The Mean, Median and Mode are the three measures of central tendency. Mean is the arithmetic average of a data set. This is found by adding the numbers in a data set and dividing by the number of observations in the data set. The median is the middle number in a data set when the numbers are listed in either ascending or descending order. The mode is the value that occurs the most often in a data set and the range is the difference between the highest and lowest values in a data set.

The Mean

\[\large \overline{x}=\frac{\sum x}{N}\]

Here,
∑ represents the summation
X represents observations
N represents the number of observations .

In the case where the data is presented in a tabular form, the following formula is used to compute the mean

Mean = ∑f x / ∑f

Where ∑f = N

The Median

If the total number of observations (n) is an odd number, then the formula is given below:

\[\large Median=\left(\frac{n+1}{2}\right)^{th}observation\]

If the total number of the observations (n) is an even number, then the formula is given below:

\[\large Median=\frac{\left(\frac{n}{2}\right)^{th}observation+\left(\frac{n}{2}+1\right )^{th}observation}{2}\]

Consider the case where the data is continuous and presented in the form of a frequency distribution, the median formula is as follows.

Find the median class, the total count of observations ∑f.

The median class consists of the class in which (n / 2) is present. 

Here

l = lesser limit belonging to the median class

c = cumulative frequency value of the class before the median class

f = frequency possessed by the median class

h = size of the class

The Mode

\[\large The\;mode\;is\;the\;most\;frequently\;occuring\;observation\;or\;value.\]

Consider the case where the data is continuous and the value of mode can be computed using the following steps.

a] Determine the modal class that is the class possessing the maximum frequency.

b] Calculate the mode using the below formula

l = lesser limit of modal class

= frequency possessed by the modal class

= frequency possessed by the class before the modal class

= frequency possessed by the class after the modal class

h = width of the class

How Are Mean, Median And Mode Related?

The 3 estimates of central tendency that is the mean, median and mode are related by the following empirical relationship.

2 Mean + Mode = 3 Median

For example, if it is required to compute the mean, median and mode of the data that is continuous grouped, then the values of the mean and median can be found using the above formulae. The value of the mode can be found using the empirical formula.

If the value of the mode is 65 and the median = 61.6, then find the value of the mean.

The value of the mean can be calculated using the formula,

2 Mean + Mode = 3 Median

2 Mean = (3 × 61.6) – 65

2 Mean = 119.8

Mean = 119.8 / 2

Mean = 59.9

Solved Examples

Question 1: Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

Solution:

Given data: 13, 18, 13, 14, 13, 16, 14, 21, 13

The mean is the usual average.

Mean = {13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13} / {9} = 15

(Note that the mean is not a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.)

The median is the middle value, so to rewrite the list in ascending order as given below:

13, 13, 13, 13, 14, 14, 16, 18, 21

There are nine numbers in the list, so the middle one will be

{9 + 1} / {2} = {10} / {2} = 5

= 5th number

Hence, the median is 14.

The mode is the number that is repeated more often than any other, so 13 is the mode.
The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.

Mean = 15
Median = 14
Mode = 13
Range = 8

Question 2: The value of the mean of five numbers is observed to be 18. If one number is not included, the mean is 16. Find the number that is excluded.

Answer: 

From the question,

There are 5 observations that mean n = 5.

The value of the mean = 18

x̄ = 18

x̄ = ∑ x / n

∑ x = 5 * 18 = 90

The sum of the five observations is 90.

Assume the excluded number to be “a”

The sum of four observations = 90 – a

Mean of 4 observations = (90 – a) / 4

16 = (90 – a) / 4

90 – a = 64

a = 26

⇒ The excluded number is 26.

More topics in Mean Median Mode Formula
Arithmetic Mean Formula	Geometric Mean Formula
Harmonic Mean Formula	Sample Mean Formula
Weighted Mean Formula	Effect Size Formula