3.5 Variability, range, variances, standard deviation

Measures of variability represent how spread out a group of scores is and provide us with more information about individual differences.

Three commonly used measures of variability are the range, variance, and standard deviation.

The range

The first thing to compute the range, we subtract the lowest score in the distribution from the highest score.

How does calculating the range of a distribution of scores help us? Let us say that an elementary school decides to administer a math test to all incoming second- grade students to measure their math skills. There are 150 questions on the test. When we calculate the average math test score of the students in each of the two second-grade classes, we find that each class has a mean of 100 correct answers. Does this mean that students in both second-grade classes have about the same math skills? Yes and no. Yes, all of the classes have the same mean math knowledge. But no, the individuals in each class do not necessarily all have similar math skills. Although the mean informs us about the average math knowledge of each class, it tells us nothing about how varied the math knowledge is within each of the classes. It is very possible that students in one of the second-grade classes scored as low as 50 and as high as 150 (a range of 100), and the other second-grade class had scores as low as 90 and as high as 110 (a range of 20).Although the mean of these two classes may have been the same, one second-grade class has a larger range and is going to require more varied math instruction than the other second-grade class.

3.5 Variability, range, variances, standard deviation

Although the range is easy to calculate, be careful using it when a distribution of scores has outlying low and/or high scores. The low and/or high scores do not accurately represent the entire distribution of scores and may misrepresent the true range of the distribution.

Variance and standard deviation tell us about the spread in a distribution of scores. However, the variance and standard deviation are more satisfactory indexes of variability than is the range. The variance tells us whether individual scores tend to be similar to or substantially different from the mean. In most cases, a large variance tells us that individual scores differ substantially from the mean, and a small variance tells us that individual scores are very similar to the mean. What is a "large" variance, and what is a "small" variance?

Large and small depend on the range of the test scores. If the range of test scores is 10, then 7 would be considered a large variance and 1 would be considered a small variance. In most cases, a large variance tells us that individual scores differ substantially from the mean. In some cases, however, a large variance may be due to outliers. For example, if there are 100 scores and 99 of them are close to the mean and one is very far from the mean, there may be a large variance due to this one outlier score.

The formula (in summation notation) for the variance(σ ²) in a population is

3.5 Variability, range, variances, standard deviation

When the variance(σ ²) is computed in a sample, the statistic


where

∑ is a sum of the values

X a raw score

M is a mean of the scores

N is a number of test scores

The variance is computed as the average squared deviation of each number from its mean. For example, for the numbers 1, 2, and 3, the mean is 2 and the variance is:

3.5 Variability, range, variances, standard deviation

which gives an unbiased estimate of σ ². Since samples are usually used to estimate parameters, σ ² is the most commonly used measure of variance. Calculating the variance is an important part of many statistical applications and analyses. It is the first step in calculating the standard deviation.