Question
If each observation of a raw data whose standard deviation is $\sigma $ is multiplied by a, then write the S.D. of the new set of observations.
✓
Answer
Standard deviation, $\sigma=\sqrt{\frac{\sum\limits_{\text{i}}(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
Here, $\overline{\text{x}}$ represents the arithmetic mean.
Multiplying each $x_i$ by a:
$\overline{\text{x}}_{\text{new}}=\frac{1}{\text{n}}\sum\limits_{\text{i}}\text{a}.\text{x}_\text{i}$
$=\text{a}\times\frac{1}{\text{n}}\sum_\text{i}\text{x}_\text{i}$
$=\text{a}.\overline{\text{x}}_{\text{old}}$
New standard deviation, $\sigma=\sqrt{\frac{\sum\limits_{\text{i}}(\text{a}.\text{x}_\text{i}-\text{a}.\overline{\text{x}})^2}{\text{n}}}$
$=\sqrt{\frac{\sum\limits_{\text{i}}\text{a}^2.(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
$=|\text{a}|\sqrt{\frac{\sum\limits_{\text{i}}(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
$=|\text{a}|.\sigma$
Here, $\overline{\text{x}}$ represents the arithmetic mean.
Multiplying each $x_i$ by a:
$\overline{\text{x}}_{\text{new}}=\frac{1}{\text{n}}\sum\limits_{\text{i}}\text{a}.\text{x}_\text{i}$
$=\text{a}\times\frac{1}{\text{n}}\sum_\text{i}\text{x}_\text{i}$
$=\text{a}.\overline{\text{x}}_{\text{old}}$
New standard deviation, $\sigma=\sqrt{\frac{\sum\limits_{\text{i}}(\text{a}.\text{x}_\text{i}-\text{a}.\overline{\text{x}})^2}{\text{n}}}$
$=\sqrt{\frac{\sum\limits_{\text{i}}\text{a}^2.(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
$=|\text{a}|\sqrt{\frac{\sum\limits_{\text{i}}(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
$=|\text{a}|.\sigma$
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