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Tuesday, April 29, 2014

Standard Deviation

According to Howell (2013), the standard deviation is “defined as the positive square root of the variance” (p. 41) and it is commonly used to measure the average deviation of all scores from the mean.  The standard deviation is the most important statistic for measuring variability. By using the distribution’s mean as an anchor point, standard deviation measures data variability based on each score’s distance to mean.  It is important to know the standard deviation for a given sample because “the standard deviation provides a measure of the standard, or average, distance from the mean, and describes whether the scores are clustered closely around the mean or are widely scattered” (Gravetter & Wallnau, 2013, p. 106).

From knowledge of the standard deviation, researchers learn score standardization about a normal distribution.  In a normal distribution, data are symmetrically distributed with the highest frequency at the central point and lower occurrences on each side.  In research, z-scores are often utilized to analyze sections of data with a normal distribution.  Gravetter and Wallnau (2013) explained that “z-scores measure positions in a distribution in terms of standard deviations from the mean” (p. 170) thus a z-score of value 1 is exactly one standard deviation away from the mean.  Using the sample’s mean and standard deviation, a given score can be converted to the standard z-score so that is can be determined whether or not this score is significantly departed from the central mean.  For example, in a sample of given size, we know that its mean is M = 40; we also calculated its standard deviation s=5.  Then we want to find out if a score of X= 48 is an extreme value.  To answer this question, we need to compute the z-score first:

    z = (X – M)/ s = (55 – 40) / 5 = 3

From “the Unit Normal Table” (Gravetter & Wallnau, 2013, p. 699), it can be found that this z-score makes Proportion in Tail as .0013 which is lower than an alpha level of 5%, therefore a score of X = 55 is identified as an extreme value.

In summary, standard deviation is an important statistic for the measurement of variability.  The measure of test score’s variability can be standardized using the z-test.   The z-scores distribute normally with z = 1 at one standard deviation from the mean.  When a score is converted to z-value, it can be determined that if the score is extreme or normal.


Gravetter, F. J. & Wallnau, L. B. (2013). Statistics for the behavioral sciences (9th ed.). Belmont, CA: Wadsworth.

Howell, D. C. (2013). Statistical Methods for Psychology (8th ed.).  Belmont, CA: Wadsworth.

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