Statistics – Assertion and Reason Questions

Q 1 :

Assertion (A): Median of first eleven prime numbers is 13.

Reason (R): Median of a grouped frequency distribution is given by $M e d i a n = l + (\frac{\frac{n}{2} - c f}{f}) \times h, w h e r e (l) =$ ower limit of the median class, cf = cumulative frequency of the class preceding the median class, f = frequency of the median class, h = class width and n= total frequency.

Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A)
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.

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(2)

First eleven prime numbers are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31$

$H e r e, (n = 11) (o d d)$

$∴ Median = {(\frac{n + 1}{2})}^{th} observation = {(\frac{11 + 1}{2})}^{th} observation$

$= 6^{th} observation = 13 A s s e r t i o n i s t r u e .$

$M e d i a n o f a g r o u p e d f r e q u e n c y d i s t r i b u t i o n i s g i v e n b y$

$Median = l + (\frac{\frac{n}{2} - c f}{f}) \times h$

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

Q 2 :

Assertion (A): Median of first 9 terms of Fibonacci sequence is 3.

Reason (R): Median of a grouped frequency distribution is given by $Median = l + (\frac{\frac{n}{2} - c f}{f}) \times h, w h e r e l, n, f, c f$ are lower limit of the median class, total frequency, frequency of the median class and cumulative frequency of the class preceding the median class, respectively.

Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.

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(3)

$F i b o n a c c i s e q u e n c e : 0, 1, 1, 2, 3, 5, 8, 13, 21 H e r e, n = 9, w h i c h i s o d d . ∴ Median = {(\frac{n + 1}{2})}^{th} observation = 5^{th} observation = 3 ∴ Assertion (A) is true. M e d i a n o f g r o u p e d f r e q u e n c y d i s t r i b u t i o n i s g i v e n b y Median = l + (\frac{\frac{n}{2} - c f}{f}) \times h ∴ Reason (R) is false. T h u s, A s s e r t i o n (A) i s t r u e a n d R e a s o n (R) i s f a l s e .$

Q 3 :

Assertion (A): If the median and mode of a frequency distribution are 50 and 60, respectively, then its mean is 45.

Reason (R): Mean, median and mode of a frequency distribution are related as:

Mode = 3 (Median) − 2 (Mean)

Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A)
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.

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(1)

Given, median and mode of a frequency distribution are 50 and 60, respectively.

The empirical relationship between mean, mode and median is:
Mode = 3 × Median − 2 × Mean

∴ 60 = 3 × 50 − 2 (Mean)

⇒ Mean = (150 − 60) / 2 = 90 / 2 = 45

Therefore, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Q 4 :

Assertion (A): If the values of mode and mean are 60 and 66, respectively, then the value of median is 64.

Reason (R): $M e d i a n = 1 / 2 (M o d e + 2 \times M e a n)$

Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A)
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.

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(3)

Using the given formula:

$M e d i a n = 1 / 2 (60 + 2 \times 66) = \frac{1}{2} (60 + 132) = \frac{192}{2} = 96$

So, the median is 96, not 64.

Q 5 :

Assertion (A) : If the values of mode and mean are 60 and 66, respectively, then the value of median is 64.

Reason (R) : $M e d i a n = 1 / 2 (M o d e + 2 \times M e a n)$

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

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(3)

$B y u \sin g f o r m u l a, m e d i a n = 1 / 3 (M o d e + 2 \times M e a n) = 1 / 3 (60 + 2 \times 66) = 1 / 3 (60 + 132) = 64 S o, A s s e r t i o n (A) i s t r u e . B u t t h e f o r m u l a g i v e n i n R e a s o n (R) i s f a l s e .$

Topic Question Set

Q 3 :

Assertion (A): If the median and mode of a frequency distribution are 50 and 60, respectively, then its mean is 45.

Reason (R): Mean, median and mode of a frequency distribution are related as:

Mode = 3 (Median) − 2 (Mean)

Q 4 :

Assertion (A): If the values of mode and mean are 60 and 66, respectively, then the value of median is 64.

Reason (R): $M e d i a n = 1 / 2 (M o d e + 2 \times M e a n)$

Q 5 :

Assertion (A) : If the values of mode and mean are 60 and 66, respectively, then the value of median is 64.

Reason (R) : $M e d i a n = 1 / 2 (M o d e + 2 \times M e a n)$

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