Measures of Central Tendency

A measure of central tendency is a single value that describes a set of data by identifying the central position within the set of data. The mean, median and mode are the measures of central tendency.

Mean

The mean or average is the most commonly used measure if tendency. It is equal to the sum of all observations (values in data set) divided by the number of observations. If we have, x₁, x₂, x₃……..x_n, n values in a data set, then the mean is given as:

A data set has a unique mean. It is useful when comparing sets of data.

Median

The median is the mid value of a distribution arranges in order of magnitude. The same number of values are less than and greater than the median. If there is an odd number of values in a data set, the median is the middle value [(n+1)^th term]. If there is an even number of values in a data set, the median is the average of the two middle values [(n/2)^th and (n/2 + 1)^th terms]. Median is also a unique value for a data set. This can also be used to compare data sets. Median is not as popular as mean.

Mode

The mode is the most frequently appearing value in the data set. It is not necessary to have a unique value of mode. It may be more than one value. When no values repeat in the data set, the mode is every value and it is useless.

Example 1: (odd number of values)

Find mean, median and mode of the following data:

8, 3, 6, 8, 1, 9, 7

Mean = (8 + 3 + 6 + 8 + 1 + 9 + 7)/7

= 42/7

= 6

Median

Arrange in order,

1, 3, 6, 7, 8, 8, 9

There is odd (7) number of terms, so median is the middle term, i.e 4^th term

Median = 7

Mode

As we see in the list of data (8, 3, 6, 8, 1, 9, 7), 8 is appearing most number of times. So mode of the given set of data is 8.

Example 2:

Find the median and mode of the data set:

6, 4, 7, 2, 4, 5, 6, 3

Arrange in order:

2, 3, 4, 4, 5, 6, 6, 7

Median = (4 + 5)/2

= 9/2 = 4.5

Mode = 4 and 6

 

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