Q.The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.

The key concept here is how adding or subtracting a constant affects measures of spread. Variance and standard deviation measure how spread out the data are around the mean. If you shift every data point by the same amount (like adding 7, then subtracting 5 — which is just adding 2 overall), the distances between points stay exactly the same. The mean shifts, but the spread doesn’t. So the variance and standard deviation are unchanged.

Let’s walk through it step by step.

1. Understand the transformation

   The problem says: first add 7 to each observation, then subtract 5 from each result.

   That’s equivalent to adding $7 - 5 = 2$ to each original observation.

   So the new observation $y_i = x_i + 2$, where $x_i$ are the original values.

2. Recall the property of variance under addition of a constant

   If every value in a dataset is increased or decreased by the same constant $c$, the variance does not change.

   Mathematically:

This is because variance depends on deviations from the mean, and both the data and the mean shift by the same amount, so deviations are unchanged.

3. Apply to the given data

   Original variance = 16.

   After adding 2 to every observation, the variance remains 16.

4. Convert variance to standard deviation

   Standard deviation is the square root of variance:

The key idea here is that standard deviation measures spread, not location. If you add the same number to every value in a dataset, the entire distribution shifts sideways — the distances between points stay exactly the same, so the standard deviation is unchanged.

Let’s see why this works step by step.

1. Recall the definition of standard deviation For a set of numbers $x_1, x_2, \dots, x_n$ with mean $\bar{x}$, the standard deviation is

It depends only on the deviations $x_i - \bar{x}$ — how far each point is from the mean.

2. What happens when we add a constant $c$ to every value? Let $y_i = x_i + c$. Then the new mean is

The deviation of any $y_i$ from its mean is

The deviations are identical! So the sum of squared deviations is the same, and therefore the standard deviation is unchanged.

3. Apply to our problem The first set is $0,1,2,\dots,9$. The second set is $10,11,12,\dots,19$. Notice that each number in the second set is exactly $10$ more than the corresponding number in the first set:

So the second set is just the first set shifted by $+10$.

4. Conclusion Since adding a constant does not change the standard deviation, the standard deviation of the second set is exactly the same as that of the first set, namely $k$.

Concept & Intuition

Variance measures spread — how far data points are from the mean. If you stretch every data point by a factor $k$ (multiply by $k$), the distances from the mean also stretch by $k$. Since variance is an average of squared distances, it scales by $k^2$. This is why multiplying by 3 makes the variance 9 times larger, not 3 times.

Step-by-step reasoning

1. Recall the definition of variance For $n$ observations $x_1, x_2, \dots, x_n$ with mean $\bar{x}$, the variance is

2. Apply the transformation Each observation is multiplied by 3: new observations are $y_i = 3x_i$. The new mean is

3. Compute the new variance

Factor out the constant 9:

4. Plug in the given variance Original variance $\sigma^2 = 8$, so

Concept: effect of a linear transformation on variance. If y = a x + b, then Var(y) = a^2 Var(x). Adding a constant does not change the spread; multiplying by -1 does not either (a^2 = 1).

Original data: 1, 2, ..., 10. Its variance is (n^2 - 1)/12 = (100 - 1)/12 = 99/12 = 8.25.

(Check directly: mean = 5.5; sum of squares = 385; variance = 385/10 - 5.5^2 = 38.5 - 30.25 = 8.25.)

New data: y = -x + 1, i.e. a = -1, b = 1.

Var(y) = (-1)^2 * 8.25 = 8.25.

ANSWER: B

Step 1 — Count the observations.

The data are the consecutive integers from $31$ to $47$:

Step 2 — Use the shift-invariance of variance.

A key property: adding a constant to every observation leaves the variance (and hence the SD) unchanged, because variance measures spread about the mean and both the data and the mean shift by the same amount.

So subtract $30$ from every number. The data $31, 32, \ldots, 47$ become $1, 2, \ldots, 17$ — the first 17 natural numbers — with exactly the same standard deviation.

Step 3 — Variance of the first $n$ natural numbers.

For $1, 2, \ldots, n$:

Step 4 — Substitute $n = 17$.

Step 5 — Why the distractors fail.

Option (B), $\sqrt{\dfrac{47^2-1}{12}}$, is the trap: it applies the formula with $n = 47$, but the data are not $1$ through $47$ — only 17 of those integers are present. The correct $n$ is the count of observations, $17$.

Numerically $2\sqrt6 = 4.899$, which is a sensible spread for 17 integers spanning a range of 16. ✓

ANSWER: C

Step 1 — Check the distribution is valid.

Step 2 — Mean.

Step 3 — Second moment.

Step 4 — Variance (why this formula).

Variance is the mean squared deviation, $\sigma^{2}=E\big[(X-\mu)^2\big]$; expanding gives the computational form

Step 5 — Standard deviation.

Step 6 — Match the options.

$\sqrt{1/3}\approx0.577$ (no); $\sqrt{5/36}\approx0.373$ (no); $\frac13\sqrt{5/2}\approx0.527$ ✓.

ANSWER: B