NIMCET 2021 — Mathematics PYQ

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Question

NIMCET 2021 — Mathematics PYQ

NIMCET | Mathematics | 2021

The standard deviation of 20 number is 30. If each of the numbers is increased by 4, then the new standard deviation will be

Choose the correct answer:

Explanation

1. Understanding the Property

A fundamental property of measures of dispersion (like Range, Mean Deviation, and Standard Deviation) is that they are independent of the change of origin.

This means that if a constant value $k$ is added to or subtracted from every observation in a data set, the standard deviation remains unchanged.

2. Mathematical Explanation

Let the original $n$ numbers be $x_1, x_2, \dots, x_n$ with mean $\bar{x}$ . The original standard deviation $(\sigma)$ is given by:

\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}

If each number is increased by $k = 4$ , the new numbers become $y_i = x_i + 4$ .

The new mean $(\bar{y})$ will also increase by $4$ :

\bar{y} = \bar{x} + 4

Now, calculate the new standard deviation $(\sigma_{new})$ :

\sigma_{new} = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n}}

\sigma_{new} = \sqrt{\frac{\sum ((x_i + 4) - (\bar{x} + 4))^2}{n}}

\sigma_{new} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}

3. Conclusion

As shown in the formula, the constant $+4$ and $-4$ cancel each other out within the deviation calculation.

\sigma_{new} = \sigma

Since the original standard deviation was $30$ , the new standard deviation after increasing each number by $4$ remains $30$ .

Correct Option: B) $30$

Choose the correct answer:

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