Statistics — JEE MCQs with answers

Answer: D — D is correct: p.260 writes the mean deviation as the sum of the absolute values of the deviations from $a$ divided by the number of observations, and §13.4.1 lets the $n$ observations be $x_1, x_2, \dots, x_n$. A is wrong: repeated observations each have their own deviation, so every observation is counted. B is wrong: the chapter always divides by the number of observations, as Example 1 divides by 8 for its 8 observations. C is wrong: that position, such as the 6th observation in Example 3, only locates the median; Example 3 divides by all 11 observations.

Answer: C — Here $n = 7$ and $\bar{x} = \frac{280}{7} = 40$. The deviations $x_i - \bar{x}$ are $-9, -6, -3, 1, 3, 5, 9$, so $\sum_{i=1}^{7}(x_i - \bar{x})^2 = 242$. Hence $\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 = \frac{242}{7} = 34.57$. So option C is correct. Option A ($35.57$) squares the deviations from the median $M = 41$, giving $\frac{249}{7}$, instead of the deviations from the mean $\bar{x} = 40$. Option B ($6$) is $\frac{36}{6}$: it adds the absolute deviations $|x_i - \bar{x}|$, as for the mean deviation, and divides by $n - 1 = 6$; the variance uses squared deviations and divides by $n = 7$. Option D ($41.5$) squares the deviations from the median $M = 41$ instead of from the mean $\bar{x} = 40$, and divides their sum, $249$, by $n - 1 = 6$ instead of by $n = 7$.

Answer: A — A is correct: every observation counts, including repeats. In order the data are 3, 3, 3, 5, 8, 9, 10; $n=7$ is odd, so the median is the 4th observation, 5. B is wrong: This lists each distinct value only once (3, 5, 8, 9, 10) and takes the middle one; repeated observations must each be counted, which gives $8$. C is wrong: This takes the middle entry of the list as written, without first arranging the data in order, which gives $3$. D is wrong: This stops at the position $\frac{n+1}{2} = 4$ instead of taking the observation in that position, which gives $4$.

Answer: A — A is correct: p.259 says the chapter will study all of the listed measures of dispersion except the quartile deviation. B is wrong: Mean deviation is studied in §13.4. C is wrong: Range is studied in §13.3. D is wrong: Standard deviation is studied in §13.5.

Answer: C — C is correct: without 42 the maximum is 37 and the minimum is still 8, so the range is $37-8=29$. A is wrong: This removes the smallest value instead, giving $42-16$, which gives $26$. B is wrong: This also drops the minimum, giving $37-16$, which gives $21$. D is wrong: This ignores the removal and keeps 42 as the maximum, $42-8$, which gives $34$.

Answer: A — Let $y_i=x_i-12$. By Example 15 with $a=-12$, $\bar{y}=\bar{x}-12$, so $\bar{x}=6.4+12=18.4$; the variance is unaffected by the shift, so it was 3.1. So option A is correct. Option B (Mean $-5.6$, variance $15.1$) subtracts 12 from the mean a second time and also adds 12 to the variance. Option C (Mean $-5.6$, variance $3.1$) subtracts 12 a second time instead of adding it back. Option D (Mean $18.4$, variance $15.1$) adds 12 back to the variance as well.

Answer: A — A is correct. By definition $\text{M.D.}(a) = \frac{1}{n}\sum_{i=1}^{n} |x_i - a|$ with $a = 10$ and $n = 5$. The absolute deviations $|x_i - 10|$ are 6, 1, 3, 8, 12, which add up to 30, so $\text{M.D.}(10) = \frac{30}{5} = 6$. B is wrong: 5.4 is the mean deviation about the median $\text{M} = 13$; the question fixes the central value at $a = 10$. C is wrong: 5 divides the total 30 by $n + 1 = 6$ instead of $n = 5$. D is wrong: 7.5 divides the total 30 by $n - 1 = 4$ instead of $n = 5$.

Answer: B — B is correct: in order the data are 6, 9, 12, 15, 21, 27, 33; $n=7$ is odd, so the median is the $\left(\frac{7+1}{2}\right)^{\text{th}}=4$th observation, 15. A is wrong: This takes the middle entry of the list as written, without first arranging the data in order, which gives $12$. C is wrong: This stops at the position $\frac{n+1}{2} = 4$ instead of taking the observation in that position, which gives $4$. D is wrong: This takes the 5th observation in order instead of the 4th, which gives $21$.

Answer: A — A is correct: p.259 takes the difference of the maximum and minimum values of each series and calls it the range: $\text{Range of a series} = \text{Maximum value} - \text{Minimum value}$. B is wrong: The listed order is arbitrary; the range uses the maximum and minimum values wherever they occur. C is wrong: The range is a difference, not a sum. D is wrong: The range does not involve the mean; it uses the maximum and the minimum.

Answer: A — A is correct: p.259 opens with the dispersion or scatter in a data. B is wrong: Average is not used in the opening sentence of §13.2. C is wrong: Frequency is not used in the opening sentence of §13.2. D is wrong: Tendency appears only in the phrase measure of central tendency.