Measures of Central Tendency

Definition

Measures of central tendency are numbers that indicate the centre of a set of ordered numerical data.

The three common measures of central tendency are the mean, the median and the mode.

Notes

The mean gives each element of a data set equal weight. When there are no extreme numbers in the data set (no very low or very high numbers), the mean is a good choice for a measure of central tendency. Statisticians state that "the mean is the most unbiased measure of central tendency".

The median gives the greatest weight to elements in the middle of the ordered data. When there are extreme numbers in the data set (very low or very high numbers), the median is a good choice for a measure of central tendency. The extreme numbers have less effect (or no effect at all) on the median.

The mode is a good choice for a measure of central tendency when the data has many identical data values.

Example One

The data below are the hourly sales of songs for an on-line music store over a ten hour period.

RAW DATA: { 11, 10, 13, 15, 73, 69, 67, 66, 14, 12 }

ORDERED DATA: { 10, 11, 12, 13, 14, 15, 66, 67, 69, 73 }

Mean: 35

Median: 14.5

Mode: There is no mode.

Given the way the data is distributed in this example (with many small and many large numbers), the arithmetic mean is probably the most appropriate measure of central tendency.

The mean number of songs sold at an on-line music store over a ten hour period is 35. [Open a demonstration(with the data of this example pre-entered).]

Example Two

The data below are the yearly wages (in dollars) of ten people working at an on-line music store.

DATA: { 41 000, 41 000, 41 000, 41 000, 43 000, 45 000, 48 000, 50 000, 50 000, 250 000 }

Mean: 65 000

Median: 44 000

Mode: 41 000

Given the way the data is distributed in this example (with one persons yearly wage being so large), the median is probably the best measure of central tendency.

NOTE: Nine people are below the mean and one person is above the mean, so the mean is probably not the most appropriate measure of central tendency.

NOTE: The majority of people working at the store (four in this case) are new employees who earn "starting wages". The mode, therefore, is probably not the most appropriate measure of central tendency.

The median yearly wage of ten people working at an on-line music store is $44 000.00. [Open a demonstration (with the data of this example pre-entered).]

Example Three

The data below are the seventeen shoe sizes of one type of shoe sold in one day at a local shoe store.

DATA: { 5, 6, 7, 7, 7, 7, 7, 7, 8, 9, 9, 10, 11, 12, 13, 13, 15 }

Mean: 9

Median: 8

Mode: 7

Given the way the data is distributed in this example (with so many size seven shoes being sold), the mode is probably the best measure of central tendency.

The mode shoe size of one type of shoe sold at a local shoe store is size seven. [Open a demonstration (with the data of this example pre-entered).]