Standard Deviation Calculator

Our standard deviation calculator allows you to find the standard deviation, variance, mean, coefficient of variance, and margin of error for a data set. Provide your data set values and the tool will show the calculations needed.

What Is Standard Deviation In Statistics?

“It is the square root of the variance of any statistical data set”

Standard Deviation Formula:

Our standard deviation calculator utilises the following formulas to calculate standard deviation:

Population Standard Deviation:

\(σ = \sqrt{\dfrac{1}{N} \sum_{i=1}^N\left(x_{i} – μ\right)^2}\)
Where:
xi = Individual Value
μ = Average Mean Value
N = Total Number Of Values

Sample Standard Deviation:

\(s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}\)
Where:
xi= Each Single Value In the Data Set
x = sample mean
N = total sample size

How To Find Standard Deviation? (Step-by-Step):

Let’s resolve an example to make your concept broader. Just stay focused!

Example # 01:

How to find the standard deviation for the following sample data set below:

\( 3, 4, 9, 7, 2, 5 \)

Solution:

Step 1(Calculate Mean Value):

\(\bar{x} = {\dfrac{3 + 4 + 9 + 7 + 2 + 5}{6}}\)

\(\bar{x} = {30}{6}\)

\(\bar{x} = 5\)

Step 2 (Calculate The Value Of \(\left(x_{i} – \bar{x}\right)\):

\(x_1-\bar{x} = 3 – 5 = -2\)

\(x_2-\bar{x} = 4 – 5 = -1\)

\(x_3-\bar{x} = 9 – 5 = 4\)

\(x_4-\bar{x} = 7 – 5 = 2\)

\(x_5-\bar{x} = 2 – 5 = -3\)

\(x_6-\bar{x} = 5 – 5 = 0\)

Now we have:

\((x_1-µ)^2 = (-2)^2 = 4\)

\((x_2-µ)^2 = (-1)^2 = 1\)

\((x_3-µ)^2 = (-4)^2 = 16\)

\((x_4-µ)^2 = (2)^2 = 4\)

\((x_5-µ)^2 = (-3)^2 = 9\)

\((x_6-µ)^2 = (0)^2 = 0\)

The variance and standard deviation calculator automatically determine all of these values without creating any difficulty for you.

Step 3 (Calculate Sample Standard Deviation):

Now we know that:

\(s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}\)

\(s = \sqrt {\dfrac { 4+1+16+4+9+0}{ 6-1}}\)

\(s = \sqrt {\dfrac { 34 }{5}}\)

\(s = \sqrt {6.8}\)

\(s = 2.60\)

How Does Standard Deviation Calculator Work?

The simple interface of the STD calculator makes your calculations super fast and 100% accurate! What you need to do includes:

Inputs:

• Select either sample or population data set
• Enter the values for the dataset
• Tap Calculate

Outputs:

• Standard deviation, variance, and mean of the dataset
• Total count & sum of data values
• Coefficient of Variance
• Standard Error of Mean (SE)
• Sum of Squares of the numbers
• Frequency table for the given dataset
• Step-by-Step calculation

References:

From the source of Wikipedia : General understanding and basic examples

From the site of scribbr.com : Formulas for population and sample standard deviation

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