2.6 Measures of the spread of the data  (Page 3/7)

In these formulas, $f$ represents the frequency with which a value appears. For example, if a value appears once, $f$ is 1. If a value appears three times in the data set or population, $f$ is 3.

Sampling variability of a statistic

In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of $n=\mathrm{20}$ fifth grade students. The ages are rounded to the nearest half year:

• 9
• 9.5
• 9.5
• 10
• 10
• 10
• 10
• 10.5
• 10.5
• 10.5
• 10.5
• 11
• 11
• 11
• 11
• 11
• 11
• 11.5
• 11.5
• 11.5

The average age is 10.53 years, rounded to 2 places.

The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating $s$ .

The sample variance, $s^2$ , is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 - 1):

$s^2=\frac{9.7375}{20-1}=0.5125$

The sample standard deviation $s$ is equal to the square root of the sample variance:

$s=\sqrt{0.5125}=.0715891$ Rounded to two decimal places, $s=0.72$

Typically, you do the calculation for the standard deviation on your calculator or computer . The intermediate results are not rounded. This is done for accuracy.