Lesson 6 will consist of the following topics
For lesson 6, read pages 67-75 in Practical Statistics for Educators,
Third Edition by Ruth Ravid (2005, University Press of America)
or read pages 115-131 in Basic Statistics for Behavioral Science Research
2nd ed by Mary B. Harris (1998, Allyn and Bacon)
or
read pages 103-111 in Practical Statistics for
Educators, 2nd Edition by Ruth Ravid (2000, University Press of America)
or read pages 73-95 in Practical Statistics for Educators
by Ruth Ravid (1994, University Press of America).
In our last lesson we considered measures of central tendency (mean, median, and mode), as numerical methods for summarizing data. In this lesson we will continue our investigation of descriptive statistics by looking at measures of variability. The measures of variability we will consider are the range, the variance, and the standard deviation.
Consider the following two sets of scores:
Set A: 4, 5, 6, 7, 8
Set B: 2, 4, 6, 8, 10
We can see that the means for these two sets of scores are the same:
For Set A the mean is (4 + 5 + 6 + 7 + 8)/5 = 6
For Set B the mean is (2 + 4 + 6 + 8 + 10)/5 = 6
Although these two sets of scores have the same mean, they differ in how spread out the scores are. The scores in Set A vary over a smaller set of values (4 through 8) than does Set B which varies over score values from 2 through 10.
We can represent this variability or spreadoutedness by a numerical index of variability called the range. The range is the difference between the largest score and the smallest score plus 1 for whole number data.
Range = Highest Score - Lowest Score + 1
For Set A the Range = (8 - 4) + 1 = 4 + 1 = 5
For Set B the Range = (10 - 2) + 1 = 8 + 1 = 9
As we expected the set of scores with the greater spreadoutedness (Set B), has a larger range than the set with less variability (Set A).
The range is very easy to calculate, but we would also like to have a measure of variability that, like the mean, considers every score in its calculation. There are two measures of variability that do this, the variance and the standardard deviation.
The variance is the average squared deviation of the scores from the mean. Don't try to memorize this definition at the present time as we will look at it in more detail as we investigate how to calculate the variance. At that time the definition will make more sense.
The standard deviation is the square root of the squared deviation of the scores from the mean and is thus the average deviation of the scores from the mean. As we will find out when we consider the calculation of the standard deviation, it is the square root of the variance and the variance is the standard deviation squared (multiplied by itself).
When we were considering the mean, we found that the mean for a sample, was calculated in the same way as the mean for a population. Only the symbols used to represent the mean (and the number of scores) differed. For the variance and the standard deviation this is no longer true, not only are the symbols different, but they are calculated in a slightly different way.
We will consider two different methods to calculate the variance for a population (the deviation score method and the raw score method), two ways to calculate the variance for a sample, and two methods to calculate the standard deviation for a population and the standard deviation for a sample.
Please send electronic mail to the course instructor if you have any questions about this lesson or other concerns.