Mean, Median and Mode

Introduction to Mean, Median and Mode: Often in statistics, we tend to represent a set of data by a representative value which would approximately define the entire collection. This representative value is called the measure of central tedency, and the name suggests that it is a value around which the data is centred. These central tendencies are mean, median and mode.

We are all interested in cricket but have you ever wondered during the match why the run rate of the particular over is projected and what does the run rate mean? Or, when you get your examination result card, you mention the aggregate percentage. Again what is the meaning of aggregate? All these quantities in real life make it easy to represent a collection of data in terms of a single value. It is called Statistics.

Statistics deals with the collection of data and information for a particular purpose. The tabulation of each run for each ball in cricket gives the statistics of the game. The representation of any such data collection can be done in multiple ways, like through tables, graphs, pie-charts, bar graphs, pictorial representation etc.

Now consider a 50 over ODI match going between India and Australia. India scored 370 runs by the end of the first innings. How do you decide whether India put a good score or not? It’s pretty simple, right; you find the overall run rate, which is good for such a score. Thus, here comes the concept of mean, median and mode in the picture. Let us learn in detail each of the central tendencies.

Measures of central tendency

The measures of central tendencies are given by various parameters but the most commonly used ones are mean, median and mode. These parameters are discussed below.

What is Mean?

Mean is the most commonly used measure of central tendency. It actually represents the average of the given collection of data. It is applicable for both continuous and discrete data.

It is equal to the sum of all the values in the collection of data divided by the total number of values.

Suppose we have n values in a set of data namely as x1, x2, x3, …, xn, then the mean of data is given by:

\(\begin{array}{l}\bar{x} = \frac{x_{1}+x_{2}+x_{3}+……..+x_{n}}{n}\end{array} \)

It can also be denoted as:

\(\begin{array}{l}\bar{x} = \frac{\sum_{i=1}^{n}x_{i}}{n}\end{array} \)

For grouped data, we can calculate the mean using three different methods of formula.

Direct method	Assumed mean method	Step deviation method
Mean \(\begin{array}{l}\overline{x}=\frac{\sum_{i=1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}\end{array} \) Here, ∑fi = Sum of all frequencies	Mean \(\begin{array}{l}(\overline{x})=a+\frac{\sum f_id_i}{\sum f_i}\end{array} \) Here, a = Assumed mean di = xi – a ∑fi = Sum of all frequencies	Mean \(\begin{array}{l}(\overline{x})=a+h\frac{\sum f_iu_i}{\sum f_i}\end{array} \) Here, a = Assumed mean ui = (xi – a)/h h = Class size ∑fi = Sum of all frequencies

What is Median?

Generally median represents the mid-value of the given set of data when arranged in a particular order.

Median: Given that the data collection is arranged in ascending or descending order, the following method is applied:

• If number of values or observations in the given data is odd, then the median is given by [(n+1)/2]th observation.
• If in the given data set, the number of values or observations is even, then the median is given by the average of (n/2)th  and  [(n/2) +1]th observation.

The median for grouped data can be calculated using the formula,

\(\begin{array}{l}Median = l + \left ( \frac{\frac{N}{2}-cf}{f} \right )\times h\end{array} \)

What is Mode?

The most frequent number occurring in the data set is known as the mode.

Consider the following data set which represents the marks obtained by different students in a subject.

Name	Anmol	Kushagra	Garima	Ashwini	Geetika	Shakshi
Marks Obtained (out of 100)	73	80	73	70	73	65

The maximum frequency observation is 73 ( as three students scored 73 marks), so the mode of the given data collection is 73.

We can calculate the mode for grouped data using the below formula:

\(\begin{array}{l}Mode = l + \left ( \frac{f_1-f_0}{2f_1-f_0-f_2} \right )\times h\end{array} \)

Example of Mean, Median and Mode

Let us see the difference between the mean median and mode through an example.

Example: The given table shows the scores obtained by different players in a match. What is mean, median and mode of the given data?

S.No	Name	Runs Scored
1	Sachin	80
2	Yuvraj	52
3	Virat	40
4	Sehwag	52
5	Rohit	70
6	Harbhajan	1
7	Dhoni	6

Solution:

i)  The mean is given by:

\(\begin{array}{l}\bar{x} = \frac{\sum_{i=1}^{n}}{n}\end{array} \)

\(\begin{array}{l}\bar{x} = \frac{80 + 52 + 40 + 52 + 70 + 1 + 6}{7}\end{array} \)

\(\begin{array}{l}\bar{x} = 43\end{array} \)

The mean of the given data is 43.

ii) To find out the median let us first arrange the given data in ascending order

Name	Harbhajan	Dhoni	Virat	Yuvraj	Sehwag	Rohit	Sachin
Runs	1	6	40	52	52	70	80

As the number of items in the data is odd. Hence, the median is [(n+1)/2]th observation.

⇒ Median = [(7+1)/2]th observation = 52

iii)  Mode is the most frequent data, which is 52.

Relation of Mean Median Mode

The relation between mean, median and mode that means the three measures of central tendency for moderately skewed distribution is given the formula:

Mode = 3 Median – 2 Mean

This relation is also called an empirical relationship. This is used to find one of the measures when the other two measures are known to us for certain data. This relationship is rewritten in different forms by interchanging the LHS and RHS.

Range

In statistics, the range is the difference between the highest and lowest data value in the set. The formula is:

Range – Highest value – Lowest value

Solved Problem

Question: Find the mean, median, mode and range for the given data:

90, 94, 53, 68, 79, 94, 53, 65, 87, 90, 70, 69, 65, 89, 85, 53, 47, 61, 27, 80

Solution:

Given,

90, 94, 53, 68, 79, 94, 53, 65, 87, 90, 70, 69, 65, 89, 85, 53, 47, 61, 27, 80

Number of observations = 20

Mean = (Sum of observations)/ Number of observations

= (90 + 94 + 53 + 68 + 79 + 94 + 53 + 65 + 87 + 90 + 70 + 69 + 65 + 89 + 85 + 53 + 47 + 61 + 27 + 80)/20

= 1419/20

= 70.95

Therefore, mean is 70.95.

Median:

The ascending order of given observations is:

27, 47, 53, 53, 53, 61, 65, 65, 68, 69, 70, 79, 80, 85, 87, 89, 90, 90, 94,94

Here, n = 20

Median = 1/2 [(n/2) + (n/2 + 1)]th observation

= 1/2 [10 + 11]th observation

= 1/2 (69 + 70)

= 139/2

= 69.5

Thus, the median is 69.5.

Mode:

The most frequently occurred value in the given data is 53.

Therefore, mode = 53

Range = Highest value – Lowest value

= 94 – 27

= 67

&

Win