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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 1. If a value appears three times in the data set or population, $f$ is 3.
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:
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 $s$ .
Data | Freq. | Deviations | ${\mathrm{Deviations}}^{2}$ | (Freq.)( ${\mathrm{Deviations}}^{2}$ ) |
---|---|---|---|---|
$x$ | $f$ | $(x-\overline{x})$ | ${(x-\overline{x})}^{2}$ | $\left(f\right){(x-\overline{x})}^{2}$ |
$9$ | $1$ | $9-10.525=-1.525$ | ${(-1.525)}^{2}=2.325625$ | $1\times 2.325625=2.325625$ |
$\mathrm{9.5}$ | $2$ | $9.5-10.525=-1.025$ | ${(-1.025)}^{2}=1.050625$ | $2\times 1.050625=2.101250$ |
$\mathrm{10}$ | $4$ | $10-10.525=-0.525$ | ${(-0.525)}^{2}=0.275625$ | $4\times .275625=1.1025$ |
$\mathrm{10.5}$ | $4$ | $10.5-10.525=-0.025$ | ${(-0.025)}^{2}=0.000625$ | $4\times .000625=.0025$ |
$\mathrm{11}$ | $6$ | $11-10.525=0.475$ | ${\left(0.475\right)}^{2}=0.225625$ | $6\times .225625=1.35375$ |
$\mathrm{11.5}$ | $3$ | $11.5-10.525=0.975$ | ${\left(0.975\right)}^{2}=0.950625$ | $3\times .950625=2.851875$ |
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.
Verify the mean and standard deviation calculated above on your calculator or computer.
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