Question

The standard deviation of the numbers $31,32,33,...,46,47$ is

1. $\sqrt{\dfrac{17}{12}}$
2. $\sqrt{\dfrac{({{47}^{2}}-1)}{12}}$
3. $2\sqrt{6}$
4. $4\sqrt{3}$

Answer 1

In this type of question we first need to know the definition of standard deviation and we must know the formula of standard deviation and that is given as $\sqrt{\dfrac{\sum\limits_{i=1}^{n}{{{({{X}_{i}}-\bar{X})}^{2}}}}{n-1}}$ and with the help of above formula we will find the standard deviation of given numbers in question.

Complete step-by-step solution:
Defining the term Standard Deviation:
Standard deviation is a statistical measurement in finance that, when applied to the annual rate of return of an investment, sheds light on that investment's historical volatility. The greater the standard deviation of securities, the greater the variance between each price and the mean, which shows a larger price range. For example, a volatile stock has a high standard deviation, while the deviation of a stable blue-chip stock is usually rather low.
The Formula for Standard Deviation:
$S\tan dard\\_Deviation=\sqrt{\dfrac{\sum\limits_{i=1}^{n}{{{({{X}_{i}}-\bar{X})}^{2}}}}{n-1}}$
Where,
${{X}_{i}}$ is the value of ${{i}^{th}}$ point in the data set.
$\bar{X}$ is the mean value of the data set.
$n$ is the number of data points in the data set.

Calculating the Standard Deviation
Standard deviation is calculated as follows:
The mean value is calculated by adding all the data points and dividing by the number of data points.
The variance for each data point is calculated by subtracting the mean from the value of the data point. Each of those resulting values is then squared and the results summed. The result is then divided by the number of data points less than one.
The square root of the variance—result from no. $2$—is then used to find the standard deviation.
Using the Standard Deviation:
Standard deviation is an especially useful tool in investing and trading strategies as it helps measure market and security volatility—and predict performance trends. As it relates to investing, for example, an index fund is likely to have a low standard deviation versus its benchmark index, as the fund's goal is to replicate the index.
Now coming to our question,
Since the standard deviation of $31,32,33,...,47$ will be the same as those of $1,2,3,...,17$ (decreasing each item by $30$ ).
Therefore,

& \Rightarrow \sqrt{\dfrac{{{17}^{2}}-1}{12}} \\\ & \Rightarrow \sqrt{\dfrac{288}{12}} \\\ & \Rightarrow \sqrt{24} \\\ & \Rightarrow 2\sqrt{6} \\\ \end{aligned}$$ **Therefore our final answer is option $$(3)$$ .** **Note:** Standard deviation comes due to a historical accident: in 1893, the great Karl Pearson introduced the term "standard deviation" for what had been known as "root mean square error". The confusion started then people thought it meant mean deviation.