variance-and-standard-deviation MCQs for UPSC Prelims

35 practice questions on variance-and-standard-deviation from the Statistics section of the UPSC Prelims syllabus. 35 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.

10 Easy 18 Medium 7 Hard

Sample questions

Q1
medium

For a set of eight observations, $\sum x_i = 40$ and $\sum x_i^2 = 272$. Find the standard deviation.

  1. A $5.83$
  2. B $9$
  3. C $34$
  4. D $3$
Show answer and explanation

Correct answer: D - $3$

The mean is $\frac{40}{8} = 5$, so the variance is $\frac{272}{8} - 5^2 = 34 - 25 = 9$ and the standard deviation is $3$. The value $9$ is the variance itself. The value $34$ is $\frac{\sum x_i^2}{n}$ with the $\bar{x}^2$ term forgotten, and $5.83$ is $\sqrt{34}$, the same slip carried into the square root.

Q2
easy

Find the standard deviation of the five observations $9, 9, 9, 9, 9$.

  1. A $3$
  2. B $9$
  3. C $45$
  4. D $0$
Show answer and explanation

Correct answer: D - $0$

Every observation equals the mean $9$, so each deviation is zero and the variance is zero, giving a standard deviation of $0$. The value $9$ is the common observation and also the mean, not a measure of spread. The value $3$ wrongly takes the square root of the mean, and $45$ is the total of the observations.

Q3
medium

Find the variance of the eight observations $3, 5, 5, 5, 6, 6, 8, 10$.

  1. A $32$
  2. B $2$
  3. C $4$
  4. D $6$
Show answer and explanation

Correct answer: C - $4$

The mean is $\frac{48}{8} = 6$. The squared deviations are $9, 1, 1, 1, 0, 0, 4, 16$, adding to $32$, so the variance is $\frac{32}{8} = 4$. The value $2$ is the standard deviation, $6$ is the mean, and $32$ is the sum of squared deviations before dividing.

Q4
easy

Every observation $x$ of a data set is replaced by $d = x - 30$. How is the variance of the $d$ values related to the variance of the $x$ values?

  1. A The variance of $d$ is zero
  2. B The variance of $d$ is $30$ times the variance of $x$
  3. C The two variances are equal
  4. D The variance of $d$ is $30$ less than the variance of $x$
Show answer and explanation

Correct answer: C - The two variances are equal

Subtracting $30$ from every observation also lowers the mean by $30$, so each deviation from the mean is unchanged and the variance is unchanged. Nothing is subtracted from the variance itself, so the second option confuses a shift of the data with a shift of the spread. The third option applies multiplication where only subtraction occurred, and the variance of $d$ is zero only if the data has no spread at all.

Q5
easy

The heights of a group of students are measured in centimetres. In which unit is the standard deviation of these heights expressed?

  1. A $\text{cm}$
  2. B $\text{cm}^2$
  3. C It is a pure number with no unit
  4. D $\sqrt{\text{cm}}$
Show answer and explanation

Correct answer: A - $\text{cm}$

Squaring the deviations gives $\text{cm}^2$, and taking the square root at the end brings the measure back to $\text{cm}$, the unit of the data. That is exactly why the standard deviation is preferred to the variance for reporting spread. The variance, not the standard deviation, carries $\text{cm}^2$. A ratio such as the coefficient of variation is unit free, and $\sqrt{\text{cm}}$ never arises.

Q6
hard

For nine observations the mean is $12$ and the sum of the squares of the deviations from the mean is $108$. A tenth observation, also equal to $12$, is now included. Find the variance of the ten observations.

  1. A $10.8$
  2. B $12$
  3. C $13.5$
  4. D $108$
Show answer and explanation

Correct answer: A - $10.8$

The new observation equals the old mean, so the mean of the ten observations stays $12$ and the extra squared deviation is zero, leaving the total at $108$. The variance is therefore $\frac{108}{10} = 10.8$. The value $12$ is the old variance $\frac{108}{9}$, which ignores the change in the count, $13.5$ divides by $8$, and $108$ is the total before dividing.

Q7
hard

The mean of five observations is $6$ and their variance is $8$. Three of the observations are $2$, $4$ and $6$. Find the remaining two observations.

  1. A $5$ and $13$
  2. B $9$ and $9$
  3. C $7$ and $11$
  4. D $8$ and $10$
Show answer and explanation

Correct answer: D - $8$ and $10$

The five values total $30$, so the missing pair totals $18$, which every option satisfies. From $\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2$, $\sum x_i^2 = 5(8 + 36) = 220$, and the known values contribute $56$, so the pair must satisfy $a^2 + b^2 = 164$. Only $8$ and $10$ give $64 + 100 = 164$; the other pairs give $170$, $194$ and $162$.

Q8
medium

For a set of ten observations, $\sum x_i = 60$ and $\sum x_i^2 = 460$. Find the variance of the ten observations.

  1. A $10$
  2. B $40$
  3. C $46$
  4. D $3.16$
Show answer and explanation

Correct answer: A - $10$

The mean is $\frac{60}{10} = 6$, and the variance is $\frac{\sum x_i^2}{n} - \bar{x}^2 = 46 - 36 = 10$. The value $40$ comes from subtracting the mean instead of its square, $46$ forgets to subtract anything, and $3.16$ is the standard deviation $\sqrt{10}$.

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