Chapter 13 Statistics - Assertion Reason Class X

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✦ NCF 2023 ALIGNED • CBSE CLASS X ✦

Chapter 13: Statistics

Assertion-Reason Question Bank

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Standard CBSE Options Framework

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Q1Assertion-Reason

Assertion (A)

If \(N = 50\), the median class corresponds to cumulative frequency equal to \(50\).

Reason (R)

Median class corresponds to cumulative frequency just greater than or equal to \(N/2 = 25\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false (it corresponds to \(N/2 = 25\), not \(N = 50\)). Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q2Assertion-Reason

Assertion (A)

If class limits are \(10-20, 20-30, 30-40\), class size \(h = 10\).

Reason (R)

Class size \(h = \text{Upper class limit} - \text{Lower class limit} = 20 - 10 = 10\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true class size formulas.

Correct Answer: (B)

Both true.

Q3Assertion-Reason

Assertion (A)

Mode of data \(2, 3, 4, 3, 5, 3, 6\) is \(3\).

Reason (R)

Value \(3\) has the maximum frequency of \(3\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true mode definitions.

Correct Answer: (B)

Both true.

Q4Assertion-Reason

Assertion (A)

Step-deviation method simplifies mean calculation when class size \(h\) is uniform.

Reason (R)

\(u_i = \frac{x_i - a}{h}\), where \(a\) is assumed mean and \(h\) is class width.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true step-deviation method facts.

Correct Answer: (B)

Both true.

Q5Assertion-Reason

Assertion (A)

Mean of grouped data can be found by direct method \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\).

Reason (R)

In direct method, \(x_i\) represents upper class limit of \(i^{\text{th}}\) class interval.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (C)

Assertion (A) is true. Reason (R) is false because \(x_i\) represents class mark (mid-point), not upper limit.

Correct Answer: (C)

Assertion true; Reason false.

Q6Assertion-Reason

Assertion (A)

The mode of a frequency distribution can be determined graphically from a histogram.

Reason (R)

The highest rectangle in a histogram represents the modal class, and its vertex intersection gives the mode.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (A)

Mode is found graphically using the histogram modal class peak. Reason explains Assertion.

Correct Answer: (A)

Reason explains Assertion.

Q7Assertion-Reason

Assertion (A)

Mean can be determined graphically from an ogive.

Reason (R)

Mean cannot be determined graphically; only Median (from ogives) and Mode (from histogram) can be determined graphically.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false (mean cannot be found graphically). Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q8Assertion-Reason

Assertion (A)

The algebraic sum of deviations of a set of observations from their mean is always \(0\).

Reason (R)

Sum of deviations \(\sum (x_i - \bar{x}) = \sum x_i - n \bar{x} = n\bar{x} - n\bar{x} = 0\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (A)

Proof that sum of deviations from mean is always \(0\). Reason explains Assertion.

Correct Answer: (A)

Reason explains Assertion.

Q9Assertion-Reason

Assertion (A)

Mean of a grouped data is always equal to one of the class marks.

Reason (R)

Mean is a calculated average and need not equal any of the individual class marks.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false. Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q10Assertion-Reason

Assertion (A)

The median of a frequency distribution can be determined graphically from cumulative frequency curves (ogives).

Reason (R)

The x-coordinate of the point of intersection of 'less than' and 'more than' ogives corresponds to \(N/2\), which is the median.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (A)

Intersection of less-than and more-than ogives gives median on x-axis. Reason explains Assertion.

Correct Answer: (A)

Reason explains Assertion.

Q11Assertion-Reason

Assertion (A)

The modal class is the class interval having maximum frequency.

Reason (R)

The median class is the class interval having minimum cumulative frequency.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (C)

Assertion (A) is true. Reason (R) is false because median class corresponds to cumulative frequency just greater than \(N/2\).

Correct Answer: (C)

Assertion true; Reason false.

Q12Assertion-Reason

Assertion (A)

Median of first \(6\) prime numbers \(2, 3, 5, 7, 11, 13\) is \(7\).

Reason (R)

Since \(n = 6\) (even), median is average of \(3^{\text{rd}}\) and \(4^{\text{th}}\) terms \(= (5 + 7)/2 = 6\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false (median is \(6\), not \(7\)). Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q13Assertion-Reason

Assertion (A)

If median \(= 20\) and mean \(= 22.5\), then mode \(= 15\).

Reason (R)

Empirical formula is \(\text{Mode} = 3 \text{ Median} - 2 \text{ Mean} = 3(20) - 2(22.5) = 60 - 45 = 15\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true calculations.

Correct Answer: (B)

Both true.

Q14Assertion-Reason

Assertion (A)

Class size of interval \(15 - 25\) is \(10\).

Reason (R)

Class mark of interval \(15 - 25\) is \(10\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (C)

Assertion (A) is true (\(25 - 15 = 10\)). Reason (R) is false because class mark is mid-point \((15+25)/2 = 20\), not \(10\).

Correct Answer: (C)

Assertion true; Reason false.

Q15Assertion-Reason

Assertion (A)

Median of data \(5, 3, 1, 4, 2\) is \(3\).

Reason (R)

Median is always the middle term of unsorted raw data.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (C)

Assertion (A) is true (sorted: \(1, 2, 3, 4, 5\), middle term is \(3\)). Reason (R) is false because data MUST be arranged in ascending/descending order first.

Correct Answer: (C)

Assertion true; Reason false.

Q16Assertion-Reason

Assertion (A)

Empirical relation between central tendencies is \(\text{Mean} = 3\text{Median} - 2\text{Mode}\).

Reason (R)

The correct empirical formula is \(\text{Mode} = 3\text{Median} - 2\text{Mean}\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false. Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q17Assertion-Reason

Assertion (A)

Mean of \(5\) numbers is \(18\). If one number is excluded, mean becomes \(16\). The excluded number is \(26\).

Reason (R)

Excluded number = New sum - Old sum.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (C)

Assertion (A) is true (Old sum \(= 5 \times 18 = 90\), New sum \(= 4 \times 16 = 64\), Excluded \(= 90 - 64 = 26\)). Reason (R) is false because Excluded number = Old sum - New sum.

Correct Answer: (C)

Assertion true; Reason false.

Q18Assertion-Reason

Assertion (A)

In formula \(\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}\), \(d_i = x_i + a\).

Reason (R)

The correct deviation formula is \(d_i = x_i - a\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false (\(d_i = x_i - a\), not \(+\)). Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q19Assertion-Reason

Assertion (A)

Mean of \(10\) observations is \(15\). If each observation is multiplied by \(2\), new mean is \(30\).

Reason (R)

If each observation is multiplied by constant \(k\), the new mean becomes \(k\) times the original mean.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true scaling properties of mean.

Correct Answer: (B)

Both true.

Q20Assertion-Reason

Assertion (A)

The mean of first \(n\) natural numbers is \(\frac{n + 1}{2}\).

Reason (R)

Sum of first \(n\) natural numbers is \(\frac{n(n+1)}{2}\), so \(\text{Mean} = \frac{S_n}{n} = \frac{n(n+1)}{2n} = \frac{n+1}{2}\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (A)

\(S_n/n = [n(n+1)/2] / n = (n+1)/2\). Reason explains Assertion.

Correct Answer: (A)

Reason explains Assertion.

Q21Assertion-Reason

Assertion (A)

The mode of a frequency distribution is always unique.

Reason (R)

A distribution can be bimodal or multimodal if two or more classes have the same maximum frequency.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false (can have multiple modes). Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q22Assertion-Reason

Assertion (A)

If mode \(= 12\) and mean \(= 15\), then median \(= 14\).

Reason (R)

Formula is \(3 \text{ Median} = \text{Mode} + \text{Mean}\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (C)

Assertion (A) is true (\(3 \text{Median} = 12 + 2(15) = 42 \implies \text{Median} = 14\)). Reason (R) is false because formula is \(3\text{Median} = \text{Mode} + 2\text{Mean}\), missing factor \(2\) on Mean.

Correct Answer: (C)

Assertion true; Reason false.

Q23Assertion-Reason

Assertion (A)

Assumed mean \(a\) is chosen from class marks \(x_i\), usually the middle value.

Reason (R)

Deviation \(d_i = a - x_i\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (C)

Assertion (A) is true. Reason (R) is false because deviation is \(d_i = x_i - a\), not \(a - x_i\).

Correct Answer: (C)

Assertion true; Reason false.

Q24Assertion-Reason

Assertion (A)

In step-deviation method, \(u_i = h(x_i - a)\).

Reason (R)

The correct step-deviation formula is \(u_i = \frac{x_i - a}{h}\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (D)

Assertion (A) is false (\(u_i = (x_i-a)/h\), divided by \(h\)). Reason (R) is true.

Correct Answer: (D)

Assertion false; Reason true.

Q25Assertion-Reason

Assertion (A)

If mean of observations \(x, x+2, x+4, x+6, x+8\) is \(9\), then the mean of the first three observations is \(7\).

Reason (R)

Sum \(= 5x + 20 = 45 \implies x = 5\). First three observations are \(5, 7, 9\), so their mean is \((5+7+9)/3 = 21/3 = 7\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (A)

Sum \(= 5x + 20 = 45 \implies x = 5\). First three: \(5, 7, 9 \implies \text{Mean} = 21/3 = 7\). Reason explains Assertion.

Correct Answer: (A)

Reason explains Assertion.

Q26Assertion-Reason

Assertion (A)

The empirical relationship between the three measures of central tendency is \(3 \text{ Median} = \text{Mode} + 2 \text{ Mean}\).

Reason (R)

Empirical formula connects Mode, Median, and Mean as \(\text{Mode} = 3 \text{ Median} - 2 \text{ Mean}\), which rearranges to \(3 \text{ Median} = \text{Mode} + 2 \text{ Mean}\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (A)

Empirical formula \(\text{Mode} = 3 \text{Median} - 2 \text{Mean}\) rearranges directly to \(3 \text{Median} = \text{Mode} + 2 \text{Mean}\). Reason explains Assertion.

Correct Answer: (A)

Reason explains Assertion.

Q27Assertion-Reason

Assertion (A)

Cumulative frequency table is useful in determining the median.

Reason (R)

The class corresponding to cumulative frequency just greater than \(N/2\) is called median class.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true median class facts.

Correct Answer: (B)

Both true.

Q28Assertion-Reason

Assertion (A)

Mean is a measure of central tendency.

Reason (R)

Mode is the observation that occurs most frequently in data.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true definitions.

Correct Answer: (B)

Both true.

Q29Assertion-Reason

Assertion (A)

Construction of a cumulative frequency table is required to find median.

Reason (R)

Median divides the frequency distribution into two equal parts.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (B)

Both A and R are true median facts.

Correct Answer: (B)

Both true.

Q30Assertion-Reason

Assertion (A)

Class mark \(x_i\) of a class interval is given by \(x_i = \frac{\text{Upper class limit} + \text{Lower class limit}}{2}\).

Reason (R)

Class mark represents the mid-value of a class interval used in calculating mean of grouped data.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Correct Option: (A)

Class mark definition and formula. Reason explains Assertion.

Correct Answer: (A)

Reason explains Assertion.

Live Practice – Statistics AR

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