Mean, Median, Mode Definition, Formulas & Solved Examples

Mean, Median, and Mode are some of the fundamentals of Statistics in Mathematics. There are different categories of data for which the Mean, Median, and Mode are calculated. There are several methods and formulas for the same.

What are Mean, Median, Mode?

In mathematics, mean, median, and mode are ways to describe the center or average of a set of numbers.

• Mean is what we usually call the average. You get it by adding up all the numbers and dividing the total by how many numbers there are.
• Median is the middle value when all the numbers are arranged from smallest to largest. If there are two middle numbers, the median is the average of those two.
• Mode is the number that shows up most often in the data set.

The mean gives a general idea of what most values are like in the data. It usually falls somewhere between the smallest and largest numbers in the group. This helps us understand the overall trend or center of the data.

Measures of Central Tendency 

Measures of central tendency help us find the middle or average value of a group of numbers. They give us an idea of what a “typical” number looks like in a set of data. The three most common measures are Mean, Median, and Mode.

1. Mean is the average. Add all the numbers and divide by how many numbers there are.
   Example: For 5, 10, 15 → Mean = (5 + 10 + 15) ÷ 3 = 10
2. Median is the middle value when numbers are arranged in order.
   Example: 3, 5, 8 → Median = 5
   If there are two middle numbers, take their average.
3. Mode is the number that appears the most.
   Example: 2, 4, 4, 6, 7 → Mode = 4

Graphs to Include:

• Bar Graph for Mode (to show frequency)
• Line Plot or Dot Plot to show Median position
• Pie Chart or Number Line for Mean illustration

Arithmetic Mean of Raw Data

The arithmetic mean of a set of values is given by dividing the sum of all the values in the set by the total number of values in that set.

The mean of n numbers

=

=

Here, x represents the mean, (known as sigma) represents the sum of all values. x represents the observations, and n represents the total number of observations.

Learn about Mean Median and Mode in this video!

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Arithmetic Mean of Tabulated Data

Tabulated data means when we are given discrete frequency distribution. In such a case, we have 3 methods to find out the arithmetic mean.

1. Direct Method
2. Short-cut Method
3. Step-deviation Method

Direct Method: Following are the steps to calculate the arithmetic mean of tabulated data using the direct method:

Step 1: First and foremost you have to prepare a frequency table with three columns in the given format:

1. In the first column, write the values of the variate (x) in the given order from top to bottom.
2. In the second column, write the corresponding frequency (f) of each variate that you wrote in step a.
3. In the third column, write the product of each x with its corresponding frequency f, i.e., write fx for each value in the first and second columns.

Step 2: Calculate f, which is the sum of all the frequencies in the second column.

Step 3: Calculate fx by adding all the entries of the third column.

Step 4: Calculate the required mean which is

Note: Direct method is used when the values of f and x are small.

Short-cut Method/Assumed Mean Method : When the values of x are large, then using the direct method can be time-consuming as it will take time and will be difficult to calculate fx for each variate. So, for complex values, we use the short-cut method.

Following are the steps to calculate the arithmetic mean of tabulated data using the short-cut method:

Step 1. First and foremost you have to prepare a frequency table with four columns in the given format:

• In the first column, write the values of the variate (x) in the given order from top to bottom.
• In the second column, write the corresponding frequency (f) of each variate that you wrote in step a.

Step 2. Assume a number A(preferably from the given values of variate (x in the first column). This number A is known as the assumed mean. Now calculate the deviation (d) for each variate by the formula d=x-A, and write them in the third column.

Step 3. Calculate fd by multiplying all the entries of the second and third columns. Write these values of fd obtained in the fourth column.

Step 4. Calculate fd by adding all the values of fd written in the fourth column. Also, find f by adding all the frequencies written in the second column.

Step 5. Calculate the required mean using the following formula:

Note: Short-cut method is used when the values of f are small.

Step-deviation Method: We use the short-cut method when the values of x are large. But what if when even the values of f are large? In such a case, we make use of the step-deviation method.

Following are the steps to calculate the arithmetic mean of tabulated data using the step-deviation method:

Step 1. First and foremost you have to prepare a frequency table with five columns in the given format:

1. In the first column, write the values of the variate (x) in the given order from top to bottom.
2. In the second column, write the corresponding frequency (f) of each variate that you wrote in step a.

Step 2. Assume a number A(preferably from the given values of variate (x in the first column). This number Ais known as the assumed mean.

Now calculate the deviation (d) for each variate by the formula d=x-A, and write them in the third column.

Step 3. From the third column, choose the largest number (i), that can divide each value of d in the third column. Divide each value of d by i, i.e., di=x-Aidenote these values by t and write them in the fourth column.

Step 4. Calculate ft by multiplying all the values of f from the second column and the corresponding values of t written in the fourth column. Write each value of ft in the fifth column.

Step 5. Calculate ft by adding all the values in the fifth column. Also find f by adding all the frequencies written in the second column.

Step 6. Calculate the required mean using the following formula:

Note: Step-deviation method is used when the values of f are large.

Learn about Assumed Mean Method 

Methods to Calculate Mean of Grouped Data

Method	Formula	Terms Used	When to Use
Direct Method	x̄ = ∑fᵢxᵢ / ∑fᵢ	- fᵢ: Frequency of each class - xᵢ: Class mark (mid-point) - ∑fᵢ: Total frequency	Use when values are simple and easy to compute directly.
Assumed Mean Method	x̄ = a + (∑fᵢdᵢ / ∑fᵢ)	- a: Assumed mean - dᵢ = xᵢ − a - fᵢ: Frequency - ∑fᵢ: Total frequency	Use when values are large or difficult to work with directly.
Step Deviation Method	x̄ = a + h × (∑fᵢuᵢ / ∑fᵢ)	- a: Assumed mean - h: Class size - uᵢ = (xᵢ − a) / h - fᵢ: Frequency	Use when class sizes are equal and numbers are large for easier calculation.

What is Definition of Median?

When the values in a set of data are arranged in ascending or descending order, then the value of the middle term in that set is known as the Median of that set. The median can be calculated for both raw and tabulated data using the median formula.

Median For Raw Data: To find the median of a given set of numbers, first arrange them in either ascending or descending order. Then apply the median formula. Let there be n terms in a given data set, then,

If n is an odd number

Median=

If n is an even number

Median For Tabulated Data: To calculate the median for tabulated data, we need to calculate the cumulative frequency. Then apply the median formula to find the answer from the table. Check the example below to understand the procedure.

What is Definition of Mode?

For a given set of data, the value which occurs most frequently is known as the Mode of that given data. It is very easy to find the mode value for a given set of observations.

Mode For Raw Data : When you get just a raw data, i.e., a list of values, then the value that occurs the most in that list is simply the mode of that list

Mode For Tabulated Data : When you get data in tabulated form, i.e., when you are provided with the frequency of each value, then the value with the maximum frequency is the mode of the entire data.

Learn about Rolle’s Theorem and Lagrange’s Mean Value Theorem.

How Mean, Median, and Mode Are Related

Mean, median, and mode are three different ways to describe the center or average of a group of numbers. These three are connected through a simple formula, especially when the data is continuous and grouped.

The relationship is:

Mean – Mode = 3 (Mean – Median)
Or
2 × Mean + Mode = 3 × Median

This formula is useful when we know two of the three values and want to find the third.

Example:

Suppose we know:

• Mode = 65
• Median = 61.6

We can find the mean using the formula:

2 × Mean + Mode = 3 × Median

Now, put the values into the formula:

2 × Mean + 65 = 3 × 61.6
2 × Mean + 65 = 184.8
2 × Mean = 184.8 – 65 = 119.8
Mean = 119.8 ÷ 2 = 59.9

How Mean, Median, and Mode Are Used in Real Life

In real life, we often use mean, median, and mode without even realizing it. These three are ways to understand and analyze numbers in everyday situations:

• Mean (Average):
  The mean helps us find the general or typical value in a group.
  For example, if you want to know the average marks in a class or the average income of people in a city, you use the mean by adding all values and dividing by how many there are.
• Median (Middle Value):
  The median is helpful when there are very high or very low values that can change the average.
  For example, in income data, the median gives a better picture of what most people earn, especially if some people earn a lot more than others. In real estate, the median house price is used to show the middle range of prices.
• Mode (Most Frequent Value): The mode tells us what happens most often. For example, in shopping, stores may track which shoe size is sold the most to stock up accordingly. In factories, mode can help find the most common defect in products to improve quality.

Differences between Mean, Median, and Mode

Feature	Mean	Median	Mode
What it means	Mean is the total of all numbers divided by how many there are.	Median is the middle number when the data is arranged in order.	Mode is the number that appears the most in the data.
Outliers effect	Mean can change a lot if one number is very big or small.	Median stays almost the same even if there's a very large number.	Mode is not affected by very big or small numbers.
How to find	Add all the numbers and divide by how many numbers there are.	Arrange the numbers and pick the middle one.	Find the number that shows up the most times.
Is the value in the data?	Mean might or might not be one of the original numbers.	Median is always from the given data.	Mode is always from the given data.

Solved Examples on Mean Median & Mode

Example 1: Find the mode from the following frequency distribution:

Number	7	8	9	10	11	12	13
Frequency	3	8	12	15	14	17	9

Solution: Since, the frequency of number 12 is maximum

Therefore,

Mode=12

Example 2: Find the mode of the following data: 3, 9, 4, 7, 8, 7, 6, 1, 7, 9, 1, 8, 7, 5, and 7

Solution: in the given data, 7 occurs the most.

Therefore,

Mode=7

Example 3: The weights of 45 people in society were recorded, to the nearest kg, as follows:

Wt. (in nearest kg)	46	48	50	52	53	54	55
No. of people	7	5	8	12	10	2	1

Calculate the median weight.

Solution: Construct the cumulative frequency table as given below:

To calculate the cumulative frequency (c.f.), the c.f. of the first value remains the same. For the second value, we add frequency of first value and frequency of second value. For third value, we add the result of the second value, i.e., c.f. of the second value with the frequency of the third value. Check the example below, to understand clearly.

Weight (x)	No. of people (f)	Cumulative frequency (c.f.)
46	7	7
48	5	7+5=12
50	8	12+8=20
52	12	20+12=32
53	10	32+10=42
54	2	42+2=44
55	1	44+1=45
n=45

The total number of people (n)=45, which is odd,

Therefore,

Median weight =weight of 23thperson

According to the above-obtained table, we can observe that the weight of each person from 21st person to 32nd person is 52kg.

Therefore, The weight of the 23rd person = 52kg Hence, The Median weight = 52 kg

Example 4: The following numbers are written in ascending order of their values:

20, 22, 25, 30, x-11, x-8, x-3, 52, 60, 68.

If their median is 39, find the value of x.

Solution: Number of terms n=10 (even)

Therefore,

78=x-8+x-11

2x=97

x=48.5

Example 5: Find the mean of the following frequency distribution using the step-deviation method.

x	10	30	50	70	90	110
f	135	187	240	273	124	151

Solution: Let the assume mean A=70 and i=20

x	f	d=x-A	t=x-Ai=x-7020	ft
10	135	-60	-3	-405
30	187	-40	-2	-374
50	240	-20	-1	-240
A=70	273	0	0	0
90	124	20	1	124
110	151	40	2	302
f=1110	ft= -593

Therefore,

=70+-593111020

=59.32

Example 6: The weights of 25 women in an office are given in the following table:

Weight in kg	65	66	67	68	69
Number of women	8	6	4	4	3

Find the mean weight of all the women using the short-cut method.

Solution: Let the assumed mean A=67.

Now make the table according to the above steps:

Weight in kg (x)	Number of women (f)	d=x-A =x-67	fd
65	8	65-67= -2	-16
66	6	66-67= -1	-6
A=67	4	67-67=0	0
68	4	68-67=1	4
69	3	69-67=2	6
f=25	fd= -12

Therefore,

=67+-1225=67-0.48

x=66.52

Example 7: Find the arithmetic mean of the following frequency distribution, using the direct method:

x	5	15	25	35	44.5
f	14	20	20	30	20

Solution:

First of all make the table according to the steps above:

x	f	fx
5	14	70
15	16	240
25	20	500
35	30	1050
44.5	20	890
f=100	fx=2750

Mean (x)=27.50

Example 8: The weights (in kilogram) of 6 children are 35,39,41,40,42, and 47. Find the arithmetic mean of their weights.

Solution: According to the above formula:

We hope that this article on Mean, Median, and Mode was beneficial for you. Hopefully, it helped you understand all the basic concepts behind the formulas. For any more query, you can contact us or you can download the Testbook App, for free, and start preparing for any competitive exam using this app.

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