# Numerical Measures of Central Tendency and Variability

## Measures of Variability

### Standard Deviation

The *standard deviation* is simply the square root of the variance. This makes the standard deviations of the two quiz distributions and . 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, of the distribution is within one standard deviation of the mean, and approximately of the distribution is within two standard deviations of the mean. Therefore, if you had a normal distribution with a mean of and a standard deviation of , then of the distribution would be between and . Similarly, about of the distribution would be between and . The symbol for the population standard deviation is σ; the symbol for an estimate computed in a sample is . Figure 2 shows two normal distributions. The red distribution has a mean of and a standard deviation of ; the blue distribution has a mean of and a standard deviation of . For the red distribution, of the distribution is between and ; for the blue distribution, is between and .

Figure 2. Normal distributions with standard deviations of and .

##### R code

q1=c(9,9,9,8,8,8,8,7,7,7,7,7,6,6,6,6,6,6,5,5)

IQR(q1, type = 6)

[1] 2

x=c(1,2,4,5)

var(x)

[1] 3.333333

sd(q1)

[1] 1.256562

q2=c(10,10,9,9,9,8,8,8,7,7,7,6,6,6,5,5,4,4,3,3)

sd(q2)

[1] 2.202869