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In these formulas, f represents the frequency with which a value appears. For example, if a value appears once, f is one. If a value appears three times in the data set or population, f is three.
The statistic of a sampling distribution was discussed in Descriptive Statistics: Measuring the Center of the Data . How much the statistic varies from one sample to another is known as the sampling variability of a statistic . You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean in the chapter The Central Limit Theorem (not now). The notation for the standard error of the mean is where σ is the standard deviation of the population and n is the size of the sample.
In practice, USE A CALCULATOR OR COMPUTER SOFTWARE TO CALCULATE THE STANDARD DEVIATION. If you are using a TI-83, 83+, 84+ calculator, you need to select the appropriate standard deviation σ x or s x from the summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean. (The calculator instructions appear at the end of this example.)
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 = 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 two 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 .
| Data | Freq. | Deviations | Deviations 2 | (Freq.)( Deviations 2 ) |
|---|---|---|---|---|
| x | f | ( x – ) | ( x – ) 2 | ( f )( x – ) 2 |
| 9 | 1 | 9 – 10.525 = –1.525 | (–1.525) 2 = 2.325625 | 1 × 2.325625 = 2.325625 |
| 9.5 | 2 | 9.5 – 10.525 = –1.025 | (–1.025) 2 = 1.050625 | 2 × 1.050625 = 2.101250 |
| 10 | 4 | 10 – 10.525 = –0.525 | (–0.525) 2 = 0.275625 | 4 × 0.275625 = 1.1025 |
| 10.5 | 4 | 10.5 – 10.525 = –0.025 | (–0.025) 2 = 0.000625 | 4 × 0.000625 = 0.0025 |
| 11 | 6 | 11 – 10.525 = 0.475 | (0.475) 2 = 0.225625 | 6 × 0.225625 = 1.35375 |
| 11.5 | 3 | 11.5 – 10.525 = 0.975 | (0.975) 2 = 0.950625 | 3 × 0.950625 = 2.851875 |
| The total is 9.7375 |
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):
The sample standard deviation s is equal to the square root of the sample variance:
which is 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.
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