# 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 $\mathrm{1.257}$ 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 $\mathrm{10}$.

##### 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