Sten scores
# Sten scores

The results for some scales of some psychometric instruments are returned as sten scores, sten being an abbreviation for 'Standard Ten' and thus closely related to stanine scores.

## Definition

A sten score indicates an individual's approximate position (as a range of values) with respect to the population of values and, therefore, to other people in that population. The individual sten scores are defined by reference to a standard normal distribution. Unlike stanine scores, which have a midpoint of five, sten scores have no midpoint (the midpoint is the value 5.5). Like stanines, individual sten scores are demarcated by half standard deviations. Thus, a sten score of 5 includes all standard scores from -.5 to zero and is centered at -0.25 and a sten score of 4 includes all standard scores from -1.0 to -0.5 and is centered at -0.75. A sten score of 1 includes all standard scores below -2.0. Sten scores of 6-10 "mirror" scores 5-1. The table below shows the standard scores that define stens and the percent of individuals drawn from a normal distribution that would receive sten score.

Standard/z scores, percentages, percentiles, and sten scores :

| z-scores | < −2.0 | −2.0 … −1.5 | −1.5 … −1.0 | −1.0 … −0.5 | −0.5 … −0.0 | +0.0 … +0.5 | +0.5 … +1.0 | +1.0 … +1.5 | +1.5 … +2.0 | > +2.0 |
| - | - | - | - | - | - | - | - | - | - | - |
| Percent | 2.28% | 4.41% | 9.18% | 14.99% | 19.15% | 19.15% | 14.99% | 9.18% | 4.41% | 2.28% |
| Percentile | 1.14 | 4.48 | 11.27 | 23.36 | 40.43 | 59.57 | 76.64 | 88.73 | 95.52 | 98.86 |
| Sten | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Percentiles are the percentile of the sten score (which is the mid-point of a range of z-scores).

Sten scores (for the entire population of results) have a mean of 5.5 and a standard deviation of 2.

## Calculation of sten scores

When the score distribution is approximately normally distributed, sten scores can be calculated by a linear transformation: (1) the scores are first standardized; (2) then multiplied by the desired standard deviation of 2; and finally, (3) the desired mean of 5.5 is added. The resulting decimal value may be used as-is or rounded to an integer.

For example, suppose that scale scores are found to have a mean of 23.5, a standard deviation of 4.2, and to be approximately normally distributed. Then sten scores for this scale can be calculated using the formula, ${\frac {(s-23.5)}{4.2}}2+5.5$. It is also usually necessary to truncate such scores, particularly if the scores are skewed.

An alternative method of calculation requires that the scale developer prepare a table to convert raw scores to sten scores by apportioning percentages according to the distribution shown in the table. For example, if the scale developer observes that raw scores 0-3 comprise 2% of the population, then these raw scores will be converted to a sten score of 1 and a raw score of 4 (and possibly 5, etc.) will be converted to a sten score of 2. This procedure is a non-linear transformation that will normalize the sten scores and usually the resulting stens will only approximate the percentages shown in the table. The 16PF Questionnaire uses this scoring method.

