## Formula for Calculating Standard Deviation

The formula for calculating standard deviation is:

σ = √( Σ(x – μ)^2 / N )

where:

- σ is the standard deviation
- x is each data point
- μ is the mean of the data
- N is the number of data points

The standard deviation measures how much variation or dispersion there is from the mean in a set of data.

A low standard deviation indicates that the data is clustered closely around the mean, while a high standard deviation indicates that the data is spread out more widely.

## Expanding the Formula

We can expand the term (x – μ)^2 as follows:

(x – μ)^2 = x^2 – 2xμ + μ^2

Substituting this into the formula for standard deviation, we get:

σ = √( Σ(x^2 – 2xμ + μ^2) / N )

We can then simplify this expression as follows:

σ = √( ( Σx^2 – 2μ Σx + Nμ^2 ) / N )

σ = √( ( Σx^2 / N ) – ( 2μ / N ) Σx + μ^2 )

σ = √( variance – μ^2 + μ^2 )

σ = √( variance )

**Therefore, the standard deviation is the square root of the variance.**

## Simple Real Life Example on How to Calculate Standard Deviation

Let’s consider a real-life example to illustrate the calculation of standard deviation:

Suppose we have a dataset of the heights of 10 students in a class, in inches:

[65, 68, 70, 72, 75, 77, 79, 80, 83, 85]

**Calculate the Mean (μ)**

Mean is the sum of all values divided by the number of values in a dataset. In this case:

μ = (65 + 68 + 70 + 72 + 75 + 77 + 79 + 80 + 83 + 85) / 10

= 753 / 10

= 75.3

**Calculate the Variance**

Variance is the average of the squared differences from the mean.

For each data point, we first find the difference from the mean, square it, and then add up all the squared differences and divide by the number of data points:

Variance = [(65 – 75.3)^2 + (68 – 75.3)^2 + (70 – 75.3)^2 + (72 – 75.3)^2 + (75 – 75.3)^2 + (77 – 75.3)^2 + (79 – 75.3)^2 + (80 – 75.3)^2 + (83 – 75.3)^2 + (85 – 75.3)^2] / 10

= 53.29 / 10

= 5.33

**Calculate the Standard Deviation (σ)**

Standard deviation is the square root of the variance:

σ = √5.33

= 2.31

Therefore, the standard deviation of the heights of the students in this class is 2.31 inches.

This value indicates that the heights of the students are somewhat spread out around the mean height of 75.3 inches.