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 STANDARD DEVIATION FORMULA

Standard deviation is defined as, “The deviation of the values  or data from an average mean”

Standard Deviation helps us to know how the values of a particular data are dispersed. Lower standard deviation concludes that the values are very close to their average. Whereas higher values mean  the values are far from the mean value. Standard deviation value can never be in negative.

Standard Deviation

Standard Deviation is of two types:
1. Population Standard Deviation
2. Sample Standard Deviation

The formula for Population Standard Deviation is: 

\[\large Population\;Standard\;Deviation= \sqrt{\frac{\sum_{i=1}^{n}\left(x-x_{i}\right)^{2}}{n}}\]

We calculate sample standard deviation, we don’t use the population formula. The formula for calculating the sample standard deviation is given as

\[\large Sample\;Standard\;Deviation=\sqrt{\frac{\sum_{i=1}^{n}\left(x-x_{i}\right)^{2}}{n-1}}\]

Where,
$x_{i}$ = Terms given in the data
$\overline{x}$ = Mean
N = Total number of Terms

Let’s solve an example to see how to find the standard deviation.

Question: During a survey, 6 students were asked that how many hours per day they study on an average ? Their answers were as follows: 2, 6, 5, 3, 2, 3.

Evaluate the standard deviation?

Step 1: Find the mean of the data:

$\frac{\left(2+6+5+3+2+3\right)}{6}=3.5$

Step 2: Construct the table:

$x_{1}$

$x_{1}-\overline{x}$ 

$\left(x_{1}-\overline{x}\right)^{2}$

2

-1.5 2.25

6

2.5

6.25

5

1.5

2.25

3

0.5

0.25

2

-1.5

2.25

3

-0.5

0.25

= 13.5

Step 3: Now, use the  Standard Deviation formula

$Sample\;Standard\;Deviation=\sqrt{\frac{\sum_{i=1}^{n}\left(x-x_{i}\right)^{2}}{n-1}}$

$\sqrt{\frac{13.5}{6}}=\sqrt{2.25}$

$=1.5$