Measures of Central Tendency

3.6. Measures of Central Tendency#

A measure of central tendency is a single number that describes the center of a particular data set.

Three measures of central tendency are: mean, median, and mode.

The two most widely used measures of the “center” of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center. [1]

Mean#

Note

The words mean and average are often used interchangeably. The substitution of one word for the other is common practice. The technical term is “arithmetic mean” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.” [1]

Mean of a Population#

For a population of size N, the mean \(\mu\) is written as:

\[ \mu = \frac{1}{N}\sum_{i=1}^N \]

Tip

The Greek letter \(\mu\) (pronounced “mew”) represents the mean of a population.

Mean of a Sample#

For a sample of size n, the mean \(\bar{x}\) is written as:

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^n \]

Tip

The letter used to represent the mean of a sample is an x with a bar over it (pronounced “x bar”): \(\bar{x}\)

One of the requirements for the mean of a sample (\(\bar{x}\)) to be a good estimate of the mean of the population (\(\mu\)) it is sampled from is for the sample taken to be truly random.

Worked Example

GIVEN:

A data set considered a sample contains the values:

1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4

FIND:

The mean of the sample, \(\bar{x}\).

SOLUTION:

Count the number of values in the data set:

\[ n = 11 \]

Calculate the mean of the sample:

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^n \]
\[ \bar{x} = \frac{1+1+1+2+2+3+4+4+4+4+4}{11} = 2.7 \]
\[ \bar{x} = 2.7 \]

Median#

The median is the value in the center of a data set when the set is arranged in ascending or descending order.

If there are an even number of data points, the median is the midpoint of the two central points.

Mode#

The mode is one or more sets of numbers that occur with the greatest frequency in a data set.

Not all data sets will have a mode.

  • If it has a single mode it is uni-modal.

  • If it has 2 it is bimodal.

  • If it has 3 it is trimodal.

  • If all values occur the same number of times, there is no mode.

Worked Example

GIVEN:

A data set contains the numbers below:

15 22 23 24 27 27 28 28 28 29 30 31 33 34 34 35 35 35 35 35 37 40 41 46 47 51 55 56 72 74

FIND:

The mean, median and mode for the data set. Assume the data set reprents a sample.

SOLUTION:

MEAN

\[ \bar{x} = \frac{15+22+23+23+...+72+74}{30} = \frac{1107}{30} \]
\[ \bar{x} = 36.9 \]

MEDIAN

List the values in the data set from smallest to largest. The two central values are 34 and 35.

\[ median = \frac{34+35}{2} = 34.5 \]

MODE

Find the value that occurs with the greatest frequency. The value that occurs most often is 35.

\[ mode = 35 \]
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