Numerical Measures of Central Tendency and Variability

The standard deviation is simply the square root of the variance. This makes the standard deviations of the two quiz distributions \(\mathrm{1.257}\) and \(\mathrm{2.203}\). The standard deviation is an especially useful measure of variability when the distribution is normal or approximately normal (see Chapter on Normal Distributions) because the proportion of the distribution within a given number of standard deviations from the mean can be calculated. For example, \(\mathrm{68\%}\) of the distribution is within one standard deviation of the mean, and approximately \(\mathrm{95\%}\) of the distribution is within two standard deviations of the mean. Therefore, if you had a normal distribution with a mean of \(\mathrm{50}\) and a standard deviation of \(\mathrm{10}\), then \(\mathrm{68\%}\)  of the distribution would be between \(50 - 10 = 40\) and \(50 +10 =60\). Similarly, about \(\mathrm{95\%}\) of the distribution would be between \(50 - 2 \times 10 = 30\) and \(50 + 2 \times 10 = 70\). The symbol for the population standard deviation is σ; the symbol for an estimate computed in a sample is \(\mathrm{s}\). Figure 2 shows two normal distributions. The red distribution has a mean of \(\mathrm{40}\) and a standard deviation of \(\mathrm{5}\); the blue distribution has a mean of \(\mathrm{60}\) and a standard deviation of \(\mathrm{10}\). For the red distribution, \(\mathrm{68\%}\) of the distribution is between \(\mathrm{35}\) and \(\mathrm{45}\); for the blue distribution, \(\mathrm{68\%}\) is between \(\mathrm{50}\) and \(\mathrm{70}\).

Figure 2. Normal distributions with standard deviations of \(\mathrm{5}\) and \(\mathrm{10}\).

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