What is the main difference between Mean, Median, and Mode?

Mean is the arithmetic average. Median is the middle value in sorted data. Mode is the most frequently occurring value. Each gives a different perspective on the data's center.

When should I use Median instead of Mean?

Use median when the data has extreme values (outliers). For example, in income data, a few very high values can distort the mean, but median gives a better central measure.

Can a data set have more than one mode?

Yes. One mode = Unimodal Two modes = Bimodal More than two = Multimodal Some datasets may also have no mode if all values are unique.

Is Mean always greater than Median and Mode?

Not always. It depends on the distribution: Symmetrical distribution: Mean = Median = Mode Positively skewed: Mean > Median > Mode Negatively skewed: Mean < Median < Mode

Can the Mean, Median, and Mode be equal?

Yes. In a perfectly symmetrical distribution (like a normal distribution or evenly spaced values), they can all be equal.

Which of Mean, Median, and Mode is most affected by outliers?

Mean is most affected by extreme values. Median and Mode are more resistant to outliers.

Are these concepts important for JEE Advanced?

Yes, especially in JEE Advanced, where combined concepts (like mean + variance, or using mode in a grouped frequency table) are tested. Knowing formulas and interpretation is key.

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Mean, Median, Mode Questions

Test your Knowledge

question 1 of 4

In statistics, what descriptive term is often used synonymously with the Mean?

1.Average

2.Most diverse value

3.Most frequent value

4.Middle value

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Mean, Median, Mode Questions

Q: Can the Mean, Median, and Mode be equal?

Yes. In a perfectly symmetrical distribution (like a normal distribution or evenly spaced values), they can all be equal.

Q: Which of Mean, Median, and Mode is most affected by outliers?

Mean is most affected by extreme values. Median and Mode are more resistant to outliers.

Q: Are these concepts important for JEE Advanced?

Yes, especially in JEE Advanced, where combined concepts (like mean + variance, or using mode in a grouped frequency table) are tested. Knowing formulas and interpretation is key.

Statistics is a vital branch of mathematics that deals with the collection, analysis, interpretation, and presentation of data. Among the most fundamental concepts in statistics are mean, median, and mode. These measures of central tendency help summarize large sets of data into a single representative value.

1.0What is Mean?

The mean, often called the average, is a measure of central tendency used in statistics. It represents the typical value in a set of numbers and gives an overall idea of the data's distribution.

Formula for Mean (Ungrouped Data):

When data is not grouped into classes, it's called ungrouped.

We use the basic average formula:

$Mean = \frac{Sum of all observations}{Number of observations} = \frac{\sum x _{i}}{n}$

Where:

$x_{i}$ represents each data point
n is the total number of data points

Formula for Mean (Grouped Data):

When data is grouped in class intervals, we can't use the simple average.
We calculate class marks (midpoints) first:

$x_{i} = \frac{Lower limit + Upper limit}{2}$

Then use this formula:

$\overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$

Where:

$f_{i} $ = frequency of class
$x_{i}$ = class mark

2.0What is the Median?

The median is the middle value of a data set when the numbers are arranged in ascending or descending order. It divides the data into two equal halves and is especially useful when the data set contains outliers.

Formula for Median (Ungrouped Data):

If the number of observations (n) is odd:

$Median = Value at position \frac{n + 1}{2}$

If n is even:

$Median = Average of values at positions \frac{n}{2} and \frac{n}{2} + 1$

Formula for Median (Grouped Data):

$Median = L + (\frac{\frac{N}{2} - F}{f}) \cdot h$

Where:

L = lower boundary of the median class
N = total frequency $(\sum f_{i})$
F = cumulative frequency before the median class
f = frequency of the median class
h = class width (class size)

3.0What is Mode?

The mode is the value that appears most frequently in a data set. It represents the most common or popular item.

If one number appears more than others, it's the mode.
If two or more numbers appear with the same highest frequency, the data can be bimodal or multimodal.
If no number repeats, the data has no mode.

How to Find the Mode(Ungrouped Data):

Look at the list of numbers.
Count how many times each number appears.
The number that appears most often is the mode.

Formula of Mode (Grouped Data):

$Mode = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \cdot h$

Where:

L = Lower boundary of the modal class
$f_{1} $ = Frequency of the modal class
$f_{0}$ = Frequency of the class preceding the modal class
$f_{2}$ = Frequency of the class succeeding the modal class
h = Class width (class size)

4.0Solved Examples on Mean, Median and Mode

Mean, Median, Mode Questions for Competitive Exams

Example 1: The mean of 6 numbers is 12. If one number is removed, the new mean becomes 11. Find the number removed.

Solution:
Let sum of 6 numbers = 6 × 12 = 72

New sum (5 numbers) = 5 × 11 = 55

Number removed = 72 – 55 = 17

Example 2: The median of the data: 7, 9, 4, 5, x, 12, 15 is 9. Find x.

Solution:
Arrange: 4, 5, 7, x, 9, 12, 15

Since median is 9, x must be in correct position so that 4th term is 9:
So, x = 9

Example 3: The mode of the following frequency distribution is 30. Find the missing frequency x.

Class Interval	Frequency
10 – 20	5
20 – 30	x
30 – 40	20
40 – 50	10
50 – 60	5

Solution (Step Sketch):

Mode class = 30–40

Use Mode formula for grouped data:

$Mode = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \cdot h$

Where:

L = 30,
$f_{1}$ = 20,
$f_{0}$ = x,
$f_{2}$ = 10,
h = 10

$30 = 30 + (\frac{20 - x}{2 ( 20 ) - x - 10}) \cdot 10 \Rightarrow 0 = \frac{20 - x}{30 - x} \Rightarrow 20 - x = 0 \Rightarrow x = 20$

Example 4: The average of 10 numbers is 20. If 5 of them have an average of 15, what is the average of the remaining 5?

Answer:
Total sum = 10 × 20 = 200

Sum of first 5 = 5 × 15 = 75

Sum of remaining 5 = 200 – 75 = 125

$Average = \frac{125}{5} = 25$

Example 5: Find the median of the following data:

x	10	20	30	40	50
f	3	5	7	4	1

Answer:
Cumulative frequency: 3, 8, 15, 19, 20

Total frequency = 20 ⇒ Median position = 10.5th term

Falls in class 30 ⇒ Median = 30

Example 6: If the mode of the data set {a, a + 2, a + 4, a + 6, a + 6, a + 6, a + 8} is 26, find a.

Answer:
Mode is the most frequent term = a + 6

Given: a + 6 = 26 ⇒ a = 20

Example 7: A student records the following marks in 5 subjects: 95, 85, 75, 65, and x. If the mean, median, and mode are all equal, find the value of x.

Answer:
Sorted data: 65, 75, 85, 95, x

Let’s assume mode = median = mean = M

Median = 85 ⇒ M = 85
Mean = $\frac{65 + 75 + 85 + 95 + x}{5} = 85$

⇒ 320 + x = 425 ⇒ x = 105

Mode must also be 85 ⇒ So 85 should appear at least twice
Update data: 65, 75, 85, 85, 95 ⇒ Now it works!

Example 8: Find the mode of the following grouped data:

Class Interval	Frequency
0 – 10	6
10 – 20	8
20 – 30	10
30 – 40	25
40 – 50	18
50 – 60	10

Answer:
Modal class = 30–40

$L = 30, f_{1} = 25, f_{0} = 10, f_{2} = 18, h = 10$

$Mode = 30 + (\frac{25 - 10}{2 \cdot 25 - 10 - 18}) \cdot 10 Mode = 30 + (\frac{15}{22}) \cdot 10 Mode = 30 + 6.82 Mode = 36.82$

Example 9: Five observations have mean 10 and variance 8. If three of them are 5, 10, and 15, find the other two.

Hint: Use:

$Mean formula: \frac{\sum x}{5} = 10 Variance formula: \frac{\sum x ^{2}}{5} - (mean)^{2} = 8$

5.0Practice Questions on Mean, Median and Mode

The mean of 6 numbers is 50. Five of them are 45, 55, 60, 35, and 50. Find the sixth.
Find the mean from the following frequency distribution:

Marks (x)	Frequency (f)
10	2
20	3
30	5
40	4

3. Mean of:

Value (x)	Frequency (f)
5	1
10	2
15	3
20	4

4. Find the median of class intervals:

Class Interval	Frequency
0-10	3
10-20	7
20-30	10
30-40	5

6.0Tips to solve problems related to mean median and mode

Pay attention to frequency tables.
Be cautious with even vs odd number of terms.
Mode can be more than one value (bimodal or multimodal data).

7.0Key Formulas

Measure	Formula
Mean (Ungrouped)	$\frac{\sum x _{i}}{n}$
Mean (Grouped)	$\frac{\sum f x}{\sum f}$
Median (Ungrouped)	Middle value (or average of two middle terms)
Median (Grouped)	$l + (\frac{\frac{n}{2} - CF}{f}) \times h$
Mode (Grouped)	$l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h$

8.0 9.0Related Questions

1. Is it necessary to memorize formulas for grouped data?

Ans: Yes. For JEE, remember:

$Mean (Grouped): \overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}} Median (Grouped): Median = L + (\frac{\frac{N}{2} - F}{f}) \cdot h Mode (Grouped): Mode = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \cdot h$

Also Read:

Mathematical Reasoning	Probability and Statistics PYQs	Inferential Statistics
Probability and Statistics	Conditional Probability Questions	Hypothesis Testing in Statistics
Systematic Random Sampling	Standard Deviation	Tangent and Normal

Test your Knowledge

question 1 of 4

In statistics, what descriptive term is often used synonymously with the Mean?

1.Average

2.Most diverse value

3.Most frequent value

4.Middle value

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Mean, Median, Mode Questions

Statistics is a vital branch of mathematics that deals with the collection, analysis, interpretation, and presentation of data. Among the most fundamental concepts in statistics are mean, median, and mode. These measures of central tendency help summarize large sets of data into a single representative value.

1.0What is Mean?

The mean, often called the average, is a measure of central tendency used in statistics. It represents the typical value in a set of numbers and gives an overall idea of the data's distribution.

Formula for Mean (Ungrouped Data):

When data is not grouped into classes, it's called ungrouped.

We use the basic average formula:

$Mean = \frac{Sum of all observations}{Number of observations} = \frac{\sum x _{i}}{n}$

Where:

$x_{i}$ represents each data point
n is the total number of data points

Formula for Mean (Grouped Data):

When data is grouped in class intervals, we can't use the simple average.
We calculate class marks (midpoints) first:

$x_{i} = \frac{Lower limit + Upper limit}{2}$

Then use this formula:

$\overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$

Where:

$f_{i} $ = frequency of class
$x_{i}$ = class mark

2.0What is the Median?

The median is the middle value of a data set when the numbers are arranged in ascending or descending order. It divides the data into two equal halves and is especially useful when the data set contains outliers.

Formula for Median (Ungrouped Data):

If the number of observations (n) is odd:

$Median = Value at position \frac{n + 1}{2}$

If n is even:

$Median = Average of values at positions \frac{n}{2} and \frac{n}{2} + 1$

Formula for Median (Grouped Data):

$Median = L + (\frac{\frac{N}{2} - F}{f}) \cdot h$

Where:

L = lower boundary of the median class
N = total frequency $(\sum f_{i})$
F = cumulative frequency before the median class
f = frequency of the median class
h = class width (class size)

3.0What is Mode?

The mode is the value that appears most frequently in a data set. It represents the most common or popular item.

If one number appears more than others, it's the mode.
If two or more numbers appear with the same highest frequency, the data can be bimodal or multimodal.
If no number repeats, the data has no mode.

How to Find the Mode(Ungrouped Data):

Look at the list of numbers.
Count how many times each number appears.
The number that appears most often is the mode.

Formula of Mode (Grouped Data):

$Mode = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \cdot h$

Where:

L = Lower boundary of the modal class
$f_{1} $ = Frequency of the modal class
$f_{0}$ = Frequency of the class preceding the modal class
$f_{2}$ = Frequency of the class succeeding the modal class
h = Class width (class size)

4.0Solved Examples on Mean, Median and Mode

Mean, Median, Mode Questions for Competitive Exams

Example 1: The mean of 6 numbers is 12. If one number is removed, the new mean becomes 11. Find the number removed.

Solution:
Let sum of 6 numbers = 6 × 12 = 72

New sum (5 numbers) = 5 × 11 = 55

Number removed = 72 – 55 = 17

Example 2: The median of the data: 7, 9, 4, 5, x, 12, 15 is 9. Find x.

Solution:
Arrange: 4, 5, 7, x, 9, 12, 15

Since median is 9, x must be in correct position so that 4th term is 9:
So, x = 9

Example 3: The mode of the following frequency distribution is 30. Find the missing frequency x.

Class Interval	Frequency
10 – 20	5
20 – 30	x
30 – 40	20
40 – 50	10
50 – 60	5

Solution (Step Sketch):

Mode class = 30–40

Use Mode formula for grouped data:

$Mode = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \cdot h$

Where:

L = 30,
$f_{1}$ = 20,
$f_{0}$ = x,
$f_{2}$ = 10,
h = 10

$30 = 30 + (\frac{20 - x}{2 ( 20 ) - x - 10}) \cdot 10 \Rightarrow 0 = \frac{20 - x}{30 - x} \Rightarrow 20 - x = 0 \Rightarrow x = 20$

Example 4: The average of 10 numbers is 20. If 5 of them have an average of 15, what is the average of the remaining 5?

Answer:
Total sum = 10 × 20 = 200

Sum of first 5 = 5 × 15 = 75

Sum of remaining 5 = 200 – 75 = 125

$Average = \frac{125}{5} = 25$

Example 5: Find the median of the following data:

x	10	20	30	40	50
f	3	5	7	4	1

Answer:
Cumulative frequency: 3, 8, 15, 19, 20

Total frequency = 20 ⇒ Median position = 10.5th term

Falls in class 30 ⇒ Median = 30

Example 6: If the mode of the data set {a, a + 2, a + 4, a + 6, a + 6, a + 6, a + 8} is 26, find a.

Answer:
Mode is the most frequent term = a + 6

Given: a + 6 = 26 ⇒ a = 20

Example 7: A student records the following marks in 5 subjects: 95, 85, 75, 65, and x. If the mean, median, and mode are all equal, find the value of x.

Answer:
Sorted data: 65, 75, 85, 95, x

Let’s assume mode = median = mean = M

Median = 85 ⇒ M = 85
Mean = $\frac{65 + 75 + 85 + 95 + x}{5} = 85$

⇒ 320 + x = 425 ⇒ x = 105

Mode must also be 85 ⇒ So 85 should appear at least twice
Update data: 65, 75, 85, 85, 95 ⇒ Now it works!

Example 8: Find the mode of the following grouped data:

Class Interval	Frequency
0 – 10	6
10 – 20	8
20 – 30	10
30 – 40	25
40 – 50	18
50 – 60	10

Answer:
Modal class = 30–40

$L = 30, f_{1} = 25, f_{0} = 10, f_{2} = 18, h = 10$

$Mode = 30 + (\frac{25 - 10}{2 \cdot 25 - 10 - 18}) \cdot 10 Mode = 30 + (\frac{15}{22}) \cdot 10 Mode = 30 + 6.82 Mode = 36.82$

Example 9: Five observations have mean 10 and variance 8. If three of them are 5, 10, and 15, find the other two.

Hint: Use:

$Mean formula: \frac{\sum x}{5} = 10 Variance formula: \frac{\sum x ^{2}}{5} - (mean)^{2} = 8$

5.0Practice Questions on Mean, Median and Mode

The mean of 6 numbers is 50. Five of them are 45, 55, 60, 35, and 50. Find the sixth.
Find the mean from the following frequency distribution:

Marks (x)	Frequency (f)
10	2
20	3
30	5
40	4

3. Mean of:

Value (x)	Frequency (f)
5	1
10	2
15	3
20	4

4. Find the median of class intervals:

Class Interval	Frequency
0-10	3
10-20	7
20-30	10
30-40	5

6.0Tips to solve problems related to mean median and mode

Pay attention to frequency tables.
Be cautious with even vs odd number of terms.
Mode can be more than one value (bimodal or multimodal data).

7.0Key Formulas

Measure	Formula
Mean (Ungrouped)	$\frac{\sum x _{i}}{n}$
Mean (Grouped)	$\frac{\sum f x}{\sum f}$
Median (Ungrouped)	Middle value (or average of two middle terms)
Median (Grouped)	$l + (\frac{\frac{n}{2} - CF}{f}) \times h$
Mode (Grouped)	$l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h$

8.0 9.0Related Questions

1. Is it necessary to memorize formulas for grouped data?

Ans: Yes. For JEE, remember:

$Mean (Grouped): \overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}} Median (Grouped): Median = L + (\frac{\frac{N}{2} - F}{f}) \cdot h Mode (Grouped): Mode = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \cdot h$

Also Read:

Mathematical Reasoning	Probability and Statistics PYQs	Inferential Statistics
Probability and Statistics	Conditional Probability Questions	Hypothesis Testing in Statistics
Systematic Random Sampling	Standard Deviation	Tangent and Normal

Test your Knowledge

question 1 of 4

In statistics, what descriptive term is often used synonymously with the Mean?

Frequently Asked Questions

What is the main difference between Mean, Median, and Mode?

When should I use Median instead of Mean?

Can a data set have more than one mode?

Is Mean always greater than Median and Mode?

Can the Mean, Median, and Mode be equal?

Which of Mean, Median, and Mode is most affected by outliers?

Are these concepts important for JEE Advanced?

Join ALLEN!

Mean, Median, Mode Questions

1.0What is Mean?

2.0What is the Median?

3.0What is Mode?

4.0Solved Examples on Mean, Median and Mode

Mean, Median, Mode Questions for Competitive Exams

5.0Practice Questions on Mean, Median and Mode

6.0Tips to solve problems related to mean median and mode

7.0Key Formulas

8.0

9.0Related Questions

Table of Contents

Test your Knowledge

question 1 of 4

In statistics, what descriptive term is often used synonymously with the Mean?

Frequently Asked Questions

What is the main difference between Mean, Median, and Mode?

When should I use Median instead of Mean?

Can a data set have more than one mode?

Is Mean always greater than Median and Mode?

Can the Mean, Median, and Mode be equal?

Which of Mean, Median, and Mode is most affected by outliers?

Are these concepts important for JEE Advanced?

Join ALLEN!

Mean, Median, Mode Questions

1.0What is Mean?

2.0What is the Median?

3.0What is Mode?

4.0Solved Examples on Mean, Median and Mode

Mean, Median, Mode Questions for Competitive Exams

5.0Practice Questions on Mean, Median and Mode

6.0Tips to solve problems related to mean median and mode

7.0Key Formulas

8.0

9.0Related Questions

Table of Contents