# Standard deviation practice exercises and solutions

Standard deviation is a measure of variability. It is the square root of variance. Variance is a measure of variability, the degree to which scores vary. More precisely, variance measures the degree scores deviate from the mean.

### The problem

A college coach recorded in seconds the time it took ten freshmen to run 100 meters. He wanted to know the degree scores vary. He calculated the mean of these scores. Next he calculated the variance, that is how far from the mean the scores were located..Finlay, he calculated the standard deviation

$X$
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To calculate the mean we add up all the scores and divide by the number of scores. Here is the formula for the mean.
$$\bar{X} ={\sum{X} \over {n}}$$ $$mean={sum of scores \over number of scores}$$

Next he subtracted each score from the mean, to have the the degree of deviation of each score from the mean.
$$X-\bar{X}$$
Next he squared these deviations $${(X-\bar{X})^2}$$
Next, he added up these squared deviation. This is the sum of squared deviations, or SS.
$$\sum{(X-\bar{X})^2}$$

Next, he divided the sum of squares by the number of scores minus 1. This is variance, The formula for variance is:
$$s^2 ={\sum{({X}-{\bar{X})}}^2 \over {n-1}}$$
$$variance={sum of squared deviations\over the number os scores minus 1}$$
Finally he calculated the standard deviation.
$$σ =\sqrt{\sum{({X}-{\bar{X})}}^2 \over {n-1}}$$ $$standard deviation ={square root of sum of squares \over number of scores minus 1}$$

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