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Measures of Variability

Variability describes the spread or dispersion of scores for a set of research data or distribution. Measures of variability include the range, variance, and standard deviation. The range is the largest score minus the smallest score. It is a rough measure of dispersion. For the same set of quiz scores (5, 6, 7, 7, 7, 8, 8, 9, 9, 10), the range is 5. Variance and standard deviation (SD) indicate the degree to which scores differ from each other and vary around the mean value for the set. Variance and standard deviation indicate both how much scores group together and how dispersed they are. Variance is determined by computing the difference between each value and the mean, squaring the difference between each value and the mean (to eliminate negative signs), summing the squared differences, then taking the average of the sum of squared differences. The standard deviation of the distribution is the square root of the variance. For a different set of quiz scores (6, 7, 8, 8, 8, 8, 8, 8, 9, 10), the variance is 1 and the SD is 1. Standard deviation must fall between 0 and half the value of the range. If the standard deviation approaches 0, scores are very similar to each other and very close to the mean. If the standard deviation approaches half the value of the range, scores vary greatly from the mean. Frequency polygons with the same mean and the same range, but a different standard deviation, that are plotted on the same axes show a difference in variability by their shapes. The taller and narrower frequency polygon shows less variability and has a lower standard deviation than the short and wider one.

Figure 6.1 (a) The normal distribution or bell curve. (b) Negatively skewed distribution—skewed to the left. (c) Positively skewed distribution—skewed to the right.

Since you don’t bring a calculator to the exam, you won’t be required to figure out variance or standard deviation.

Correlation

Scores can be reported in different ways. One example is the standard score or z score. Standard scores enable psychologists to compare scores that are initially on different scales. For example, a z score of 1 for an IQ test might equal 115, while a z score of 1 for the SAT I might equal 600. The mean score of a distribution has a standard score of zero. A score that is one standard deviation above the mean has a z score of 1. A standard score is computed by subtracting the mean raw score of the distribution from the raw score of interest, then dividing the difference by the standard deviation of the distribution of raw scores. Another type of score, the percentile score, indicates the percentage of scores at or below a particular score. Thus, if you score at the 90th percentile, 90% of the scores are the same or below yours. Percentile scores vary from 1 to 99.

A statistical measure of the degree of relatedness or association between two sets of data, X and Y, is called the correlation coefficient. The correlation coefficient (r) varies from –1 to +1. One indicates a perfect relationship between the two sets of data. If the correlation coefficient is –1, that perfect relationship is inverse; as one variable increases, the other variable decreases. If the correlation coefficient (r) is +1, that perfect relationship is direct; as one variable increases the other variable increases, and as one variable decreases, the other variable decreases. A correlation coefficient (r) of 0 indicates no relationship at all between the two variables. As the correlation coefficient approaches +1 or –1, the relationship between variables gets stronger. Correlation coefficients are useful because they enable psychologists to make predictions about Y when they know the value of X and the correlation coefficient. For example, if r = .9 for scores of students in an AP Biology class and for the same students in AP Psychology class, a student who earns an A in biology probably earns an A in psychology, whereas a student who earns a D