Contents	Introduction	Descriptive	Hypothesis	Tables	Appendix

Range: The difference between the highest and lowest score (high-low). It describes the span of scores but cannot be compared to distributions with a different number of observations. In the table below, the range is 41 (65-24).

Variance:The average of the squared deviations between the individual scores and the mean. The larger the variance the more variability there is among the scores. When comparing two samples with the same unit of measurement (age), the variances are comparable even though the sample sizes may be different. Generally, however, smaller samples have greater variability among the scores than larger samples. The sample variance for the data in the table below is 251.57. The formula is almost the same for estimating population variance. See formula in Appendix.

Standard deviation: The square root of variance. It provides a representation of the variation among scores that is directly comparable to the raw scores. The sample standard deviation in the following table is 15.86 years.
 

Variableè	Age			ç 44.29
	24	-20.29	411.68
	32	-12.29	151.04
	32	-12.29	151.04	ç squared deviations
	42	-2.29	5.24
	55	10.71	114.70
	60	15.71	246.80
	65	20.71	428.90
n=	7		1509.43	ç
			251.57	ç sample variance
			15.86	ç sample standard deviation

Standardized Z-Score

A standardized z-score represents both the relative position of an individual score in a distribution as compared to the mean and the variation of scores in the distribution. A negative z-score indicates the score is below the distribution mean. A positive z-score indicates the score is above the distribution mean. Z-scores will form a distribution identical to the distribution of raw scores; the mean of z-scores will equal zero and the variance of a z-distribution will always be one, as will the standard deviation.

To obtain a standardized score you must subtract the mean from the individual score and divide by the standard deviation. Standardized scores provide you with a score that is directly comparable within and between different groups of cases.

Variableè	Age			Z
Doug	24	-20.29	-20.29/15.86	-1.28
Mary	32	-12.29	-12.29/15.86	-0.77
Jenny	32	-12.29	-12.29/15.86	-0.77
Frank	42	-2.29	-2.29/15.86	-0.14
John	55	10.71	10.71/15.86	0.68
Beth	60	15.71	15.71/15.86	0.99
Ed	65	20.71	20.71/15.86	1.31

As an example of how to interpret z-scores, Ed is 1.31 standard deviations above the mean age for those represented in the sample. Another simple example is exam scores from two history classes with the same content but difference instructors and different test formats. To adequately compare student A's score from class A with Student B's score from class B you need to adjust the scores by the variation (standard deviation) of scores in each class and the distance of each student's score from the average (mean) for the class.