### Chapter 1. Descriptive Statistics: Measures of Variability

### Variance and Standard Deviation

The previous section introduced the s*um of squares *which serves as the basis for two of the most important measures of variability, namely the *variance *and *standard deviation*.

Variance

**Definition**

**Variance **is the average of the squared deviation scores.

The population and sample standard variance are denoted as #\sigma^2# and #s^2#, respectively.

The sample variance is calculated slightly differently than the population variance in order to make it an *unbiased estimator**.

**Formulas**

\[\begin{array}{rcl}

\sigma^2 &=&\dfrac{SS}{N}= \dfrac{\sum{(X - \mu)^2}}{N}\\

\\

s^2&=&\dfrac{SS}{n-1}= \dfrac{\sum{(X - \bar{X})^2}}{n - 1}\\

\end{array}\]

Calculating the Sample Variance with Statistical Software

To calculate the *sample* *variance* in Excel, make use of the following function:

VAR(array)

array: The array or cell range of numeric values for which you want to calculate the sample variance.

To calculate the *sample **variance* in R, make use of the following function:

var(x)

x: The numeric vector whose sample variance you wish to calculate.

#\phantom{x}#

One downside of the variance as a measure of variability is that it is not expressed in the same units as the original measurement. As a side effect of squaring the deviation scores, the units of measurement have also been squared. For example, if the original measurement is taken in meters, then the variance would produce a value expressed in square meters.

Squared measurement units make it difficult to compare the variance to other statistical measures that do have the same units as the original scores, such as the mean or the interquartile range. For this reason, the *standard deviation *is generally the preferred measure of variability.

#\phantom{x}#

Standard Deviation

**Definition**

The **standard deviation **is a measure of variability that is expressed in the same units as the original measurement. It is calculated by taking the square root of the variance.

The population and sample standard deviation are denoted as #\sigma# and #s#, respectively.

**Formulas**

\[\begin{array}{rcl}

\sigma &=& \sqrt{\sigma^2} = \sqrt{\dfrac{\sum{(X-\mu)^2}}{N}}

\\

s &=& \sqrt{s^2} = \sqrt{\dfrac{\sum{(X-\bar{X})^2}}{n-1}}

\end{array}\]

Calculating the Sample Standard Deviation with Statistical Software

To calculate the *sample **standard deviation* in Excel, make use of the following function:

STDEV(array)

array: The array or cell range of numeric values for which you want to calculate the sample standard deviation.

To calculate the *sample* *standard deviation* in R, make use of the following function:

sd(x)

x: The numeric vector whose sample standard deviation you wish to calculate.

#\phantom{x}#

A useful rule of thumb is that the majority of all scores is located within one standard deviation on either side of the mean.

**Pass Your Math**independent of your university. See pricing and more.

Or visit omptest.org if jou are taking an OMPT exam.