Mean Median Mode

Mean, median, and mode are the three measures of central tendency in statistics. We identify the central position of any data set while describing a set of data. This is known as the measure of central tendency. We come across data every day. We find them in newspapers, articles, in our bank statements, mobile and electricity bills. The list is endless; they are present all around us. Now the question arises if we can figure out some important features of the data by considering only certain representatives of the data. This is possible by using measures of central tendency or averages, namely mean, median, and mode.

Let us understand mean, median, and mode in detail in the following sections using solved examples.

1.	What is Mean, Median and Mode in Statistics?
2.	Mean
3.	Median
4.	Mode
5.	Mean, Median, and Mode Formulas
6.	Relation between Mean, Median, and Mode
7.	Difference between Mean and Average
8.	Difference between Mean and Median
9.	Differences Between Mean, Median, and Mode
10.	FAQs on Mean, Median, and Mode

Mean, median, and mode are the measures of central tendency, used to study the various characteristics of a given set of data. A measure of central tendency describes a set of data by identifying the central position in the data set as a single value. We can think of it as a tendency of data to cluster around a middle value. In statistics, the three most common measures of central tendencies are Mean, Median, and Mode.

• Mean: The mean is also known as the average, and it is calculated by adding up all the values in a data set and dividing by the total number of values.
• Median: The median is the middle value of a data set, which separates the highest and lowest values equally. It is calculated by arranging the data set in order from lowest to highest and finding the value in the exact middle.
• Mode: The mode is the value that appears most frequently in a data set.

Choosing the best measure of central tendency depends on the type of data we have. Let’s begin by understanding the meaning of each of these terms.

The arithmetic mean of a given data is the sum of all observations divided by the number of observations. For example, a cricketer's scores in five ODI matches are as follows: 12, 34, 45, 50, 24. To find his average score in a match, we calculate the arithmetic mean of data using the mean formula:

Mean = Sum of all observations/Number of observations

Mean = (12 + 34 + 45 + 50 + 24)/5

Mean = 165/5 = 33

Mean is denoted by x̄ (pronounced as x bar). Mean is calculated a little differently when the data is grouped or ungrouped. Let's find the mean in both cases.

Mean of Ungrouped Data

Let x₁, x₂, x₃ , . . . , x_n be n observations. We can find the arithmetic mean using the mean formula:

Mean, x̄ = (x₁ + x₂ + ... + x_n)/n

Example: If the heights of 5 people are 142 cm, 150 cm, 149 cm, 156 cm, and 153 cm.

Find the mean height.

Mean height, x̄ = (142 + 150 + 149 + 156 + 153)/5

= 750/5

= 150

Mean, x̄ = 150 cm

Thus, the mean height is 150 cm.

Mean of Grouped Data

When the data is present in tabular form, we use the following formula:

Mean, x̄ = (x₁f₁ + x₂f₂ + ... + x_nf_n)/(f₁ + f₂ + ... + f_n)

Consider the following example.

Example 1: Find the mean of the following distribution:

x	4	6	9	10	15
f	5	10	10	7	8

Solution:

Calculation table for arithmetic mean:

xi	fi	xifi
4	5	20
6	10	60
9	10	90
10	7	70
15	8	120
	∑ fi = 40	∑ xi fi = 360

Mean, x̄ = (∑x_i f_i)/(∑f_i)

= 360/40

= 9

Thus, Mean = 9

Example 2: Here is an example where the data is in the form of class intervals. The following table indicates the data on the number of patients visiting a hospital in a month. Find the average number of patients visiting the hospital in a day.

Number of patients	Number of days visiting hospital
0-10	2
10-20	6
20-30	9
30-40	7
40-50	4
50-60	2

Solution:

In this case, we find the classmark (also called as mid-point of a class) for each class.

Note: Class mark = (lower limit + upper limit)/2

Let x₁, x₂, x₃ , . . . , x_n be the class marks of the respective classes.

Hence, we get the following table:

Class mark (xi)	frequency (fi)	xifi
5	2	10
15	6	90
25	9	225
35	7	245
45	4	180
55	2	110
Total	∑ fi = 30	∑ fixi = 860

Mean, x̄ = (∑ x_if_i)/(∑ f_i)

= 860/30

= 28.67

x̄ = 28.67

Challenging Question on Mean:

Let the mean of x₁, x₂, x₃ … x_n be A, then what is the mean of:

• (x₁ + k), (x₂ + k), (x₃ + k), … , (x_n + k)
• (x₁ - k), (x₂ - k), (x₃ - k), … , (x_n - k)
• kx₁, kx₂, kx₃, … , kx_n

The value of the middlemost observation, obtained after arranging the data in ascending or descending order, is called the median of the data. For example, consider the data: 4, 4, 6, 3, 2. Let's arrange this data in ascending order: 2, 3, 4, 4, 6. There are 5 observations. Thus, median = middle value i.e. 4.

Median of Ungrouped Data

• Step 1: Arrange the data in ascending or descending order.
• Step 2: Let the total number of observations be n.

To find the median, we need to consider if n is even or odd. If n is odd, then use the formula:

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

Example 1: Let's consider the data: 56, 67, 54, 34, 78, 43, 23. What is the median?

Solution:

Arranging in ascending order, we get: 23, 34, 43, 54, 56, 67, 78. Here, n (number of observations) = 7

So, (7 + 1)/2 = 4

∴ Median = 4^th observation

Median = 54

If n is even, then use the formula:

Median = [(n/2)^th obs.+ ((n/2) + 1)^th obs.]/2

Example 2: Let's consider the data: 50, 67, 24, 34, 78, 43. What is the median?

Solution:

Arranging in ascending order, we get: 24, 34, 43, 50, 67, 78.

Here, n (no.of observations) = 6

6/2 = 3

Using the median formula,

Median = (3^rd observation + 4^th observation) / 2

= (43 + 50)/2

Median = 46.5

Median of Grouped Data

When the data is continuous and in the form of a frequency distribution, the median is found as shown below:

Step 1: Find the median class.

Let n = total number of observations i.e. ∑ f_i

Note: Median class is the class where (n/2) lies.

Step 2: Use the following formula to find the median.

Median = \( l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)

where,

• l = lower limit of median class
• c = cumulative frequency of the class preceding the median class
• n = total number of observations
• f = frequency of the median class
• h = class size

Let's consider the following example to understand this better.

Example: Find the median marks for the following distribution:

Classes	0-10	10-20	20-30	30-40	40-50
Frequency	2	12	22	8	6

Solution:

We need to calculate the cumulative frequencies to find the median.

Calculation table:

Classes	Number of students	Cumulative frequency
0-10	2	2
10-20	12	2 + 12 = 14
20-30	22	14 + 22 = 36
30-40	8	36 + 8 = 44
40-50	6	44 + 6 = 50

N = 50

N/2 = 50/2 = 25

Median Class = (20 - 30)

l = 20, f = 22, c = 14, h = 10

Using Median formula:

Median = \(l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)

= 20 + (25 - 14)/22 × 10

= 20 + (11/22) × 10

= 20 + 5 = 25

∴ Median = 25

The value which appears most often in the given data i.e. the observation with the highest frequency is called a mode of data.

Case 1: Ungrouped Data

For ungrouped data, we just need to identify the observation which occurs maximum times.

Mode = Observation with maximum frequency

For example in the data: 6, 8, 9, 3, 4, 6, 7, 6, 3, the value 6 appears the most number of times. Thus, mode = 6. An easy way to remember mode is: Most Often Data Entered. Note: A data may have no mode, 1 mode, or more than 1 mode. Depending upon the number of modes the data has, it can be called unimodal, bimodal, trimodal, or multimodal.

The example discussed above has only 1 mode, so it is unimodal.

Case 2: Grouped Data

When the data is continuous, the mode can be found using the following steps:

• Step 1: Find modal class i.e. the class with maximum frequency.
• Step 2: Find mode using the following formula:

Mode = \(L + [\dfrac {f_m-f_1}{2f_m-f_1-f_2}]\times h\)

where,

• L = lower limit of modal class,
• f_m = frequency of modal class,
• f₁ = frequency of class preceding modal class,
• f₂ = frequency of class succeeding modal class,
• h = class width

Mode formula equivalently is written as follows as well:

Consider the following example to understand the formula.

Example: Find the mode of the given data:

Marks Obtained	0-20	20-40	40-60	60-80	80-100
Number of students	5	10	12	6	3

Solution:

The highest frequency = 12, so the modal class is 40-60.

L = lower limit of modal class = 40

f_m = frequency of modal class = 12

f₁ = frequency of class preceding modal class = 10

f₂ = frequency of class succeeding modal class = 6

h = class width = 20

Using the mode formula,

Mode = \(L + [\dfrac {f_m-f_1}{2f_m-f_1-f_2}]\times h\)

= 40 + \([\dfrac{12-10}{2 \times 12 - 10-6} ]\times 20\)

= 40 + (2/8) × 20

= 45

∴ Mode = 45

We covered the formulas and methods to find the mean, median, and mode for a grouped and ungrouped set of data. Let us summarize and recall them using the list of mean, median, and mode formulas given below,

• Mean formula for ungrouped data: Sum of all observations/Number of observations
• Mean formula for grouped data: x̄ = (x₁f₁ + x₂f₂ + ... + x_nf_n)/(f₁ + f₂ + ... + f_n)
• Median formula for ungrouped data: If n is odd, then use the formula: Median = (n + 1)/2^th observation. If n is even, then use the formula: Median = [(n/2)^th obs.+ ((n/2) + 1)^th obs.]/2
• Median formula for grouped data: Median = \( l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\), where
  • l = lower limit of median class
  • c = cumulative frequency of the class preceding the median class
  • f = frequency of the median class
  • h = class size
• Mode formula for ungrouped data: Mode = Observation with maximum frequency
• Mode formula for grouped data: Mode = \(L + [\dfrac {f_m-f_1}{2f_m-f_1-f_2}]\times h\), where
  • L = lower limit of modal class,
  • f_m = frequency of modal class,
  • f₁ = frequency of class preceding modal class,
  • f₂ = frequency of class succeeding modal class,
  • h = class width

Take a quick look at the figure below with mean mode median formulas.

The three measures of central values i.e. mean, median, and mode are closely connected by the following relations (called an empirical relationship).

Mean - Mode = 3 (Mean - Median) (or) 2Mean + Mode = 3Median

For instance, if we are asked to calculate the mean, median, and mode of continuous grouped data, then we can calculate mean and median using the formulas as discussed in the previous sections and then find mode using the empirical relation.

For example, we have data whose mode = 65 and median = 61.6.

Then, we can find the mean using the above mean, median, and mode relation.

2Mean + Mode = 3 Median

⇒ 2Mean = 3 × 61.6 - 65

⇒ 2Mean = 119.8

⇒ Mean = 119.8/2

⇒ Mean = 59.9

The term average is frequently used in everyday life to denote a value that is typical for a group of quantities. Average rainfall in a month or the average age of employees of an organization is a typical example. We might read an article stating "People spend an average of 2 hours every day on social media. " We understand from the use of the term average that not everyone is spending 2 hours a day on social media but some spend more time and some less.

However, we can understand from the term average that 2 hours is a good indicator of the amount of time spent on social media per day. Most people use average and mean interchangeably even though they are not the same.

• Average is the value that indicates what is most likely to be expected.
• They help to summarise large data into a single value.

An average tends to lie centrally with the values of the observations arranged in ascending order of magnitude. So, we call it an average measure of the central tendency of the data. Averages are of different types. What we refer to as mean i.e. the arithmetic mean is one of the averages. Mean is called the mathematical average whereas median and mode are positional averages.

Mean is known as the mathematical average whereas the median is known as the positional average. To understand the difference between the two, consider the following example. A department of an organization has 5 employees which include a supervisor and four executives. The executives draw a salary of ₹10,000 per month while the supervisor gets ₹40,000.

Mean = (10000 + 10000 + 10000 + 10000 + 40000)/5 = 80000/5 = 16000

Thus, the mean salary is ₹16,000.

To find the median, we consider the ascending order: 10000, 10000, 10000, 10000, 40000.

n = 5,

so, (n + 1)/2 = 3

Thus, the median is the 3^rd observation.

Median = 10000

Thus, the median is ₹10,000 per month.

Now let us compare the two measures of central tendencies.

We can observe that the mean salary of ₹16,000 does not give even an estimated salary of any of the employees whereas the median salary represents the data more effectively.

One of the weaknesses of the mean is that it gets affected by extreme values (known as outliers).

Look at the following graph to understand how extreme values affect mean and median:

So, mean is to be used when we don't have extremes in the data. If we have extreme points, then the median gives a better estimation. Here's a quick summary of the differences between the two.

Mean Vs Median	Mean	Median
Definition	Average of given data (Mathematical Average)	The central value of data (Positional Average)
Calculation	Add all values and divide by the total number of observations	Arrange data in ascending / descending order and find the middle value
Values of data	Every value is considered for calculation	Every value is not considered
Effect of extreme points	Greatly affected by extreme points	Doesn't get affected by extreme points

Now, we have got enough knowledge about mean mode and median. Let us summarize the differences between mean median mode in the table below.

Property	Mean	Median	Mode
Definition	Average value of a set of numbers.	Middle value of a set of numbers when they are arranged in order.	The most frequently occurring value in a set of numbers.
Calculation	Add up all the numbers and divide by the total number of numbers.	Arrange all the numbers in order, and the median is the middle number. If there are an even number of values, take the average of the two middle values.	Identify the value that appears most often in the set.
Affected by Outliers?	Yes	No	No

☛ Related Topics:

• Average
• Categorical Data
• Range in Statistics
• Geometric Mean

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What is the Definition of Mean, Median, and Mode?

Mean, median, and mode are measures of central tendency or, in other words, different kinds of averages in statistics. Mean is the "average", where we find the total of all the numbers and then divide by the number of numbers, while the median is the "middle" value in the list of numbers. Mode is the value that occurs most often in the given set of data.

What are Formulas to Find Mean, Median, and Mode of Ungrouped Data?

Different sets of formulas can be used to find mean, median, and mode depending upon the type of data if that is grouped or ungrouped. The following formulas can be used to find the mean median and mode for ungrouped data:

• Mean = Sum of all observations/Number of observations
• If n is odd, then use the formula: Median = (n + 1)/2^th observation. If n is even, then use the formula: Median = [(n/2)^th obs.+ ((n/2) + 1)^th obs.]/2
• Mode = Observation with maximum frequency

How to Find Mean, Median and Mode for Grouped Data?

We can find the mean, mode, and median for grouped data using the below-given formulas,

• Mean, x̄ = (x₁f₁ + x₂f₂ + ... + x_nf_n)/( f₁ + f₂ + ... + f_n)
• Median = \( l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)
• Mode = \(l + [\dfrac {f_m-f_1}{2f_m-f_1-f_2}]\times h\)

To understand the meaning of each variable here, scroll up this page.

What are the Uses of Mean Median Mode?

Here are some uses of mean median and mode along with some examples:

• The mean is often used as a measure of central tendency because it takes into account all the values in the data set. The mean is useful for comparing the average values of different data sets, for example, the average salary of employees in two different companies.
• The median is often used when the data set has a skewed distribution, where most of the values are concentrated on one side of the distribution. For example, the median income of a group of people may be a better indicator of their financial status than the mean income if there are a few extremely wealthy or extremely poor individuals in the group.
• The mode is useful for describing the most common value or category in a data set. For example, the mode of a set of test scores may indicate the most common score that students achieved on the test.

How to Find Mean Median Mode?

The mean, median, and mode for a given set of data can be obtained using the mean, median, and mode formulas. Click here to check these formulas in detail and understand their applications. For more detailed information, scroll this page up.

What Does Mean, Mode, and Median Represent?

Mean, mode, and median are the three measures of central tendency in statistics. Mean represents the average value of the given set of data, while the median is the value of the middlemost observation obtained after arranging the data in ascending or descending order. Mode represents the most common value. It tells you which value has occurred most often in the given data. On a bar chart, the mode is the highest bar. It is used with categorial data such as most sold T-shirts size.

How to Find Median Using Mean Median Mode Formula?

Median is the value of the middlemost observation, obtained after arranging the data in ascending order.

• For ungrouped data, we need to consider two cases.
  • If n is odd, then use the formula: Median = (n + 1)/2^th observation.
  • If n is even, then use the formula: Median = [(n/2)^th obs.+ ((n/2) + 1)^th obs.]/2
• For grouped data, the median is obtained using the median formula:
  • Median = \(l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)

Are Mean, Mode, and Median the Same?

No, mean, mode and median are not the same.

• Mean is the average of the given sets of numbers. We need to add the numbers up then divide their sum by the number of observations.
• For finding the mode, we find whether any number appears more than once. The number which appears most is the mode. If there are other numbers that repeat to the same level, there may be more than one mode. A set could be bimodal or trimodal. But the mean of a given data is unique.
• Median is the value of the middlemost observation, obtained after arranging the data in ascending order.