Mean, Median, Mode Questions 

Introduction 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. In this blog, we present a comprehensive guide on Mean, Median, Mode Questions, ideal for students of all levels, especially those in Class 7, as well as aspirants of competitive exams. We also provide downloadable resources such as a Mean, Median, Mode Questions PDF.

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: 

$\text{Mean}=\frac{\text{Sum of all observations}}{\text{Number of observations}}=\frac{\sum x_i}{n}$

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:

$x_i=\frac{\text{Lower limit + Upper limit}}{2}$

Then use this formula:

$\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}$

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:

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

• If n is even:

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

Formula for Median (Grouped Data):

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

Where:

• L = lower boundary of the median class
• N = total frequency $\left(\sum f_i\right)$
• 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):

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

Formula of Mode (Grouped Data):  

$\text{Mode}=L+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\cdot h$

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 Questions

Mean, Median, Mode Questions for Class 7, 8.

Example 1: The marks obtained by 5 students are: 45, 50, 55, 60, and 90. Find the mean.

Solution:

$\text{Mean}=\frac{45+50+55+60+90}{5}=\frac{300}{5}=60$

Example 2: Find the median of the numbers: 12, 15, 10, 18, 14.

Solution:
First, arrange in ascending order: 10, 12, 14, 15, 18

Number of terms = 5 (odd)

Median = 3^rd term =14 

Example 3: Find the median of: 5, 8, 12, 14, 17, 20

Solution:

Sorted: 5, 8, 12, 14, 17, 20

Median = average of 3rd and 4th terms

$\text{Median}=\frac{12+14}{2}=13$

Example 4: Find the mode of: 3, 5, 2, 3, 8, 3, 7, 5

Solution:

Count of 3 = 3 times, 5 = 2 times, others = once

Mode = 3 

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.

Solution (Step Sketch):

Mode class = 30–40

Use Mode formula for grouped data:

 $\text{Mode}=L+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\cdot h$

Where:

• L = 30,
• $f_1$ = 20,
• $f_0$ = x,
• $f_2$ = 10,
• h = 10 

$$\begin{gathered} 30=30+\left(\frac{20-x}{2(20)-x-10}\right)\cdot 10 \\ \Rightarrow 0=\frac{20-x}{30-x} \\ \Rightarrow 20-x=0 \\ \Rightarrow x=20 \end{gathered}$$

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

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

Example 5: Find the median of the following data:

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:

Answer:
Modal class = 30–40

$L=30,f_1=25,f_0=10,f_2=18,h=10$

$$\begin{gathered} \text{Mode}=30+\left(\frac{25-10}{2\cdot 25-10-18}\right)\cdot 10 \\ \text{Mode}=30+\left(\frac{15}{22}\right)\cdot 10 \\ \text{Mode}=30+6.82 \\ \text{Mode}=\boxed{36.82} \end{gathered}$$

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:

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

5.0Practice Questions on Mean, Median and Mode

1. The mean of 6 numbers is 50. Five of them are 45, 55, 60, 35, and 50. Find the sixth.
2. Median of: 14, 18, 21, 23, 25, 29, 30
3. Mode of: 2, 4, 4, 4, 6, 6, 7, 8, 8, 8, 8

Practice Table-Based Questions:

4. Find the mean from the following frequency distribution:

5. Mean of: 

6. Find the median of class intervals:

6.0Competitive Exam Tips

• 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).

7.0Summary Table of Key Formulas

8.0Final Tips for Students

• Always arrange data in order before finding the median.
• Use a calculator carefully in grouped data questions.
• Understand when to apply which formula.
• For exams, practice identifying the correct class intervals and frequencies.

9.0Related Questions

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

Ans: Yes. For JEE, remember:

$$\begin{gathered} \text{Mean (Grouped):} \\ \bar{x}=\frac{\sum f_i{x}_i}{\sum f_i} \\ \text{Median (Grouped):} \\ \text{Median}=L+\left(\frac{\frac{N}{2}-F}{f}\right)\cdot h \\ \text{Mode (Grouped):} \\ \text{Mode}=L+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\cdot h \end{gathered}$$