In a normally distributed set of data, all three measures of central tendency are approximately the same. Using the data sample below, calculate the measures of central tendency.

**{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 this example, the mean is of a *sample *which is represented by x-bar. Sometimes M is also used to represent the sample mean. For a population mean, the greek letter μ is used. See the table below for common notation.

**Median**

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. This 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

**Mode**

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.

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).

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