Statistics Class 10 Previous Year Questions: 1 Mark (Chapter 13)

1-mark questions on Statistics

This page has 33 1-mark questions on Statistics from CBSE Class 10 board papers of 2023, 2024, 2025 and 2026. Of these, 21 appeared only in Standard papers and 12 appeared only in Basic papers. Question formats: 33 MCQs. 22 of them were asked in two or more paper sets.

0 of 33 practised

All 33 Statistics 1-mark questions

Q1. The following distribution shows the number of runs scored by some batsmen in test matches :

Mark practised

Runs Scored	3000 – 4000	4000 – 5000	5000 – 6000	6000 – 7000
Number of Batsmen	5	10	9	8

The lower limit of the modal class is :

(A) 3000
(B) 4000
(C) 5000
(D) 6000

Marks: 1
Level: Basic
Asked in: 6 paper sets
Years: 2025, 2026
Topic: 13.3 Mode of Grouped Data
Source: (CBSE 2025 Mathematics Basic, Set 430/1/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 4000, because the modal class is 4000 – 5000, whose lower limit is 4000.
Why? The modal class is the class interval with the maximum frequency; here the frequency 10 is the largest, so the modal class is 4000 – 5000.

The full 1 mark is for the correct option.

Answer:

(B) 4000

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/1/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 3000, the lower limit of the first class, without comparing the frequencies 5, 10, 9 and 8.
Choosing 5000, the upper limit of the modal class, instead of its lower limit 4000.
Choosing 4500, the class mark of 4000 – 5000, because the question asks for the lower limit, not the midpoint.

Q2. The following table shows the marks scored by 23 students of a class.

Mark practised

Marks	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50
Number of Students	5	3	4	8	3

The lower limit of the modal class is :

(A) 10
(B) 20
(C) 30
(D) 40

Marks: 1
Level: Basic
Asked in: 6 paper sets
Years: 2025, 2026
Topic: 13.3 Mode of Grouped Data
Source: (CBSE 2025 Mathematics Basic, Set 430/3/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 30, because the modal class 30 – 40 has the maximum frequency 8.
Why? The modal class is the class interval with the maximum frequency, so the class 30 – 40 with frequency 8 is modal, and its lower limit is 30.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/3/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 40, the upper limit of the modal class 30 – 40, when the question asks for the lower limit.
Choosing 20 because the frequencies 3, 4, 8 increase, but the modal class is decided only by the maximum frequency, which is 8 in 30 – 40.
Choosing 10, the lower limit of 10 – 20, without comparing all frequencies; the class 30 – 40 has the highest frequency 8.

Q3. For a distribution, if mean = 15 and mode = 12, then its median is :

Mark practised

(A) 12
(B) 13
(C) 14
(D) 15

Marks: 1
Level: Basic
Asked in: 6 paper sets
Years: 2025, 2026
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Basic, Set 430/3/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 14, because 3 × Median = Mode + 2 × Mean = 12 + 2 × 15 = 42, so Median = 42 ÷ 3 = 14.
Why? The empirical relationship between the three measures of central tendency is 3 Median = Mode + 2 Mean; substituting the given values gives the median.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/3/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 15 because the mean is given as 15, without applying the empirical relationship.
Choosing 12 because the mode is given as 12, without applying the empirical relationship.
Forgetting the coefficient 2 of Mean, so writing 3 Median = Mode + Mean and dividing 12 + 15 by 3.

Q4. In the formula of mode given by mode = \(l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\), \(f_1\) denotes the :

Mark practised

(A) frequency of the modal class
(B) frequency of class preceding modal class
(C) frequency of class succeeding modal class
(D) cumulative frequency of modal class

Marks: 1
Level: Basic
Asked in: 4 paper sets
Years: 2025, 2026
Topic: 13.3 Mode of Grouped Data
Source: (CBSE 2025 Mathematics Basic, Set 430/2/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) frequency of the modal class, because f1 is the frequency of the class with the maximum frequency.
Why? In the mode formula for grouped data, f1 is defined as the frequency of the modal class, the class having the maximum frequency.

The full 1 mark is for the correct option.

Answer:

(A) frequency of the modal class

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/2/1: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing f0, the frequency of the class preceding the modal class, instead of f1.
Choosing f2, the frequency of the class succeeding the modal class.
Thinking f1 is the cumulative frequency of the modal class.

Q5. For a distribution, if mean = median = a, then its mode is :

Mark practised

(A) 3a
(B) 2a
(C) a
(D) 0

Marks: 1
Level: Basic
Asked in: 4 paper sets
Years: 2025, 2026
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Basic, Set 430/2/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) a, because 3 Median = Mode + 2 Mean gives 3a = Mode + 2a, so Mode = a.
Why? The empirical relationship links the three measures of central tendency; substituting the given values and solving gives Mode = a.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/2/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 3a by taking the left side 3 Median as the mode itself, without rearranging the relationship.
Choosing 2a by subtracting the mean once instead of twice in the rearranged relationship.
Choosing 0 because 0 appears among the options, without substituting mean = median = a into the empirical relationship.

Q6. Using empirical relationship, the mode of a distribution whose mean is 7.2 and the median 7.1, is :

Mark practised

(A) 6.2
(B) 6.3
(C) 6.5
(D) 6.9

Marks: 1
Level: Standard
Asked in: 3 paper sets
Compartment: 2023
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2023 Compartment Mathematics Standard, Set 30/C/1, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 6.9, because Mode = 3 Median − 2 Mean = 3 × 7.1 − 2 × 7.2 = 6.9.
Why? By the empirical relationship between the three measures of central tendency, 3 Median = Mode + 2 Mean, so Mode = 3 Median − 2 Mean.

The full 1 mark is for the correct option.

Answer:

(D) 6.9

✓ Answer matches the CBSE answer key, 2023 Compartment, Paper 30/C/1: option (D).

Open CBSE’s official marking scheme for the 2023 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Multiplying the mean by 3 and the median by 2, giving 7.4, which is not an option.
Averaging the mean and median instead of using the empirical relationship, giving 7.15, which is not an option.

Why this works

The empirical relationship is an approximate connection between the three measures of central tendency. Rearrange 3 Median = Mode + 2 Mean to get Mode = 3 Median − 2 Mean. Substituting Mean = 7.2 and Median = 7.1 gives Mode = 3(7.1) − 2(7.2) = 21.3 − 14.4 = 6.9. So the mode is 6.9, which is option D. A quick check: if someone swaps the coefficients, 3 × 7.2 − 2 × 7.1 = 7.4, which is not an option, so the order matters.

Key idea to remember

Empirical relationship between the three measures of central tendency: see Key ideas used on this page.

Explanation

The empirical relationship states 3 Median = Mode + 2 Mean. This solution rearranges the relation to express the mode in terms of the median and the mean, then substitutes the values given in the question.

Important formulas

3 Median = Mode + 2 Mean · Empirical relationship among the three measures of central tendency.
Mode = 3 Median − 2 Mean · Rearranged form used to find the mode when the mean and median are known.

Q7. The distribution below gives the marks obtained by 80 students on a test :

Mark practised

Marks	Less than 10	Less than 20	Less than 30	Less than 40	Less than 50	Less than 60
Number of Students	3	12	27	57	75	80

The modal class of this distribution is :

(A) \(10 - 20\)
(B) \(20 - 30\)
(C) \(30 - 40\)
(D) \(50 - 60\)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2023
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2023 Mathematics Standard, Set 30/1/1, Q11, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 30 - 40, because the class 30 - 40 has the maximum frequency 30.
Why? The modal class is the class interval having the maximum frequency, so the class with the greatest class frequency is selected.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/1/1: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 50 − 60 because its cumulative frequency 80 is the largest, without converting to class frequencies.
Taking 12 as the frequency of 10 − 20 instead of 12 − 3 = 9, which distorts the comparison.

Why this works

The given table is a less than type cumulative frequency distribution. Convert it to class frequencies by subtracting the previous cumulative frequency: the class 0−10 has frequency 3, 10−20 has 12 − 3 = 9, 20−30 has 27 − 12 = 15, 30−40 has 57 − 27 = 30, 40−50 has 75 − 57 = 18, and 50−60 has 80 − 75 = 5. The frequencies add up to 3 + 9 + 15 + 30 + 18 + 5 = 80, which matches the total number of students. The largest frequency is 30, and it belongs to the class 30−40, so the modal class is 30−40. Options A, B and D have frequencies 9, 15 and 5 respectively, all smaller than 30.

Key idea to remember

Modal class: see Key ideas used on this page.

Explanation

For grouped data, the modal class is the class interval having the maximum frequency. The mode is a value inside this class; this question asks only for the modal class, not the mode value.

Important formulas

Frequency of a class = cumulative frequency up to that class − cumulative frequency of the preceding class · Use this to change a less than type cumulative table into ordinary class frequencies.

Q8. If the value of each observation of a statistical data is increased by 3, then the mean of the data

Mark practised

(A) remains unchanged
(B) increases by 3
(C) increases by 6
(D) increases by 3n

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2023
Topic: 13.2 Mean of Grouped Data
Source: (CBSE 2023 Mathematics Standard, Set 30/1/1, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) increases by 3, because adding 3 to all n observations adds 3n to the sum, so the mean increases by 3n/n = 3.
Why? The mean is the sum of all observations divided by the number of observations, so an equal increase in every observation increases the mean by the same number.

The full 1 mark is for the correct option.

Answer:

(B) increases by 3

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/1/1: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A (remains unchanged), thinking that because the mean is an average, adding 3 to every observation averages out to no change.
Choosing D (increases by 3n) after seeing that the sum becomes old sum + 3n, but forgetting to divide the added 3n by the number of observations n.
Choosing C (increases by 6) by doubling the 3 added to each observation, instead of dividing the total increase 3n by n.

Why this works

Let the observations be x1, x2, ..., xn. The original mean is x̄ = (x1 + x2 + ... + xn)/n. After adding 3 to each observation, the new sum is (x1 + 3) + (x2 + 3) + ... + (xn + 3) = (x1 + x2 + ... + xn) + 3n. So the new mean is ((x1 + x2 + ... + xn) + 3n)/n = x̄ + 3. Quick check: the data 2, 4, 6 has mean (2 + 4 + 6)/3 = 4; after adding 3 it becomes 5, 7, 9, whose mean is (5 + 7 + 9)/3 = 7, which is the old mean 4 increased by 3.

Key idea to remember

Mean of grouped data: see Key ideas used on this page.

Explanation

The mean x̄ of observations x1, x2, ..., xn is the sum of all the values divided by the number of observations n. The NCERT formula is x̄ = Σ f_i x_i / Σ f_i, where f_i is the frequency of x_i. The same idea answers this question: every observation grows by 3, so the sum grows by 3n, and dividing by n gives 3n/n = 3.

Important formulas

x̄ = Σ f_i x_i / Σ f_i · Mean of grouped data: the sum of each observation multiplied by its frequency, divided by the total frequency.
x̄ = (x1 + x2 + ... + xn)/n · Mean of ungrouped data: the sum of the n observations divided by n.

Q9. If every term of the statistical data consisting of \(n\) terms is decreased by 2, then the mean of the data :

Mark practised

(A) decreases by 2
(B) remains unchanged
(C) decreases by \(2n\)
(D) decreases by 1

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2023
Topic: 13.2 Mean of Grouped Data
Source: (CBSE 2023 Mathematics Standard, Set 30/5/3, Q11, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) decreases by 2, because subtracting 2 from each of the n terms lowers the total by 2n, so the mean falls by 2n/n = 2.
Why? The mean is the sum of all observations divided by the number of observations, so subtracting the same number from every observation changes the mean by that same number.

The full 1 mark is for the correct option.

Answer:

(A) decreases by 2

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/5/3: option (A).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 'decreases by 2n' because the total of the data falls by 2n, without dividing the total change by the number of terms n.
Choosing 'remains unchanged' because every term is reduced by 2, without checking what happens to the average.
Choosing 'decreases by 1' because they divide the total decrease 2n by 2n instead of by n.

Why this works

Let the original terms be \(x_1, x_2, \dots, x_n\). Then the original mean is \(\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}\). After decreasing every term by 2, the new terms are \(x_1 - 2, x_2 - 2, \dots, x_n - 2\). Their sum is \((x_1 + x_2 + \dots + x_n) - 2n\). So the new mean is \(\frac{(x_1 + x_2 + \dots + x_n) - 2n}{n} = \bar{x} - 2\). Hence the mean decreases by 2. Quick check: for 4, 6, 8 the mean is \(18/3 = 6\); after subtracting 2 the data is 2, 4, 6 and the mean is \(12/3 = 4\), which is \(6 - 2\).

Key idea to remember

Mean of observations: see Key ideas used on this page.

Explanation

The mean is the sum of the values of all the observations divided by the total number of observations. When every observation is decreased by 2, the sum decreases by 2n, and dividing this decrease by n shows that the mean decreases by exactly 2.

Important formulas

\(\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}\) · Mean of n observations; in this question every term has frequency 1.
\(\bar{x}_{new} = \bar{x} - 2\) · Subtracting the same constant from every observation subtracts that constant from the mean.

Q10. If the mean and median of a data are 10 and 11 respectively, then mode of the data is :

Mark practised

(A) 12
(B) 8
(C) 20
(D) 13

Marks: 1
Level: Basic
Asked in: 3 paper sets
Years: 2024
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2024 Mathematics Basic, Set 430/3/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 13, because 3 × 11 − 2 × 10 = 13, which is the mode.
Why? The empirical relationship 3 × Median = Mode + 2 × Mean links the three measures of central tendency, so substituting the two given values gives the mode.

The full 1 mark is for the correct option.

Answer:

(D) 13

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/3/1: option (D).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Swapping the coefficients and using 3 times mean minus 2 times median, which gives 8, option B.
Using 2 times median minus mean, which gives 12, option A.
Choosing option C, 20, by guessing near the mean and median instead of applying the empirical relationship.

Q11. The middle most observation of every data arranged in order is called :

Mark practised

(A) mode
(B) median
(C) mean
(D) deviation

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2024 Mathematics Standard, Set 30/1/1, Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) median, because the median is the value of the middle-most observation when the data is arranged in order.
Why? By definition, the median is the measure of central tendency that gives the middle-most observation in data arranged in ascending order.

The full 1 mark is for the correct option.

Answer:

(B) median

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/1/1: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing mode because the word 'most' in 'middle most' suggests 'most often', but the median, not the mode, is the middle-most observation.
Choosing mean because it is the common average, while the middle-most observation is specifically the median.

Q12. If value of each observation in a data is increased by 2, then median of the new data

Mark practised

(A) increases by 2
(B) increases by 2n
(C) remains same
(D) decreases by 2

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2024 Mathematics Standard, Set 30/2/1, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) increases by 2, because the median is the middle-most observation and every observation, including the middle one, increases by 2.
Why? The median is defined as the value of the middle-most observation, so adding the same constant to every observation adds that constant to the middle value.

The full 1 mark is for the correct option.

Answer:

(A) increases by 2

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/2/1: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 'increases by 2n' because the sum of all observations increases by 2n; the median is not the sum, it is the middle position value.
Choosing 'remains same' because only positions matter; positions stay the same but the value at the middle position increases by 2.
Choosing 'decreases by 2' by reversing the direction of the change.

Why this works

The median is the value of the middle-most observation after arranging the data in ascending order. Adding the same number 2 to every observation moves the whole data set 2 units to the right on the number line, so the order stays the same. The observation that was in the middle remains in the middle, and its value is now 2 more. For an odd-sized data 3, 5, 8, the median is 5. After adding 2, the data is 5, 7, 10 and the median is 7, which is 5 + 2. For an even-sized data 2, 4, 6, 8, the median is (4 + 6)/2 = 5. After adding 2, the data is 4, 6, 8, 10 and the median is (6 + 8)/2 = 7, again 2 more than the original median.

Key idea to remember

Median: see Key ideas used on this page.

Explanation

The median is a measure of central tendency that gives the value of the middle-most observation in the data. For ungrouped data arranged in ascending order, if n is odd the median is the ((n + 1)/2)th observation; if n is even, the median is the average of the (n/2)th and ((n/2) + 1)th observations. Here, increasing every observation by 2 does not change the positions, so the middle observation, or the average of the two middle observations, also increases by 2.

Important formulas

If n is odd, Median = ((n + 1)/2)th observation · Used when the data has an odd number of observations.
If n is even, Median = ((n/2)th observation + ((n/2) + 1)th observation) / 2 · Used when the data has an even number of observations.

Q13. After an examination, a teacher wants to know the marks obtained by Maximum number of the students in her class. She requires to calculate _____ of marks.

Mark practised

(A) median
(B) mode
(C) mean
(D) range

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 13.3 Mode of Grouped Data
Source: (CBSE 2024 Mathematics Standard, Set 30/2/2, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) mode, because the mode is the value that occurs most often, so it gives the marks obtained by the maximum number of students.
Why? The mode is the value among the observations which occurs most often, that is, the value of the observation having the maximum frequency.

The full 1 mark is for the correct option.

Answer:

(B) mode

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/2/2: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing mean because it is the average of all marks, while the teacher wants the most frequent marks.
Choosing median because it is the middle value, not the value that occurs most often.
Choosing range because it tells the spread of marks, not the most frequent value.

Q14. The cumulative frequency for calculating median is obtained by adding the frequencies of all the :

Mark practised

(A) classes up to the median class
(B) classes following the median class
(C) classes preceding the median class
(D) all classes

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2025
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/4/1, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) classes preceding the median class, because the median formula uses the cumulative frequency of the class preceding the median class.
Why? The cumulative frequency of a class is, by definition, the sum of the frequencies of all classes before it; the median formula's cf is exactly this value for the class preceding the median class.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/4/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A, because the cumulative frequency up to the median class includes the frequency of the median class itself, but the formula needs only the classes before it.
Choosing D, because a cumulative frequency table is a running total of all classes, but for the median the value of cf is taken only up to the class preceding the median class.

Why this works

To find the median of grouped data, first find n/2 and locate the median class: the class whose cumulative frequency is just greater than n/2. The formula Median = l + ((n/2 − cf)/f) × h then uses cf, the cumulative frequency of the class preceding the median class. By definition, this cf is the sum of the frequencies of all the classes before the median class. In NCERT Example 7, for 51 girls, n/2 = 25.5, the median class is 145–150, and cf = 11, which is 4 + 7, the frequencies of the two classes before it. So the cumulative frequency used for the median comes only from the preceding classes.

Key idea to remember

Cumulative frequency: see Key ideas used on this page.

Explanation

The cumulative frequency of a class is the frequency obtained by adding the frequencies of all the classes preceding that class. In the median formula, cf stands for the cumulative frequency of the class preceding the median class.

Important formulas

Median = l + ((n/2 − cf)/f) × h · cf is the cumulative frequency of the class preceding the median class.

Q15. Following data shows the marks obtained by 100 students in a class test : 1

Mark practised

Marks obtained	20	29	28	33	42	38	43	25
Number of students	6	28	24	15	2	4	1	20

The median will be the average of which two observations ?

(A) 29 and 33
(B) 25 and 28
(C) 28 and 29
(D) 33 and 38

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2025
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/5/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 28 and 29, because with 100 observations the median is the average of the 50th and 51st observations, and after arranging the marks in ascending order the 50th observation is 28 and the 51st is 29.
Why? For ungrouped data with an even number n of observations, the median is the average of the n/2 th and the (n/2 + 1)th observations.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/5/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Picking 29 and 33, because these look like the middle marks of the table, without first arranging the marks in ascending order.
Choosing 28 and 28, because the cumulative frequency reaches 50 at 28, but the 51st observation is the next mark, 29.
Choosing 25 and 28, thinking the median uses the 25th and 28th observations instead of the 50th and 51st.

Why this works

Here n is 100, an even number, so the median is the average of the 50th and 51st observations. Arrange the marks in ascending order: 20, 25, 28, 29, 33, 38, 42, 43 with frequencies 6, 20, 24, 28, 15, 4, 2, 1. The cumulative frequency reaches 50 at mark 28, so the 50th observation is 28. The next 28 students scored 29, so the 51st observation is 29. Therefore the median is the average of 28 and 29. As a quick check: 6 + 20 + 24 = 50 and 50 + 28 = 78.

Key idea to remember

Median of Ungrouped Data: see Key ideas used on this page.

Explanation

The median is the middle-most observation. When n is even, no single observation is in the middle, so the median is the average of the n/2 th and the (n/2 + 1)th observations after arranging the data in ascending order. This question only asks which two observations are averaged, so we locate the 50th and 51st observations.

Important formulas

Median = ((n/2)th observation + (n/2 + 1)th observation) / 2 · For ungrouped data when n is even.

Q16. If the maximum number of students has obtained 52 marks out of 80, then

Mark practised

(A) 52 is the mean of the data.
(B) 52 is the median of the data.
(C) 52 is the mode of the data.
(D) 52 is the range of the data.

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2025
Topic: 13.3 Mode of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/6/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 52 is the mode of the data, because the mode is the value of the observation having the maximum frequency.
Why? The mode is that value among the observations which occurs most often, that is, the value of the observation having the maximum frequency.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A because 52 looks like an average mark, but the mean is the sum of all marks divided by the total number of students, not the most repeated mark.
Choosing B because 52 could be in the middle of the marks, but the median is the middle-most observation when the data is arranged in order.
Choosing D because the mark obtained by most students sounds like a spread, but the range is the highest mark minus the lowest mark.

Q17. If the mean and mode of a data are 12 and 21 respectively, then its median is :

Mark practised

(A) 6
(B) 13.5
(C) 15
(D) 14

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2026
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2026 Mathematics Standard, Set 30/1/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 15, because 3 Median = Mode + 2 Mean = 21 + 2 × 12 = 45, so Median = 45 ÷ 3 = 15.
Why? By the empirical relationship 3 Median = Mode + 2 Mean, substitute the given mean and mode and solve for the median.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/1/1: option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Giving 45, the value of Mode + 2 Mean, without dividing by 3; the median is 15.
Reversing the formula to 3 Median = Mean + 2 Mode, which would give 18.
Averaging 12 and 21 to get 16.5, instead of using the empirical relationship.

Q18. If the mean and the median of a data are 12 and 15 respectively, then its mode is :

Mark practised

(A) 13.5
(B) 21
(C) 6
(D) 14

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2023
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2023 Mathematics Standard, Set 30/5/1, Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 21, because the empirical relationship 3 Median = Mode + 2 Mean gives Mode = 3 × 15 − 2 × 12 = 21.
Why? By the empirical relationship, three times the median equals the mode plus twice the mean, so the mode is three times the median minus twice the mean.

The full 1 mark is for the correct option.

Answer:

(B) 21

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/5/1: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 13.5 by averaging the mean and the median, (12 + 15)/2, instead of using the empirical relationship.
Swapping the coefficients and computing 3 × 12 − 2 × 15, which gives 6.
Adding instead of subtracting, 3 × 15 + 2 × 12, which gives 69.

Why this works

The three measures of central tendency are connected by the empirical relationship 3 Median = Mode + 2 Mean. When the mean and median are given, rearrange it to Mode = 3 Median − 2 Mean. Substituting median 15 and mean 12: Mode = 3 × 15 − 2 × 12 = 45 − 24 = 21. Quick check: with Mode = 21, the left side of the relationship is 3 × 15 = 45, and the right side is 21 + 2 × 12 = 21 + 24 = 45. Both sides match, so the answer is correct.

Key idea to remember

Empirical relationship between the three measures of central tendency: see Key ideas used on this page.

Explanation

The empirical relationship 3 Median = Mode + 2 Mean links the mean, median and mode of a data. When the mean and the median are given, rearranging this relationship gives Mode = 3 Median − 2 Mean. Substituting the given median 15 and mean 12 gives the mode directly.

Important formulas

3 Median = Mode + 2 Mean · Empirical relationship between the three measures of central tendency.
Mode = 3 Median − 2 Mean · Rearranged form used to find the mode when the mean and median are given.

Q19. The value of \(x\) for which class mark of the class interval \(30 - x\) is 36, is :

Mark practised

(A) 38
(B) 40
(C) 36
(D) 42

Marks: 1
Level: Basic
Asked in: 2 paper sets
Compartment: 2024
Topic: 13.2 Mean of Grouped Data
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430/S/1, Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 42, because class mark = (30 + x)/2 = 36, so x = 42.
Why? The class mark of a class interval is the midpoint of its two limits, so it equals half the sum of the lower limit and the upper limit.

The full 1 mark is for the correct option.

Answer:

(D) 42

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430/S/1: option (D).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Using (30 − x)/2 as the class mark, because the dash in 30 - x is a separator, not a minus sign; the class interval is from 30 to x.
Treating the class mark as 30 + x, which would give 6 instead of 42.
Choosing 40 without solving, because it looks close to 36.

Why this works

The class interval is 30 - x, so the lower limit is 30 and the upper limit is x. The class mark is the average of the limits: (30 + x)/2. The question says this class mark is 36, so (30 + x)/2 = 36. Multiplying both sides by 2 gives 30 + x = 72. Subtracting 30 gives x = 42. A quick check: with x = 42, the class mark is (30 + 42)/2 = 36.

Key idea to remember

Class mark: see Key ideas used on this page.

Explanation

The class mark of a class interval is its mid-point. The NCERT formula is Class mark = (Upper class limit + Lower class limit)/2. Here the lower limit is 30 and the upper limit is x, so the class mark is (30 + x)/2. Equating this to the given class mark 36 and solving gives x = 42.

Important formulas

Class mark = (Upper class limit + Lower class limit)/2 · Use the two printed limits of the class interval; here they are 30 and x.

Q20. The annual rainfall record of a city for 66 days is given in the following table :

Mark practised

Rainfall (in cm) :	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60
Number of days :	22	10	8	15	5	6

The difference of upper limits of modal and median classes is :

(A) 10
(B) 15
(C) 20
(D) 30

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2024
Topic: Mixed topics
Source: (CBSE 2024 Mathematics Basic, Set 430/2/2, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 20, because the modal class is 0–10 with upper limit 10, the median class is 20–30 with upper limit 30, and 30 − 10 = 20.
Why? The modal class has the highest frequency; the median class is the class whose cumulative frequency is greater than and nearest to n/2.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/2/2: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Picking 0–10 as the median class because its frequency 22 is the largest, instead of using cumulative frequencies and n/2 = 33.
Subtracting the lower limit of the modal class (0) from the upper limit of the median class (30), giving 30, instead of the correct 30 − 10 = 20.

Why this works

For grouped data, the modal class is the class interval having the maximum frequency. The maximum frequency is 22, so the modal class is 0–10. For the median, find n/2 = 66/2 = 33. Add the frequencies in order to get the cumulative frequencies: 22, 22 + 10 = 32, 32 + 8 = 40, 40 + 15 = 55, 55 + 5 = 60, 60 + 6 = 66. The class 20–30 has cumulative frequency 40, which is greater than and nearest to 33, so it is the median class. The upper limits are 10 and 30, and 30 − 10 = 20.

Key idea to remember

Modal Class and Median Class: see Key ideas used on this page.

Explanation

For grouped data, the modal class is the class interval with the maximum frequency. The median class is the class interval whose cumulative frequency is greater than and nearest to n/2. This question asks only for the upper limits of these two classes, so the mode and median formulae are not needed.

Q21. If for a distribution, \(\sum_1^n f_i x_i = 132 + 5p\), \(\sum_1^n f_i = 20\) and the mean of the distribution is 8.1, then the value of p is :

Mark practised

(A) 3
(B) 6
(C) 4
(D) 5

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2024
Topic: 13.2 Mean of Grouped Data
Source: (CBSE 2024 Mathematics Basic, Set 430/3/2, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 6, because (132 + 5p)/20 = 8.1 gives 132 + 5p = 162, so 5p = 30 and p = 6.
Why? By the definition of the mean, mean = (Σ fᵢxᵢ)/(Σ fᵢ); substituting the given values and solving the linear equation gives p.

The full 1 mark is for the correct option.

Answer:

(B) 6

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/3/2: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing 132 + 5p = 8.1 without dividing by the total frequency 20.
Choosing 5 only because the term 5p contains 5, without solving the equation.
Subtracting 132 from 162 incorrectly and then dividing by 5.
Substituting p = 6 into 132 + 5p but forgetting to divide by 20, so the mean is not checked.

Why this works

The mean of a distribution is the sum of all the products of frequencies and values divided by the total frequency. Here mean = (Σ fᵢxᵢ)/(Σ fᵢ) = (132 + 5p)/20. Since the mean is given as 8.1, we get (132 + 5p)/20 = 8.1. Multiplying both sides by 20 gives 132 + 5p = 162. Subtracting 132 gives 5p = 30, so p = 30/5 = 6. Check: (132 + 5 × 6)/20 = 162/20 = 8.1, which matches the given mean.

Key idea to remember

Mean of grouped data: see Key ideas used on this page.

Explanation

The mean of a frequency distribution is the sum of the products of each value with its frequency, divided by the total frequency. The formula is x̄ = (Σ fᵢxᵢ)/(Σ fᵢ). In this question the values are not listed, but the sums are given, so we substitute them directly into the formula and solve for p.

Important formulas

x̄ = (Σ fᵢxᵢ)/(Σ fᵢ) · The mean of a frequency distribution, used to connect the given sums with the given mean.

Q22. The median group in the following frequency distribution is :

Mark practised

Class	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60
Frequency	5	8	20	15	7	5

(A) 10 – 20
(B) 20 – 30
(C) 30 – 40
(D) 40 – 50

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2024
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2024 Mathematics Basic, Set 430/4/1, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 20 – 30, because n/2 = 60/2 = 30, and the cumulative frequency 5 + 8 + 20 = 33 is the first cumulative frequency greater than 30.
Why? The median class is the class whose cumulative frequency is greater than and nearest to n/2; the cumulative frequencies identify that class.

The full 1 mark is for the correct option.

Answer:

(B) 20 – 30

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/1: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the cumulative frequency before the class, 13, instead of the cumulative frequency up to the class, 33, which would wrongly point to 30 – 40.
Looking for a class whose own frequency is 30; no class has frequency 30, so some students then pick a later class such as 40 – 50.

Why this works

With 60 observations, the middle is taken at n/2 = 30. Add the frequencies from the top to get the cumulative frequencies: 5, 13, 33, 48, 55, 60. The first cumulative frequency greater than 30 is 33, which belongs to the class 20 – 30. So the 30th observation lies in 20 – 30, and this is the median class. The class 10 – 20 covers only the first 13 observations, so it is below the middle; the class 30 – 40 begins after the 30th observation has already been counted. A quick check: 5 + 8 = 13 and 13 + 20 = 33, so 20 – 30 is the class that first crosses 30.

Key idea to remember

Median class: see Key ideas used on this page.

Explanation

For grouped data, the median is a value inside some class interval. To locate it, first find n/2. Then look at the cumulative frequency column: the class whose cumulative frequency is greater than and nearest to n/2 is the median class. Here n = 60, so n/2 = 30. The cumulative frequencies are 5, 13, 33, 48, 55, 60; 33 is the first cumulative frequency greater than 30, so the median class is 20 – 30.

Q23. If the mean and the mode of a distribution are 15 and 18 respectively, then the median of the distribution is :

Mark practised

(A) 17
(B) 15
(C) 16
(D) 18

Marks: 1
Level: Standard
Years: 2023
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2023 Mathematics Standard, Set 30/5/3, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 16, because 3 × Median = Mode + 2 × Mean = 18 + 2 × 15 = 48, so Median = 48 ÷ 3 = 16.
Why? By the empirical relationship between the three measures of central tendency, 3 Median = Mode + 2 Mean, so substituting the given values gives the median.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/5/3: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Thinking the median is the average of 15 and 18, i.e. 16.5.
Reversing the empirical relationship to 3 Mode = Median + 2 Mean, which makes the answer too large.

Q24. The marks obtained by 50 students in a test are tabulated below :

Mark practised

Marks :	0 – 15	15 – 30	30 – 45	45 – 60	60 – 75
Number of students :	3	9	20	15	3

The number of students who got less than 45 marks is :

(A) 12
(B) 47
(C) 32
(D) 18

Marks: 1
Level: Basic
Compartment: 2024
Topic: Mixed topics
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430(B), Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 32, because 3 + 9 + 20 = 32 students lie in the classes 0–15, 15–30 and 30–45, which are all below 45 marks.
Why? Students with less than 45 marks are counted by the cumulative frequency up to the class 30–45, which is the sum of the frequencies of the first three classes.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430(B): option (C).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 47 by adding 3 + 9 + 20 + 15, which counts students with less than 60 marks instead of less than 45.
Choosing 12 by adding only 3 + 9 and forgetting the 20 students in the 30–45 class.
Choosing 18 by adding 15 + 3, the students in the classes above 45, instead of the classes below 45.

Why this works

The three classes below 45 marks are 0–15, 15–30 and 30–45. Their frequencies are 3, 9 and 20. Add them: 3 + 9 + 20 = 32. So 32 students got less than 45 marks. Option D (18) adds only 15 + 3, the two classes above 45. Option B (47) wrongly adds 3 + 9 + 20 + 15, which includes the 45–60 class. Option A (12) adds only 3 + 9, leaving out the 20 students of the 30–45 class.

Key idea to remember

Cumulative frequency: see Key ideas used on this page.

Explanation

In a grouped frequency distribution, the cumulative frequency of a class is the total of the frequencies of that class and all classes before it. This question asks for the number of students with marks less than 45, which is the cumulative frequency of the class 30–45.

Important formulas

cf = f₁ + f₂ + … + fₖ · Here the three classes below 45 are 0–15, 15–30 and 30–45, so cf = 3 + 9 + 20 = 32.

Q25. If the difference of mode and median of a data is 24, then the difference of its median and mean is :

Mark practised

(A) 12
(B) 24
(C) 8
(D) 36

Marks: 1
Level: Standard
Years: 2024
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2024 Mathematics Standard, Set 30/4/1, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 12, because 3 Median = Mode + 2 Mean, so Median − Mean = (Mode − Median)/2 = 24/2 = 12.
Why? The empirical relationship connects all three measures, so the known difference of mode and median determines the difference of median and mean.

The full 1 mark is for the correct option.

Answer:

(A) 12

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/4/1: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 24, because you think Median − Mean is the same as Mode − Median and do not use 3 Median = Mode + 2 Mean.
Choosing 8, because you divide the given 24 by 3 and forget the factor 2 in the relationship.
Choosing 36, because you add the given 24 and the result 12 instead of halving the given difference.

Why this works

Let Median = M, Mode = Mo and Mean = x. The question gives Mode − Median = 24, so Mo = M + 24. The empirical relationship says 3 Median = Mode + 2 Mean, i.e. 3M = Mo + 2x. Putting Mo = M + 24 gives 3M = M + 24 + 2x, so 2M − 24 = 2x and x = M − 12. Hence Median − Mean = M − x = 12. Quick check: if M = 100, then Mo = 124 and x = 88; 3M = 300 and Mo + 2x = 124 + 176 = 300, so the numbers fit, and M − x = 12.

Key idea to remember

Empirical relationship between the three measures of central tendency: see Key ideas used on this page.

Explanation

For a distribution, the mean, median and mode are related by 3 Median = Mode + 2 Mean. This question uses that relationship to convert the given difference (Mode − Median) into the required difference (Median − Mean).

Important formulas

3 Median = Mode + 2 Mean · Empirical relationship between the three measures of central tendency.
Median − Mean = (Mode − Median)/2 · Rearranged form used in this question.

Q26. The formula : \(l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\), where symbols have their usual meanings, is used to calculate :

Mark practised

(A) Mean of a grouped data.
(B) Median of a grouped data.
(C) Mode of a grouped data.
(D) Range of a grouped data.

Marks: 1
Level: Basic
Compartment: 2025
Topic: 13.3 Mode of Grouped Data
Source: (CBSE 2025 Compartment Mathematics Basic, Set 430/S/2, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) Mode of a grouped data., because this formula gives the mode of grouped data.
Why? The mode of grouped data is found by this formula, where l, h, f1, f0 and f2 have their usual meanings.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/2: option (C).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing the median because both the mode and median formulas start with l + ...; the median formula uses (n/2 - cf)/f in its numerator.
Confusing f1, f0 and f2 with cumulative frequencies; they are the frequencies of the modal class and the classes before and after it.

Why this works

Here l is the lower limit of the modal class, h is the class size, f1 is the frequency of the modal class, f0 is the frequency of the class just before it, and f2 is the frequency of the class just after it. The mean of grouped data is found by a different formula, and the median also has a different formula, so this one belongs to the mode.

Key idea to remember

Mode of Grouped Data: see Key ideas used on this page.

Explanation

The mode is the value that occurs most often. For grouped data, the class with the highest frequency is called the modal class, and the mode lies inside that class. This formula finds that value using the lower limit of the modal class, the class size, and the frequencies of the modal class and its two neighbouring classes.

Important formulas

Mode = l + ((f1 - f0)/(2f1 - f0 - f2)) × h · l = lower limit of the modal class; h = class size; f1, f0, f2 = frequencies of the modal class, the preceding class and the succeeding class.

Q27. Mode and Mean of a data are 15x and 18x, respectively. Then the median of the data is :

Mark practised

(A) x
(B) 11x
(C) 17x
(D) 34x

Marks: 1
Level: Standard
Years: 2025
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/1/1, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 17x, because 3 Median = Mode + 2 Mean = 15x + 2(18x) = 51x, so Median = 51x/3 = 17x.
Why? The empirical relationship between the three measures of central tendency is 3 Median = Mode + 2 Mean; substituting the given values gives the median.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/1/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Thinking the median is the average of the mode and the mean, 16.5x, instead of using 3 Median = Mode + 2 Mean.
Swapping the coefficients, so that the mode is doubled and the mean is added once, and then dividing by 3 to get 16x.
Forgetting to divide 51x by 3 and reporting 51x as the median.

Q28. What is the mode of a data if median and mean of the same data are 9.6 and 10.5, respectively ?

Mark practised

(A) 7.8
(B) 12.3
(C) 8.4
(D) 7

Marks: 1
Level: Standard
Years: 2025
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/1/3, Q1, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 7.8, because 3 × Median = Mode + 2 × Mean, so Mode = 3 × 9.6 − 2 × 10.5 = 28.8 − 21 = 7.8.
Why? By the empirical relationship between the three measures of central tendency, 3 Median = Mode + 2 Mean; substituting the given median and mean isolates the mode.

The full 1 mark is for the correct option.

Answer:

(A) 7.8

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/1/3: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Swapping mean and median in the formula, so that mode is taken as 3 × mean − 2 × median, giving 12.3.
Taking mode as 2 × mean − 3 × median, giving −7.8.
Forgetting the factor 2 on the mean term, giving 3 × 9.6 − 10.5, which is 18.3.

Why this works

The mode is the value that occurs most often. The empirical relationship between the three measures of central tendency is 3 × Median = Mode + 2 × Mean. Rearrange it as Mode = 3 × Median − 2 × Mean. Substitute the given values: 3 × 9.6 = 28.8 and 2 × 10.5 = 21, so Mode = 28.8 − 21 = 7.8. To check, put this back into the relationship: 7.8 + 2 × 10.5 = 7.8 + 21 = 28.8, which equals 3 × 9.6. So the required mode is 7.8.

Key idea to remember

Empirical relationship between the three measures of central tendency: see Key ideas used on this page.

Explanation

For a given set of data, the mean, median and mode are connected by 3 Median = Mode + 2 Mean. When any two of the three are known, this formula can be solved for the third. Here median = 9.6 and mean = 10.5 are given, so the formula gives the mode directly.

Important formulas

3 Median = Mode + 2 Mean · Empirical relationship connecting the three measures of central tendency.
Mode = 3 Median − 2 Mean · Rearranged form used when median and mean are known.

Q29. The mean of seven observations is 17. If the mean of the first four observations is 15 and that of the last four observations is 18, then the fourth observation is :

Mark practised

(A) 14
(B) 13
(C) 12
(D) 10

Marks: 1
Level: Standard
Years: 2025
Topic: 13.2 Mean of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/3/3, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 13, because (sum of first 4) + (sum of last 4) − total sum = 4 × 15 + 4 × 18 − 7 × 17 = 13.
Why? The mean equals the sum divided by the count, so each group's sum is mean × count. Since the fourth observation lies in both the first four and the last four, adding the two partial sums counts it twice.

The full 1 mark is for the correct option.

Answer:

(B) 13

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/3/3: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the means directly, 15 + 18 − 17, instead of first finding the sums 60 and 72.
Taking 119 − 60 = 59 as the fourth observation, because only the first four group was considered.

Why this works

Let the observations be a, b, c, d, e, f, g. The total sum is 7 × 17 = 119. The first four observations add to 4 × 15 = 60, and the last four add to 4 × 18 = 72. Adding these two sums gives (a+b+c+d) + (d+e+f+g) = 60 + 72 = 132, which is the total sum plus the fourth observation d. So d = 132 − 119 = 13. Quick check: first three sum = 60 − 13 = 47, last three sum = 72 − 13 = 59, total = 47 + 13 + 59 = 119, and 119 ÷ 7 = 17.

Key idea to remember

Mean of ungrouped data: see Key ideas used on this page.

Explanation

The mean is the total of all observations divided by how many there are. Here, each partial mean gives the partial sum; the fourth observation is the overlap of the first four and the last four, so it is counted twice when the two partial sums are added.

Important formulas

Mean = (sum of values) / (number of observations) · So the sum of a group = mean × number of observations.

Q30. If mean and median of given set of observations are 10 and 11 respectively, then the value of mode is :

Mark practised

(A) 10.5
(B) 8
(C) 13
(D) 21

Marks: 1
Level: Standard
Years: 2025
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/4/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 13, because Mode = (3 × 11) − (2 × 10) = 33 − 20 = 13.
Why? The empirical relationship between the three measures of central tendency is 3 Median = Mode + 2 Mean, so solving for Mode gives Mode = (3 × Median) − (2 × Mean).

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/4/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 10.5 by averaging 10 and 11, because the mode is not the average of the mean and median.
Choosing 21 by adding 10 and 11, because the relationship subtracts twice the mean from three times the median.
Choosing 8 by using 3 Mean − 2 Median instead of 3 Median − 2 Mean.

Q31. If mode and median of given set of observations are 13 and 11 respectively, then the value of mean is :

Mark practised

(A) 17
(B) 7
(C) 10
(D) 28

Marks: 1
Level: Standard
Years: 2025
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2025 Mathematics Standard, Set 30/4/3, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 10, because by the empirical relationship, (3 × 11 − 13)/2 = 10.
Why? The empirical relationship connects the three measures of central tendency; rearranging it gives the mean as (3 × Median − Mode)/2.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/4/3: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Swapping the values: putting 11 for Mode and 13 for Median in the formula.
Writing the relationship as 3 Mean = Mode + 2 Median.
Stopping at 20 after subtracting 13 from 33, without dividing by 2.

Why this works

The question gives mode 13 and median 11. The empirical relationship connects the three measures of central tendency: three times the median equals the mode plus twice the mean. Rearranging gives the mean as (3 × median − mode)/2. Substituting, (3 × 11 − 13)/2 = 20/2 = 10. Therefore the mean is 10, so option (C) is correct.

Key idea to remember

Empirical relationship between the three measures of central tendency: see Key ideas used on this page.

Explanation

This relationship connects the mean, median and mode of a data set: three times the median equals the mode plus twice the mean. When two of the three measures are known, the third can be found by substituting and solving.

Important formulas

3 Median = Mode + 2 Mean · Empirical relationship between the three measures of central tendency.

Q32. The median and mode of a distribution are 25·2 and 26·1 respectively. The mean of the distribution is :

Mark practised

(A) 24·75
(B) 24·25
(C) 24·3
(D) 25·5

Marks: 1
Level: Standard
Years: 2026
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2026 Mathematics Standard, Set 30/3/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 24·75, because by the empirical relationship 3 Median = Mode + 2 Mean, Mean = (3 × 25.2 − 26.1)/2 = 24.75.
Why? The empirical relationship connects the three measures of central tendency, so substituting the two given values and solving for Mean gives the required mean.

The full 1 mark is for the correct option.

Answer:

(A) 24·75

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/3/1: option (A).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using 2 Median − Mode, which gives 2 × 25.2 − 26.1 = 24.3 (option C), instead of the empirical relationship.
Swapping the values, i.e., using (3 × Mode − Median)/2, which gives 26.55, not any of the given options.

Q33. The median and mode of a distribution are 25.2 and 26.1 respectively. The mean of the distribution is :

Mark practised

(A) 24.75
(B) 24.25
(C) 24.3
(D) 25.5

Marks: 1
Level: Standard
Years: 2026
Topic: 13.4 Median of Grouped Data
Source: (CBSE 2026 Mathematics Standard, Set 30/3/2, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 24.75, because 3 × 25.2 = 75.6, so 2 Mean = 75.6 − 26.1 = 49.5 and Mean = 24.75.
Why? The empirical relationship 3 Median = Mode + 2 Mean connects the three measures of central tendency; substituting the two given values leaves the mean as the only unknown.

The full 1 mark is for the correct option.

Answer:

(A) 24.75

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/3/2: option (A).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing 3 Mean = Mode + 2 Median; this gives (26.1 + 2 × 25.2)/3 = 25.5.
Forgetting to divide by 2 after subtracting, so taking the mean as 49.5 instead of 24.75.
Putting the mode in place of the median; swapping 26.1 and 25.2 in the formula gives 26.55.

1 / 33

Key ideas used on this page

Empirical relationship between the three measures of central tendency
Used in questions 5 6 10 17 18 23 25 28 30 31 33
Modal class
Used in questions 1 2 4 7
Empirical relationship between mean, median and mode
Used in questions 3 27 32
Mean of grouped data
Used in questions 8 21
Median
Used in questions 11 12
Mode
Used in questions 13 16
Cumulative frequency
Used in questions 14 24
Mean of observations
Used in question 9
Median of Ungrouped Data
Used in question 15
Class mark
Used in question 19
Modal Class and Median Class
Used in question 20
Median class
Used in question 22
Mode of Grouped Data
Used in question 26
Mean of ungrouped data
Used in question 29

How to use these questions

Try each question on paper first. Then read the steps: the mark beside each step shows what the examiner looks for. If a step is new to you, revise Section 13.2, Section 13.3 and Section 13.4 of the NCERT chapter.

To revise the lessons behind these questions, read the NCERT chapter, exactly as NCERT publishes it on its own website.

Official papers behind this page

Every question on this page comes from an official CBSE Class 10 Mathematics board paper. The board publishes the question papers on cbse.gov.in and the marking schemes on cbse.gov.in, where you can see how marks are given.

Year	Exam	Course	Question paper	Marking scheme
2026	Main	Standard (041)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 22.8 MB, Class 10 Maths Standard (041)) to see how marks are given.
2025	Main	Standard (041)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 54.5 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Main	Basic (241)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 49.9 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Compartment	Basic (241)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 2.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 18.7 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Main	Standard (041)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 35.3 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2024	Main	Basic (241)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 43.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.8 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Compartment	Basic (241)	Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 2.2 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 5.6 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Main	Standard (041)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 44 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 17.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2023	Compartment	Standard (041)	Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.6 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 3.3 MB, Class 10 Maths Standard (041)) to see how marks are given.

Sources and session

Each question is reproduced as printed in its CBSE paper, with its year, paper code and question number; only the layout is tidied. The same question printed in several paper sets counts once, and the card says how often it was printed in 2023–2026. These counts describe past papers; they are not predictions.

Every solution is original and follows NCERT Mathematics Class X (reprint 2026-27). The mark beside each step follows the official CBSE marking scheme where the scheme shows it; a mark labelled “(suggested)” follows the board’s usual pattern for that type of question.

Reference: CBSE Class 10 Mathematics board question papers via cbse.gov.in; all solutions are original. learncbse.net is an independent website and is not affiliated with CBSE or NCERT.

1-mark questions on Statistics

All 33 Statistics 1-mark questions

Q1. The following distribution shows the number of runs scored by some batsmen in test matches :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q2. The following table shows the marks scored by 23 students of a class.

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q3. For a distribution, if mean = 15 and mode = 12, then its median is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q4. In the formula of mode given by mode = \(l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\), \(f_1\) denotes the :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q5. For a distribution, if mean = median = a, then its mode is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q6. Using empirical relationship, the mode of a distribution whose mean is 7.2 and the median 7.1, is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q7. The distribution below gives the marks obtained by 80 students on a test :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q8. If the value of each observation of a statistical data is increased by 3, then the mean of the data

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q9. If every term of the statistical data consisting of \(n\) terms is decreased by 2, then the mean of the data :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q10. If the mean and median of a data are 10 and 11 respectively, then mode of the data is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q11. The middle most observation of every data arranged in order is called :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q12. If value of each observation in a data is increased by 2, then median of the new data

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q13. After an examination, a teacher wants to know the marks obtained by Maximum number of the students in her class. She requires to calculate _____ of marks.

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q14. The cumulative frequency for calculating median is obtained by adding the frequencies of all the :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q15. Following data shows the marks obtained by 100 students in a class test : 1

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q16. If the maximum number of students has obtained 52 marks out of 80, then

Step-by-step solution

Step 1: Choose the option