Standard Deviation Formula
Before learning the standard deviation formula, let us recall what is the standard deviation. Standard deviation is commonly abbreviated as SD, and it tells about the value that how much it has deviated from the mean value. If we get a low standard deviation then it means that the values tend to be close to the mean whereas a high standard deviation tells us that the values are far from the mean value. The standard deviation formula is used to find the standard deviation quickly.
What is the Standard Deviation Formula?
Here are two standard deviation formulas that are used to find the standard deviation of data and the standard deviation when variance is given.
 The standard deviation formula is given as:
\(S = \sqrt{\dfrac{1}{n1} \sum^{n}_{i=1}(x_i  \bar{x})^2} \)
 Standard deviation formula if variance is given:
\( S = \sqrt {\text{Variance}} \)
We can see the applications of the standard deviation formula in the following section.
Solved Examples using Sample Standard Deviation Formula

Example 1: If there are 39 plants in the garden. And a few plants were selected randomly and their heights in cm were recorded as follows: 51, 38, 79, 46, 57. Calculate the standard deviation of their heights using the standard deviation formula. (Use √962.73 = 31).
Solution
N = 5
Mean (\(\bar{x}\))= \(\dfrac{51+38+79+46+57}{5}\) = 54.2
Standard Deviation = \( \sqrt{\dfrac{\Sigma (x_i\bar{x})^2}{N1}} \)
= \( \sqrt{\frac{(5154.2)^2 +(3854.2)^2 +(7954.2)^2 +(4654.2)^2 +(5754.2)^2}{4}} \)
= 15.5
Answer: Standard Deviation for this data is 15.5

Example 2: In a class of 50, 4 students were selected at random and their total marks in the final assessments are recorded, which are: 812, 836, 982, 769. Find the standard deviation of their marks. (Take √23.1= 4.8).
Solution
N = 4
Sample Mean (X̄) = \( \dfrac{812+836+982+769}{4} \) = 849.75
Variance = \( \dfrac{\sum^{N}_{i=1} (X_i  \bar{X})^2}{N1} \)
=\( \dfrac{\sum^{4}_{i=1} (X_i  849.75)^2}{3} \)
= \( \dfrac{(812  849.75)^2 + (836  849.75)^2 + (982  849.75)^2 + (769  849.75)^2}{3} \) = 92.4
Using the standard deviation formula,
Standard Deviation = \(\sqrt{92.4}\) = \(2 \sqrt{23.1}\) = 9.6
Answer: Standard Deviation for this data is 9.6