
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 (bellshaped 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 xbar.
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. The is measure for skewed or nonnormal 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 "n1"
in the denominator to calculate the sample statistics (the rows in
Excel for the data were in rows 316).
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