Standard Deviation Formula

Standard deviation formula is used to find the values of a particular data is dispersed. In simple words, the standard deviation is defined as the deviation of the values or data from an average mean. 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. It should be noted that the standard deviation value can never be negative.

Standard Deviation is of two types:

  1. Population Standard Deviation
  2. Sample Standard Deviation
Standard Deviation Formula

Formula to Calculate Standard Deviation

Formulas for Standard Deviation

Population Standard Deviation Formula \(\sigma= \sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}{N}}\)
Sample Standard Deviation Formula \(\sigma= \sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}{N-1}}\)

Notations for Standard Deviation

  • \(\sigma\) = Standard Deviation
  • \(x_{i}\) = Terms Given in the Data
  • \(\bar{x}\) = Mean
  • N = Total number of Terms

Standard Deviation Formula Based on Discrete Frequency Distribution

For discrete frequency distribution of the type:

\(x:x_{1},x_{2},x_{3},…x_{n}\) and

\(f:f_{1},f_{2},f_{3},…f_{n}\)

The formula for standard deviation becomes:

\(\sigma= \sqrt{\frac{1}{N}{\sum_{i=1}^{n}f_{i}\left(x_{i}-\bar{x}\right)^{2}}}\)

Here, N is given as:

\(N=\sum_{i=1}^{n}f_{i}\)

Another Formula for Standard Deviation

There is another standard deviation formula which is derived from the variance. This formula is given as:

\(\sigma=\frac{1}{N}\sqrt{\sum_{i=i}^{n}f_{i}x_{i}^{2}-(\sum_{i=1}^{n}f_{i}x_{i})^{2}}\)

Also Check: Difference Between Variance and Standard Deviation

Example Question based on Standard Deviation Formula

Question: During a survey, 6 students were asked 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.

Solution:

Find the mean of the data:

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

Step 2: Construct the table:

x1 \(x_{1}-\overline{x}\) \((x_{1}-\overline{x})^{2}\)
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

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Practise This Question

Regarding the assertion and reason; choose the correct option.

Assertion (A): When the disorder Jaundice sets in, the skin and the mucous membranes assume a yellowish color.

Reason (R): The pigment, Bilirubin, imparts the yellow color to the blood