to occur if the theory’s moderator variables are ignored.

The theory was developed through induction rather than deduction. Thus it is an open rather than a closed theory; that is, goal setting theory is open to modification through findings from subsequent research (e.g., the discovery of the necessity for setting learning rather than performance goals). A new research frontier is the exploration of the effect of primed goals on organizational behavior and the extent to which their effects on performance are similar to or differ from the effects that have been found with goals that are consciously set.

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• Porter, R. L., & Latham, G. P. (2013). The effect of employee learning goals and goal commitment on departmental performance. Journal of Leadership and Organizational Studies, 20, 62–68.
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• Wood, R. E., Whelan, J., Sojo, V., & Wong M. (2013). Goals, goal orientations, strategies, and performance. In E. A. Locke & G. P. Latham (Eds.), New developments in goal setting and task performance. New York: Routledge.
• Vorhauser-Smith, S. (2011). The neuroscience of talent management. Employment Relations Today, 28, 17–22.
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### Notes

• 1. Wood et al. (2013) found that strategy can also moderate, in addition to mediate, the goal–performance relationship.

• 2. Interestingly, Vroom himself never conducted an empirical experiment to test any aspect of his theory (Vroom, 2015).

• 3. To the author’s knowledge, no field experiment has set goals that only 10% of the participants could attain. Most practitioners adopt the heuristic explained by Mealiea and Latham (1996), namely, SMART—that is, a goal should be specific, measurable, attainable, relevant, and have a time frame for its attainment.

• 4. Findings from laboratory settings regarding consciously set specific, high goals do have external validity for the workplace (Latham & Lee, 1986).


text:
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Products of involutory matrices over rings. (English) Zbl 0836.15006
An involution in a group $G$ is an element $a$ of order 2. Let $a \in GL_n A$ be a matrix over a commutative ring $A$ with 1 such that $(\text{det} a)^2 = 1$. If $a$ is cyclic, it can be written as a product of at most three involutions. The paper proves that when $A$ satisfies the first Bass stable range condition, then $a$ can be written as a product of at most five involutions. If in addition either $n \le 3$ or $n = 4$ and $\text{det} a = - 1$, then $a$ can be written as a product of at most four involutions. When $A$ is a Dedekind ring of arithmetic type, the number of involutions needed to express $a$ is uniformly bounded for any $n \ge 3$. When $A = \bbfC[x]$ is the polynomial ring with complex coefficients, the number of involutions is unbounded for any $n\ge 2$.
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##### References:
 [1] Ambrosiewicz, E.: Powers of sets of involutions in linear groups. Demonstratio math. 24, 311-314 (1991) · Zbl 0806.20038 [2] Ambrosiewicz, J.: On the square of sets of linear groups. Rend. sem. Mat. univ. Padova 75, 253-256 (1985) · Zbl 0591.20048 [3] . Linear and multilinear algebra 4, 69 (1976) [4] Ballantine, C. S.: Products of involutory matrices I. Linear and multilinear algebra 5, 53-62 (1977) · Zbl 0364.15017 [5] Berggren, J.: Finite groups in which every element is conjugate to its inverse. Pacific J. Math. 28, 289-293 (1969) · Zbl 0172.03101 [6] Dennis, R. K.; Vaserstein, L. N.: On a question of M. Newman on the number of commutators. J. algebra 118, 150-161 (1988) · Zbl 0649.20048 [7] Djoković, D.: Products of two involutions. Arch. math. 18, 582-584 (1967) · Zbl 0153.35502 [8] Ellers, E.: Products of two involutory matrices over skew fields. Linear algebra appl. 26, 59-63 (1979) · Zbl 0405.15010 [9] Gustafson, W. H.: On products of involutions. Paul halmos celebrating 50 years of mathematics (1991) [10] Gustafson, W. H.; Halmos, P. R.; Radjavi, H.: Products of involutions. Linear algebra appl. 13, 157-162 (1976) · Zbl 0325.15009 [11] Hoffman, F.; Paige, E. C.: Products of two involutions in the general linear group. Indiana univ. Math. J. 20, 1017-1020 (1971) · Zbl 0237.20041 [12] Knüppel, F.: $GL{\pm}n(R)$ is 5-reflectional. Abh. math. Sem. univ. Hamburg 61, 41-47 (1991) · Zbl 0747.20025 [13] Knüppel, F.; Nielsen, K.: $SL(V)$ is 4-reflectional. Geom. dedicata. 38, 301-308 (1991) · Zbl 0722.20033 [14] Laffey, T. J.: Factorization of matrices involving symmetric matrices and involutions. Current trends in matrix theory, 175-198 (1987) [15] Liu, K.: Decomposition of matrices into three involutions. Linear algebra appl. 111, 1-24 (1988) · Zbl 0659.15009 [16] Sourour, A.: A factorization theorem for matrices. Linear and multilinear algebra 19, 141-147 (1986) · Zbl 0591.15008 [17] Vaserstein, L. N.; Wheland, E.: Commutators and companion matrices over rings of stable rank 1. Liner algebra appl. 142, 263-277 (1990) · Zbl 0713.15003 [18] Waterhouse, W. C.: Solution to advanced problem 5876. Amer. math. Monthly 81, 1035 (1974) [19] Wonnenburger, M.: Transformations which are products of two involutions. J. math. Mech. 16, 327-338 (1966) · Zbl 0168.03403 [20] Wu, P. Y.: The operator factorization problems. Linear algebra appl. 117, 35-63 (1989) · Zbl 0673.47018


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# A point charge of magnitude q is at the center of a cube with sides of length L. What is the electric flux Φ through each of the six faces of the cube? What would be the flux Φ_1 through a face of the cube if its sides were of length L_{1}?

This article aims to find the electric flux in a cube having six sides. This article uses the concept of electric flux. For a closed gaussian surface electric flux is given by the formula

$\Phi_{e} = \dfrac{Q}{xi_{o}}$

Consider a cube having side length $L$ in which a size $q$ charge is placed at the center. Consider a closed Gaussian surface, which is a cube whose electric flux is $\Phi$, which is given by:

$\Phi=\dfrac{ q } {\xi_{o}}$

The number of lines of force arising from the charge will be divided into six walls. So the electric flux is given by:

$\Phi =\dfrac{q}{6\xi_{o}}$

Part (A)

The electric flux of each of the six faces of the cube is $\Phi = \dfrac{ q } { 6 \xi _{ o } }$.

Electric flux is number of field lines passing per unit area. The flux through any face of the cube is equal to the total flux of the cube divided by six.

Consider the sides of the cube $L_{1}$.

Since the electric flux depends only on the enclosed charge $q$, the flux through each surface would be the same as the previous part, even if the cube’s dimension changes. That is, the electric flux of each of the six walls of the cube, the length $L_{ 1 }$ of which

$\Phi _{1}=\dfrac{q}{6\xi_{o}}$

Part (B)

The electric flux of each of the six faces of the cube is $\Phi _{ 1 }=\dfrac{q}{6\xi _{o}}$.

Since the flux depends on the charge inside the closed surface, the flux through each surface would be same as in the previous section, even if the dimension changes.

## Numerical Result

(a) Electric flux $\Phi$ across each of the six faces of the cube is equal to $\dfrac{ q } { 6 \xi _{ o } }$.

(b) Flux $\Phi _{1}$ over the face of the cube if its sides were $L_{1}$ long is equal to $\dfrac{ q } { 6 \xi _{ o } }$.

## Example

A point charge of magnitude $Q$ is at the cube’s center with sides of length $x$. What is the electric flux $\Phi$ across each of the cube’s six faces? What would be the flux $\Phi$ over the face of the cube if its sides were long $x_{1}$?

Solution

Consider a closed Gaussian surface, which is a cube whose electric flux is $\Phi$ which is given by

$\Phi =\dfrac{Q}{\xi _{o}}$

The number of lines of force arising from the charge will be divided into six walls. So the electric flux is given by

$\Phi =\dfrac{Q}{6\xi _{o}}$

Part (A)

The electric flux of each of the six faces of the cube is $\Phi = \dfrac{Q}{6\xi _{ o }}$.

Consider the sides of the cube $x_{1}$. That is, the electric flux of each of the six walls of the cube, the length $L_{1}$ of which

$\Phi _{1}=\dfrac{Q}{6\xi _{o}}$

Part (B)

The electric flux of each of the six faces of the cube is $\Phi _{1}=\dfrac{Q}{ 6 \xi _{o}}$.


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# Rewrite the Logic to Assign the DDS Telemetry Data

XMLWordPrintable

#### Details

• Type: Story
• Status: Done
• Resolution: Done
• Fix Version/s: None
• Component/s:
• Labels:
• Story Points:
2
• Sprint:
TSSW Sprint - Jun 07 - Jun 21
• Team:
Telescope and Site
• Urgent?:
No

#### Description

Rewrite the logic to assign the DDS telemetry data. This task will check the data to DDS is useful or not, and might add new field if needed. In addition, the DDS telemetry data should come from the GUI data directly.

Some comment from Russell Owen in the slack channel to debug DM-30550:

Te-Wei Tsai and I were talking. I suggested that the first step is for Te-Wei Tsai to print one or both of strutEncoderRaw and strutEncoder_microns (to stdout, a log, or a file – whatever is easy) from the dds telemetry packet when he publishes that packet. This will tell us if the problem is in the low-level controller or in the CSC. Also the code that sets these fields is suspicious and he will probably change it. But first I think it is important to prove whether the problem is actually in the low-level controller or in the CSC.

A longer-term improvement we have tentatively agreed on is to make the telemetry sent to the CSC identical to the telemetry sent to the EUI. That way if we see a problem in one, we expect to see it in the other as well. Unfortunately the amount of data sent to the EUI is likely too much — it may overwhelm the python in the CSC. So a variant of that suggestion is to divide the telemetry into two pieces: one has all the “important” information that is sent to the CSC and the EUI. The other piece has additional information that is only sent to the EUI.
This would be a big improvement over the present situation: the telemetry packet for the CSC is computed entirely separately from the telemetry packet sent to the EUI. A bug in one will not show up in the other.

#### Attachments

1. HexapodEUI.png
60 kB
2. image (1).png
592 kB
3. loggingDataInGUI.png
72 kB
4. messages
3.24 MB
5. telemetry.txt
2.09 MB

#### Activity

Hide
Te-Wei Tsai added a comment -

Get the strut encoder raw data in DDS from the GUI telemetry directly.

Show
Te-Wei Tsai added a comment - Get the strut encoder raw data in DDS from the GUI telemetry directly.
Hide
Te-Wei Tsai added a comment - - edited

The mapping between the GUI and telemetry field is here:

The first sub-figure maps to strutEncoder_microns in DDS telemetry. The following is the mapping of CSC code by Russell:

 self.tel_actuators.set_put(  calibrated=server.telemetry.strut_encoder_microns,  raw=server.telemetry.strut_encoder_raw,  )

Show
Te-Wei Tsai added a comment - - edited The mapping between the GUI and telemetry field is here: The first sub-figure maps to strutEncoder_microns in DDS telemetry. The following is the mapping of CSC code by Russell: self.tel_actuators.set_put( calibrated=server.telemetry.strut_encoder_microns, raw=server.telemetry.strut_encoder_raw, )
Hide
Te-Wei Tsai added a comment -

The GUI can log data:

I need to write the related script to read the binary data.

Show
Te-Wei Tsai added a comment - The GUI can log data: I need to write the related script to read the binary data.
Hide
Te-Wei Tsai added a comment - - edited

Added the following log message just before the sending of DDS packet in sys.c (need to remove this before PR):

 // Print the debug message (REMOVE THIS BEFORE PR)  syslog(LOG_NOTICE,  "DDS telemetry: tlmStruct->strutEncoder_microns[0] is %f.",  ddsTlm.strutEncoder_microns[0]);    mq_send(gTlmDdsSendMsgQueue, (char *)&ddsTlm, sizeof(ddsTlm), 1);

Show
Te-Wei Tsai added a comment - - edited Added the following log message just before the sending of DDS packet in sys.c (need to remove this before PR): // Print the debug message (REMOVE THIS BEFORE PR) syslog(LOG_NOTICE, "DDS telemetry: tlmStruct->strutEncoder_microns[0] is %f." , ddsTlm.strutEncoder_microns[0]);   mq_send(gTlmDdsSendMsgQueue, ( char *)&ddsTlm, sizeof (ddsTlm), 1);
Hide
Holger Drass added a comment -

The following image shows that the low-level controller is not sending zero values:

Show
Holger Drass added a comment - The following image shows that the low-level controller is not sending zero values:
Hide
Te-Wei Tsai added a comment -

The system message in the test is: messages.
Extract the debug message of telemetry publish is: telemetry.txt.

Show
Te-Wei Tsai added a comment - The system message in the test is: messages . Extract the debug message of telemetry publish is: telemetry.txt .
Hide
Te-Wei Tsai added a comment - - edited

Added the GUI binary output file in repo. The binary order is little-endian: Write to Binary File Function.

Show
Te-Wei Tsai added a comment - - edited Added the GUI binary output file in repo. The binary order is little-endian : Write to Binary File Function .
Show
Hide
Russell Owen added a comment -

Reviewed on github

Show
Russell Owen added a comment - Reviewed on github

#### People

Assignee:
Te-Wei Tsai
Reporter:
Te-Wei Tsai
Reviewers:
Russell Owen
Watchers:
Holger Drass, Russell Owen, Te-Wei Tsai
0 Vote for this issue
Watchers:
3 Start watching this issue

#### Dates

Created:
Updated:
Resolved:
Start date:
End date:

#### Jenkins

No builds found.


text:
Heron's formula - math word problems

The Heron's formula is used to calculate the contents of a general triangle using the lengths of its sides. Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is:
$S=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$
, where
$s=\frac{a+b+c}{2}$
is the semiperimeter (half perimeter) of the triangle .

Number of problems found: 33

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The railway is to interconnect in a circular arc the points A, B, and C, whose distances are | AB | = 30 km, AC = 95 km, BC | = 70 km. How long will the track from A to C?
• Annulus from triangle
Calculate the content of the area bounded by a circle circumscribed and a circle inscribed by a triangle with sides a = 25mm, b = 29mm, c = 36mm
• Triangular prism
Calculate the surface of a triangular prism 10 cm high, the base of which is a triangle with sides 6 cm 8 cm and 8 cm
• Area and two angles
Calculate the size of all sides and internal angles of a triangle ABC, if it is given by area S = 501.9; and two internal angles α = 15°28' and β = 45°.
• Gardens
The area of the square garden is 3/4 of the area of the triangular garden with sides of 80 m, 50 m, 50 m. How many meters of the fence do we need to fence a square garden?
• The farmer
The farmer would like to first seed his small field. The required amount depends on the seed area. Field has a triangular shape. The farmer had fenced field, so he knows the lengths of the sides: 119, 111 and 90 meters. Find a suitable way to determine th
• Compute 4
Compute the exact value of the area of the triangle with sides 14 mi, 12 mi, and 12 mi long.
• Park
In the newly built park will be permanently placed a rotating sprayer irrigation of lawns. Determine the largest radius of the circle which can irrigate by sprayer P so not to spray park visitors on line AB. Distance AB = 55 m, AP = 36 m and BP = 28 m.
• Triangle
Plane coordinates of vertices: K[11, -10] L[10, 12] M[1, 3] give Triangle KLM. Calculate its area and its interior angles.
• Prism
Calculate the volume of the rhombic prism. The prism base is a rhombus whose one diagonal is 47 cm, and the edge of the base is 28 cm. The edge length of the base of the prism and height is 3:5.
• Box
Cardboard box shaped quadrangular prism with a rhombic base. Rhombus has a side 5 cm and one diagonal 8 cm long and height of the box is 12 cm. The box will open at the top. How many cm2 of cardboard we need to cover overlap and joints that are 5% of are
What is the volume of a quadrilateral oblique prism with base edges of length a = 1m, b = 1.1m, c = 1.2m, d = 0.7m, if a side edge of length h = 3.9m has a deviation from the base of 20° 35 ´ and the edges a, b form an angle of 50.5°.
• Cardboard box
We want to make a cardboard box shaped quadrangular prism with rhombic base. Rhombus has a side of 5 cm and 8 cm one diagonal long. The height of the box to be 12 cm. The box will be open at the top. How many square centimeters cardboard we need, if we ca

We apologize, but in this category are not a lot of examples.
Do you have an interesting mathematical word problem that you can't solve it? Submit a math problem, and we can try to solve it.

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text:
# GTS3

#### CreativeCubing

##### Member
I am so excited because I just ordered a GTS3 from thecubicle.us. It has ridges on the edges that makes your grip better. It is also magnetic. The GTS3 also comes with a tensioning kit so you can tension it properly. They only have it stickerless so no stickers! If you would like to purchase the GTS3 M go to this link https://thecubicle.us/moyu-weilong-gts3-p-10626.html or this one https://speedcubeshop.com/products/moyu-weilong-gts3m-3x3

I hope you have a good time with the GTS3!
I think that the ridges would be weird on a 3x3. It’s going to be weird to make the transition to a heavier cube as well because I use the Gans Air SM

#### FJT97

##### Member
saw a review from speedcubereview and he said that he didnt really notice iirc
I think that the ridges would be weird on a 3x3. It’s going to be weird to make the transition to a heavier cube as well because I use the Gans Air SM

#### PnknHead

##### Member
I think that the ridges would be weird on a 3x3. It’s going to be weird to make the transition to a heavier cube as well because I use the Gans Air SM
I still need to get a Gan air SM. I think I saw other cubes with ridges . I think Gan will start making stickerless cubes with ridges. I hope they do!

#### 40yearstosolve

##### Premium Member
Mine just arrived today. Moyu GTS 3M stickerless (my preference). Got it from the following site at half the listed price on a pre-order (bought two)...

https://www.speedcube.com.au/collections/magnetic-cubes/products/2018-moyu-weilong-gts-3m-3x3x3-magnetic-speed-cube?variant=12310365208657

My quick review..

1. Magnets are very strong so it holds its shape very well. Because of this it requires a bit of extra push to turn I think but I seemed to get used to that and I prefer the surety of the 3Ms magnets, using the 2M side-by-side it seems very lose and he magnets almost pointless on the 2M!
2. I couldn’t feel a difference between the different tension settings! Going from 1 to 8 all seems the same to me!!
3. Colours are fantastic compared to the GTS 2M (again stickerless) - I am colourblind and found the 2M hard to distinguish betweenn orange, green and yellow in some light conditions but the 3M they have made the yellow brighter and the orange and green deeper which suits my eyes. The red is deeper too.
4. The ridges, meh, I don’t really notice them to be honest which surprised me and I suppose they will help with grip so good on Moyu for trail blazing.
5. It’s a little bigger than the 2M by maybe two or three mm.
6. Smoothness is probably the same as the 2M, but my 2M I had lubed so the 3M out-of-the-box is probably smoother.

Edit: I didn’t notice a weight difference compared to the 2M.

That’s it for now, very impressed and as I always have done I love Moyu’s stickerless.

Cheers

Last edited:

#### PnknHead

##### Member
I am so excited because I just ordered a GTS3 from thecubicle.us. It has ridges on the edges that makes your grip better. It is also magnetic. The GTS3 also comes with a tensioning kit so you can tension it properly. They only have it stickerless so no stickers! If you would like to purchase the GTS3 M go to this link https://thecubicle.us/moyu-weilong-gts3-p-10626.html or this one https://speedcubeshop.com/products/moyu-weilong-gts3m-3x3

I hope you have a good time with the GTS3!
I just got it in the mail and ootb it is very dry and too fast. The magnets are very strong and the ridges are small and I don't notice them. I really like the tensioning system and I think it is a big improvement. After I put in some Galaxy lube from SpeedCubeShop, tensioned it, and did a couple of solves it became a lot more controllable. It is definitely my new main.

#### 1001010101001

##### Member
Speedcube.com.au sells it at $29.50 AUD Cubicle at$34 USD

#### 1001010101001

##### Member
It's listed for \$49.00 AUD right now on speedcube.com.au.
Pre order discount

#### SauceObtainer

##### Member
I just got it in the mail and ootb it is very dry and too fast. The magnets are very strong and the ridges are small and I don't notice them. I really like the tensioning system and I think it is a big improvement. After I put in some Galaxy lube from SpeedCubeShop, tensioned it, and did a couple of solves it became a lot more controllable. It is definitely my new main.
Which lube or sets of cosmic lube do you recommend for and sotb cube

L

#### lucarubik

##### Guest
reviews are nice and all but i just want to know where the cube stands in a tier list.

Actually i just want to know if its better than the v2 (the best v2 if there are more subtipes, i wouldn't know) because that one is easily top 3 stickerless right?

#### Keiserolsen

##### Member
Does anybody know the size and strength of the magnets?

#### CarterK

##### Member
I don't know exact strength, but they are definitely on the stronger end.

#### pglewis

##### Member
reviews are nice and all but i just want to know where the cube stands in a tier list.

Actually i just want to know if its better than the v2 (the best v2 if there are more subtipes, i wouldn't know) because that one is easily top 3 stickerless right?
My take is it's similar but different; better is hard to quantify. A well setup GTS2M corner cuts more effortlessly for me; that's sometimes a pro, sometimes a con if my accuracy is off. The GTS3M corner cuts just fine, mind you, and I have not spent any time trying to optimize the tensions, which are very configurable. I still haven't spent a lot of time trying to fine-tune it because it's just fine the way it is for me right now.

The GTS3's factory magnets are definitely stronger. I've preferred weaker magnets in the past but I actually like the stronger feel on this puzzle, especially for blind.

Stickerless shades are less "washed out" vs. the GTS2 and provide better contrast IMO. The plastic is thin enough that strong light can show through a little and look weird, but it isn't much worse than my other stickerless puzzles.

The ridges are very subtle but I love them and hope this is "a thing" for future flagship 3x3s. It's just enough grip reassurance, highly appreciated especially during blind.

It's ever-so-slightly larger than the current flagships, which might be a plus or minus for individual OH preferences (I'm a minute and a half OH solver sooooo... I hardly care lol)

It's as main-worthy as any of the other flagship stickerless 3x3s I've tried, which includes the GTS2M, Valk 3M, and HuangLong M. I like all of the above but the GTS3M has been getting the most use lately.

Last edited:

#### CarterK

##### Member
Moyu posted on FB saying that they're gonna release a non magnetic version! It's gonna be the best cube for BLD by farrr.

L

#### lucarubik

##### Guest
My take is it's similar but different; better is hard to quantify. A well setup GTS2M corner cuts more effortlessly for me; that's sometimes a pro, sometimes a con if my accuracy is off. The GTS3M corner cuts just fine, mind you, and I have not spent any time trying to optimize the tensions, which are very configurable. I still haven't spent a lot of time trying to fine-tune it because it's just fine the way it is for me right now.

The GTS3's factory magnets are definitely stronger. I've preferred weaker magnets in the past but I actually like the stronger feel on this puzzle, especially for blind.

Stickerless shades are less "washed out" vs. the GTS2 and provide better contrast IMO. The plastic is thin enough that strong light can show through a little and look weird, but it isn't much worse than my other stickerless puzzles.

The ridges are very subtle but I love them and hope this is "a thing" for future flagship 3x3s. It's just enough grip reassurance, highly appreciated especially during blind.

It's ever-so-slightly larger than the current flagships, which might be a plus or minus for individual OH preferences (I'm a minute and a half OH solver sooooo... I hardly care lol)

It's as main-worthy as any of the other flagship stickerless 3x3s I've tried, which includes the GTS2M, Valk 3M, and HuangLong M. I like all of the above but the GTS3M has been getting the most use lately.
don't get me started :3 that hardly recomend it cube reviewer still gives me PTSD, 8 years after
If we ever meet and i ask you what do you think about one cube just tell me its better than this one and worse than this one, and kind of like this other one, if you are so kind, if that makes me an idito ill be a happy idiot with a cube that i don't know how many fricking degrees can corner cut, after all they dont make piano keys with different resistance do they? skinny wemen and huge men both play the same, actually they might make them im not sure, and qwerty keyboards are the thing so no point to use that as an argument... either way, you tell me how much you like a cube ill decide how much i trust your opinion, thats all i want to know, and thats a fact.

or maye im so sick of 10 min of rambling about the cube characteristics and the lack of adjetives like good or bad or better than or worse than that i think the second is the only i care about, but i honestly think this is not the case

#### pglewis

##### Member
don't get me started :3 that hardly recomend it cube reviewer still gives me PTSD, 8 years after
If we ever meet and i ask you what do you think about one cube just tell me its better than this one and worse than this one, and kind of like this other one, if you are so kind, if that makes me an idito ill be a happy idiot with a cube that i don't know how many fricking degrees can corner cut, after all they dont make piano keys with different resistance do they? skinny wemen and huge men both play the same, actually they might make them im not sure, and qwerty keyboards are the thing so no point to use that as an argument... either way, you tell me how much you like a cube ill decide how much i trust your opinion, thats all i want to know, and thats a fact.

or maye im so sick of 10 min of rambling about the cube characteristics and the lack of adjetives like good or bad or better than or worse than that i think the second is the only i care about, but i honestly think this is not the case
k

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