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The Standard deviation calculator helps to compute the sample standard deviation and population standard deviation. It also shows the value of variance, mean, sum, and difference.

Input a data set separated by commas into the standard deviation finder, choose the sample or population, click calculate, and it will calculate the standard deviation and provide you with a thorough result.

Let’s dive deeper into crucial aspects such as standard deviation definition, measuring standard deviation, and the equation of standard deviation.

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

The standard deviation indicates the measure of variance in a data set. In statistics, it inform about how far each observed value is from the mean. It is used for determining the population variability, margin of error, and much more!

Having a low or small standard deviation indicates that the data is packed near the mean and a high or large deviation means that the data is largely dispersed from the mean, showing greater variance.

The symbol that is used to represent standard deviation is “σ”. In most of the distributions, about 95% of values lie within 2 standard deviations as defined by the empirical rule.

This rule provides a quick overview of your data set. It informs you about where most of the values are present in a normal distribution:

- Approximately 68% of values are within 1 standard deviation
- 95% of values are within 2 standard deviations
- Approximately 99.7% of values are within 3 standard deviations

The SD calculator utilizes the following formulas:

The following population standard deviation formula is used when all the members of the population can be sampled:

**\(\ σ = \sqrt{\dfrac{1}{N} \sum_{i=1}^N\left(x_{i} – μ\right)^2}\)**

Where

- \( x_{i} = Individual \ Value\)
- μ = Average Mean Value/expected value
- N = Total Number Of Values

Keep in mind, sometimes it is not possible to sample all the members of the populations so for that reason, the standard deviation equation is modified as you can see below:

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

Where

- \( x_{i} = Each\ Single\ Value\ In\ the\ Data\ Set \)
- x = sample mean
- N = total sample size

Follow the below-mentioned steps to perform the manual calculations:

- First of all, calculate the mean
- Measure the distance from the mean for all the values
- Calculate the square of the distance for each value
- Sum all the squared values
- Now divide the sum by the total values available in the data set
- At the end, find the square root of the answer

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

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

Step 1(Calculate Mean Value):

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

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

\(\bar{x} = 5\)

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 calculator automatically determines all of these values without creating any difficulty for you.

Step 3 (Calculate Sample Standard Deviation):

Now we know that:

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

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

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

\(s = \sqrt {6.8}\)

\(s = 2.60\)

It is useful because:

- It provides you with a better idea of your data’s variability
- Provides you with precise results

In a normal distribution, the high value of standard deviation informs that the values are away from the mean, and a low value indicates that the data set values are near the mean(expected value).

Variance means the average squared deviations and the square root of this, is the average standard deviation. They both reflect the variability.

Whenever you are conducting research, you consider a small sample size of the whole population and because of this, you may have different means every time. But if you take enough samples from the population, then the result will be closer to the true population, This standard deviation of the standard mean is known as the standard error.

From the source of Wikipedia: General understanding and basic examples

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