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:
\[14,\,\,\,19,\,\,\,8,\,\,\,4,\,\,\,20,\,\,\,11,\,\,\,9,\,\,\,19\]
Calculate the mean of this sample.
#\bar{X}=13#
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#:\[\begin{array}{rcl}\bar{X} &=& \dfrac{\sum{X}}{n}\\&&\color{blue}{\text{Formula for the sample mean}}\\&=& \dfrac{X_1 + X_2 + \ldots + X_{n}}{n}\\&&\color{blue}{\text{Expanded the summation sign}}\\&=& \dfrac{14+19+8+4+20+11+9+19}{8}\\&&\color{blue}{\text{Entered the values for }X_1 \text{ through } X_{8} \text{ and }n \text{ into the equation}}\\&=& \dfrac{104}{8}\\&&\color{blue}{\text{Added the scores}}\\&=& 13\\\end{array}\]
Assuming the sample scores are located in cells A1 through A8, the Excel command to calculate the mean is:AVERAGE(array)
- array: The array or cell range of numeric values for which you want to calculate the mean.
\[= \text{AVERAGE(A1:A8)}\]
This gives:
\[\bar{X} = 13\]
Thus, to calculate the mean, run the following command:mean(x)
- x: The numeric vector whose median you wish to calculate.
\[mean(x = c(14,19,8,4,20,11,9,19))\]
Looking at the output generated by R, we find:
\[\bar{X}=13\]
#\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.
