Today’s Question

1. Today’s Question • In some data sets the scores are tightly clustered around the mean. In other data sets, the scores are spread out. How can we quantify this property of distributions?

2. Measures of Spread • What is spread or dispersion? The degree to which scores are clumped around the mean.

3. How can we quantify spread? • We want to know how far, on average, an individual score is from the mean. • “how far an individual score is from the mean” (often called “deviation scores”) • “on average” Note the similarity to the standard formula for the mean:

4. Problem: The deviations from the mean sum to zero. (We proved this previously.) Recall that the deviations sum to zero because the mean is a “balancing point” for a set of scores--the point at which the “weight” of the scores above counterbalances the “weight” of the scores below.

5. One Solution: Average the absolute value of the deviation scores. Average Absolute Deviation: How far the typical (i.e., average) score is from the mean.

6. A second solution: Average the squared deviation scores Variance: The average squared deviation score.

7. A third solution: Take the square root of the average of the squared deviation scores Standard deviation: The square root of the average squared deviation score In our example, the square root of 7 is 2.65. The standard deviation is the square root of the variance. *** What does this tell us? It tells us how far people are from the mean, on average. (Ignoring whether people are above or below the mean.)

8. Measures of Spread • Of these three measures, the variance and standard deviation are used most frequently. • Why? Mathematically, it is easier to work with squared functions than absolute value functions.