2.8 Measuring the spread of the data (modified r. bloom)  (Page 2/6)

• In general, a value = mean + (#ofSTDEVs)(standard deviation)
• where #ofSTDEVs = the number of standard deviations
• 7 is one standard deviation more than the mean of 5 because: 7=5+ (1) (2)
• 1 is two standard deviations less than the mean of 5 because: 1=5+ (−2) (2)

Calculating the standard deviation

If $x$ is a data value, then the difference " $x$ - mean" is called its deviation . In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the data is for a population, in symbols a deviation is $x-\mu$ . For sample data, in symbols a deviation is $x-$ $\overline{x}$ .

The procedure to calculate the standard deviation depends on whether the data is for the entire population or comes from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is a population or a sample. The lower case letter $s$ represents the sample standard deviation and the Greek letter $\sigma$ (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then $s$ should be a good estimate of $\sigma$ .

To calculate the standard deviation, we need to calculate the variance first. The variance is an average of the squares of the deviations (the $x-$ $\overline{x}$ values for a sample, or the $x-\mu$ values for a population). The symbol ${\sigma }^2$ represents the population variance; the population standard deviation $\sigma$ is the square root of the population variance. The symbol $s^2$ represents the sample variance; the sample standard deviation $s$ is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.

If the data is from a population, when we calculate the average of the squared deviations to find the variance, we divide by N, the number of items in the population. If the data is from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n-1, one less than the number of items in the sample. You can see that in the formulas below.

Formulas for sample standard deviation

• $s=$ $\sqrt{\frac{\Sigma {(x-\overline{x})}^2}{n-1}}$ or $s=$ $\sqrt{\frac{\Sigma {f·(x-\overline{x})}^2}{n-1}}$
• For the sample standard deviation, the denominator is n-1, that is the sample size MINUS 1.

Formulas for population standard deviation

• $\sigma =$ $\sqrt{\frac{\Sigma {(x-\overline{\mu })}^2}{N}}$ or $\sigma =$ $\sqrt{\frac{\Sigma {f·(x-\overline{\mu })}^2}{N}}$
• For the population standard deviation, the denominator is N, the number of items in the population.

In these fomulas, $f$ 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, $f$ is 3.