Measures of Position

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Measures of position are used to determine how a score compares to the set of scores. For another approach to explaining z-scores, see the Behavioral Research Methods approach.

z-score
Used to get a relative comparison of data that normally might not be compared. Uses the sample mean and standard deviation to obtain a "standard score." The z-score represents the number of standard deviations the data value falls above or below the mean. This allows you to compare data that are on different scales.


 * Z_score_for_a_sample.png example: An aptitude test has a mean of 220 and a standard deviation of 10.


 * The z-score for a score of 200 would be: z=(200-220)/10=(-20)/10=-2 so the score of 220 is 2 standard deviations below the mean.


 * The z-score for a score of 232 would be: z=(232-220)/10=(12)/10=1.2 so the score of 232 is 1.2 standard deviations above the mean.


 * The z-score for a score of 220 would be: z=(220-220)/10=(0)/10=0 so the score of 220 is 0 standard deviations from the mean.


 * Another example: A student scored a 65 on a calculus test that had a mean of 50 and a standard deviation of 10. She scored a 30 on an english test with a mean of 25 and a standard deviation of 5. On which test did she do better?


 * Calculus test z-score: z=(65-50)/10=(15)/10=1.5


 * English test z-score: z=(30-25)/5=(5)/5=1


 * So she scored 1.5 standard deviations above the mean on the calculus test and only 1 standard deviation above the mean on the english test. This means she scored better on the calculus test than the english test when compared to her peers.

Percentiles
Position measures used in education and health fields. NOT a percentage but a representation of the position of a score compared to the other scores.


 * If a score of 58 fell in the 82nd percentile means 82% of the scores fell below 58.


 * The median corresponds to the 50th percentile. This is because 50% of the scores fall below the median.


 * Percentiles only go up to 99. There is no such thing as a 100th percentile because you cannot have all of the scores below a score (then that score cannot be below itself).


 * For example: A teacher gives a 20 point test to 10 students. The scores are (2, 3, 5, 6, 8, 10, 12, 15, 18, 20). There are six scores below 12, so the percentile rank of a score of 12 is: (6+0.5)/10 = (6.5)/10=0.65. So the score of 12 is in the 65th percentile.