Chapter 4 How it Works

4.1 Calculating The Mean

Here are data for the numbers of nights out socializing in a week for 20 students.

1 2 7 6 1 2 6 5 4 4 0 3 2 2 3 4 3 5 4 4

How can we calculate the mean? First, we add up all of the scores:

1 + 2 + 7 + 6 + 1 + 2 + 6 + 5 + 4 + 4 + 0 + 3 + 2 + 2 + 3 + 4 + 3 + 5 + 4 + 1 = 68

Then we divide by 20, the number of scores:

68/20 = 3.40

With the formula, we calculate:

4.2 Calculating The Median

Using the data for “nights out socializing,” how can we calculate the median? The median is either the middle score or the average of the two middle scores. We first arrange the data from lowest score to highest score:

0 1 1 2 2 2 2 3 3 3 4 4 4 4 4 5 5 6 6 7

With 20 scores (an even number), there are two middle scores, the 10th and 11th scores, which are 3 and 4. We determine the median by taking the average of 3 and 4. The median is 3.5.

4.3 Calculating The Mode

How can we calculate the mode for the “nights out socializing” data? The mode is the most common score. We can determine the mode for these data by looking at the frequency distribution. Five people have a score of 4. The mode is 4.

4.4 Calculating Variance

How can we calculate variance for the “nights out socializing” data? To calculate variance for these data, we first subtract the mean, 3.40, from every score. We then square these deviations. These calculations are shown in the table below.

X X − M (X − M)2
    1  − 2.40     5.76
    2  − 1.40     1.96
    7     3.60   12.96
    6     2.60     6.76
    1  − 2.40     5.76
    2  − 1.40     1.96
    6     2.60     6.76
    5     1.60     2.56
    4     0.60     0.36
    4     0.60     0.36
    0  − 3.40   11.56
    3  − 0.40     0.16
    2  − 1.40     1.96
    2  − 1.40     1.96
    3  − 0.40     0.16
    4     0.60     0.36
    3  − 0.40     0.16
    5     1.60     2.56
    4     0.60     0.36
    4     0.60     0.36

95

We then add all of the scores in the third column to get the sum of squared deviations, or the sum of squares. This sum is 64.80.

We can use the formula to complete the calculations:

The variance is 3.24.

4.5 Calculating Standard Deviation

How can we calculate standard deviation for the “nights out socializing” data? The standard deviation is the square root of the variance. For these data, we calculate standard deviation directly from the variance we calculated above using this formula:

The standard deviation is 1.80.