### Chapter 1. Descriptive Statistics: Measures of Central Tendency

### Mean

Mean

**Definition**

The *arithmetic mean*, commonly referred to as simply the **mean**, is calculated by dividing the sum of the scores by the total number of observations.

The sample mean of a variable #X# is denoted #\bar{X}# ('X-bar').

**Formula**

\[\bar{X} = \cfrac{\sum{X}}{n}\]

Calculating the Mean with Statistical Software

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

AVERAGE(array)

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

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

mean(x)

x: The numeric vector whose mean you wish to calculate.

Consider the following sample of scores:

\[4,\,\,\,14,\,\,\,16,\,\,\,14,\,\,\,2,\,\,\,13,\,\,\,18,\,\,\,7\]

Calculate the *mean* of this sample.

#\bar{X}=11#

There are a number of different ways we can calculate the *mean*. Click on one of the panels to toggle a specific solution.

To calculate the *sample mean* #\bar{X}#, divide the sum of scores #\sum{X}# by the sample size #n#:\[\begin{array}{rcl}\bar{X} &=& \dfrac{\sum{X}}{n}\\&&\color{blue}{\text{Formula for the sample mean}}\\&=& \dfrac{X_1 + X_2 + \ldots + X_{n}}{n}\\&&\color{blue}{\text{Expanded the summation sign}}\\&=& \dfrac{4+14+16+14+2+13+18+7}{8}\\&&\color{blue}{\text{Entered the values for }X_1 \text{ through } X_{8} \text{ and }n \text{ into the equation}}\\&=& \dfrac{88}{8}\\&&\color{blue}{\text{Added the scores}}\\&=& 11\\\end{array}\]

*mean*in Excel, make use of the following function:

Assuming the sample scores are located in cells A1 through A8, the Excel command to calculate the

AVERAGE(array)

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

*mean*is:

\[= \text{AVERAGE(A1:A8)}\]

This gives:

\[\bar{X} = 11\]

*mean*in R, make use of the following function:

Thus, to calculate the

mean(x)

x: The numeric vector whose median you wish to calculate.

*mean*, run the following command:

\[mean(x = c(4,14,16,14,2,13,18,7))\]

Looking at the output generated by R, we find:

\[\bar{X}=11\]

#\phantom{0}#

In order for the mean to be a meaningful measure of centrality, the data has to have been measured on an

*interval*or

*ratio*scale.

\[

\begin{array}{c|cccc}

&\text{Nominal}&\text{Ordinal}&\text{Interval}&\text{Ratio}\\

\hline

\text{Mode}&\green{\text{Yes}}&\green{\text{Yes}}&\green{\text{Yes}}&\green{\text{Yes}}\\

\text{Median}&\red{\text{No}}&\green{\text{Yes}}&\green{\text{Yes}}&\green{\text{Yes}}\\

\text{Mean}&\red{\text{No}}&\red{\text{No}}&\green{\text{Yes}}&\green{\text{Yes}}\\

\end{array}

\]

One thing to keep in mind when calculating the mean is the skewness of the distribution. If a distribution is skewed either heavily to the left or to the right, the mean might not be a very good measure of centrality.

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