Let S be the standard deviation of n...

MathematicsMeasures of DispersionJEE Advanced 1980Easy

Let $S$ be the standard deviation of $n$ observations. Each of the $n$ observations is multiplied by a constant $c$ . Then the standard deviation of the resulting number is

(A)

s

(B)

cs

(C)

s c

(D)

none of these

Answer:B

Visualized Solution (English)

Defining Initial Observations

Let the $n$ observations be $x_{1}, x_{2}, ..., x_{n}$ .
The standard deviation of these observations is given as $S$ .

The Standard Deviation Formula

The formula for standard deviation $S$ is:
$S = \ s q r t \ f r a c 1 n \ s u m_{i = 1}^{n} (x_{i} - \ ba r x)^{2}$
where $\ ba r x$ is the mean of the observations.

Applying the Transformation

Each observation is multiplied by a constant $c$ .
New observations: $y_{i} = c \ c d o t x_{i}$ for $i = 1, 2, ..., n$ .

Calculating the New Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \overset{y}{ˉ}

The new mean $\ ba r y$ is:
$\ ba r y = \ f r a c 1 n \ s u m_{i = 1}^{n} y_{i} = \ f r a c 1 n \ s u m_{i = 1}^{n} (c x_{i})$
$\ ba r y = c \ l e f t (\ f r a c 1 n \ s u m_{i = 1}^{n} x_{i} \ r i g h t) = c \ ba r x$

Setting up New Variance

The new variance $\ s i g m a_{y}^{2}$ is:
$\ s i g m a_{y}^{2} = \ f r a c 1 n \ s u m_{i = 1}^{n} (y_{i} - \ ba r y)^{2}$

Substituting Transformed Values

Substitute $y_{i} = c x_{i}$ and $\ ba r y = c \ ba r x$ into the variance formula:
$\ s i g m a_{y}^{2} = \ f r a c 1 n \ s u m_{i = 1}^{n} (c x_{i} - c \ ba r x)^{2}$

Factoring the Constant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c

Factor out $c$ from the squared term:
$\ s i g m a_{y}^{2} = \ f r a c 1 n \ s u m_{i = 1}^{n} [c (x_{i} - \ ba r x)]^{2}$
$\ s i g m a_{y}^{2} = \ f r a c 1 n \ s u m_{i = 1}^{n} c^{2} (x_{i} - \ ba r x)^{2}$

Isolating the Original Variance

Since $c^{2}$ is a constant, pull it out of the summation:
$\ s i g m a_{y}^{2} = c^{2} \ l e f t (\ f r a c 1 n \ s u m_{i = 1}^{n} (x_{i} - \ ba r x)^{2} \ r i g h t)$
$\ s i g m a_{y}^{2} = c^{2} S^{2}$

Finding the New Standard Deviation

The new standard deviation is the square root of the new variance:
New S.D. $= \ s q r t c^{2} S^{2} = ∣ c ∣ S$
Assuming $c > 0$ as per the options, New S.D. $= c S$

Key Takeaway and Conclusion

Key Takeaway: Scaling observations by $c$ scales the standard deviation by $∣ c ∣$ .
Note: Adding a constant to observations does not change the standard deviation.
The correct option is $cs$ .

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