Measures of Central Tendency

Mean, Median, Mode

In a normally distributed set of data, all three measures of central tendency are approximately the same. Using the set of data from a sample shown below, calculations are shown for the measures of central location.

{1, 3, 8, 3, 7, 11, 8, 3, 9, 10}

Mean (arithmetic)

Since most populations exhibit normality (bell-shaped curve) or can be assumed to be normal, the mean is the most common measure for central tendency. It is used to describe normal data.

The formula is the summation of all the values divided by the sample size:

Sum of all values: 63

n: 10 samples

Mean = 6.3

In the example, the mean is of a sample, represented by x-bar.


The median is the midpoint, the middle value or observation of the data set. If the set of data has an even count, the median is the average of the middle two values. The is measure for skewed or non-normal data.

Arrange the numbers in ascending or descending order:

{1, 3, 3, 3, 7, 8, 8, 9, 10, 11}

Since the sample is an even set of data (10 samples) and the middle two values are 7 and 8, then the average of the two middle values is 7.5.

Median = 7.5


The mode is the most commonly occurring value in the data set. Not commonly used as a measure of central location but can be found in the tallest bar of a vertical histogram chart.

Mode = 3, since it occurs more than any other value

Measures of dispersion
are numerical statistics which describe the spread of data or the width of the distribution.

Population and Sample measure symbols

Another example using Excel

Using Excel

Showing the formulas

For the Standard Deviation, Excel uses "n-1" in the denominator to calculate the sample statistics (the rows in Excel for the data were in rows 3-16).

Statistics using Excel

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