## Numerical Measures of Central Tendency and Variability

Read these sections and complete the questions at the end of each section. First, we will define central tendency and introduce mean, median, and mode. We will then elaborate on median and mean and discusses their strengths and weaknesses in measuring central tendency. Finally, we'll address variability, range, interquartile range, variance, and the standard deviation.

### 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