# Standard Deviation Calculator

Enter Data Set Values (Separated by Comma)

Table of Content
 1 What are the Formulas of Standard Deviation? 2 Formula For Sample: 3 Formula For Population: 4 Applications of Standard Deviation: 5 How to Find Standard Deviation (Step-by-Step): 6 Standard Deviation in Histograms:
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The online standard deviation calculator helps you to find the standard deviation (σ) step-by-step and other statistical measurements of the given dataset. You can easily measure the variability or volatility of the given dataset by using this std dev calculator. You just need to enter the dataset numerical values in this tool to solve your statistical questions. Remember that this SD calculator will perform standard deviation calculations for both population and sample standard deviation.

Well, in this context, calculator-online provides the standard deviation formulas that help you in step-by-step solution and much more you should beware of.

For ease, you can try our online standard error calculator that helps you to calculate the standard error of the given raw dataset.

## What is Standard Deviation?

According to statistical terms, the SD (σ) is referred to as a measure of the amount of variation or dispersion of a dataset of values. It is the square root of the variance of the dataset. Typically, it is used to measure statistical results such as margin of error.

The low value of SD represents that the values are close to the mean of the dataset, while a high SD said referred to as the values are spread out over a wider range. Also, the mean is known as the average of the numbers in the dataset.

### Formulas of Standard Deviation:

In mathematics, the positive square root of the variance is referred to as the formula of SD. The given formulas are used by this sample standard deviation calculator to perform statistical calculations:

#### Formula For Sample Standard Deviation:

It is quite complex to sample the every member from the whole population, so, the standard deviation equation for random sample from the population is:

$$s = \sqrt {\frac{\sum{(x_i-µ)^2}}{ N-1} }$$

this will be equal to the following equation:

$$s = \sqrt {\frac{((x_1-µ) + (x_2-µ) + (x_3-µ) +………+ (x_n-µ))^2}{ N-1} }$$

#### Formula For Population Standard Deviation:

When we have to do the calculation of standard deviation from the whole population, then the formula can be modified as:

$$s = \sqrt {\frac{\sum{(x_i-µ)^2}}{ N} }$$

Equals to the following following formula:

$$s = \sqrt {\frac{((x_1-µ) + (x_2-µ) + (x_3-µ) +………+ (x_n-µ))^2}{ N} }$$

Where,

• x is the value of number
• N is the total number of values
• µ is the mean of the values
• s is the standard deviation of the number

The formula for the variance of dataset of sample is:

$$σ^2 = {\frac{((x_1-µ) + (x_2-µ) + (x_3-µ) +………+ (x_n-µ))^2 }{ N-1}}$$

To avoid from the estimating of variance for population, simply replaced N with N-1. It becomes for the population:

$$σ^2 = {\frac{((x_1-µ) + (x_2-µ) + (x_3-µ) +………+ (x_n-µ))^2 }{ N}}$$

Our population standard deviation calculator considers this formula for the calculations of the standard deviation & variance.

Beside these formulas, the other statistics formulas used by this std deviation solver are as follows:

$$Sum of Squares SS = (x_1 + x_2 + x_3 +………+ x_n)^2$$

$$Mean = {\frac { x_1 + x_2 + x_3 +………+ x_n}{N}}$$

$$Count numbers = n = count(x_i)_{i=1}^n$$

Also, this simple, but highly accurate covariance calculator will efficiently estimate the covariance between two random variables X and Y during probability & statistics experiments.

## Applications of Standard Deviation:

The Standard Deviation is widely used to test the models in real-world data experimentally and in industrial settings. It can be used to find the minimum and maximum value of some product when the product is in high percentage. If the values are going outside the range, then it is necessary to change the production to improve the quality of the product. This measure of dispersion is widely used in different fields of science like weather forecasting to predict the weather, finance to measure the price fluctuations of products, and many others. You can easily determine the normal or average range of the dataset of anything by using an online standard deviation solver. SD is widely used in the field of social science for research purposes to analyze the health statistics, test scores and shows the different patterns of cultural behavior.

## How to Find Standard Deviation (Step-by-Step):

Our mean and standard deviation calculator performs instant calculations to find a statistical measures of diversity or variability in a data set that is known as S.D. You just have to follow the following points to do the exact calculations:

• Find out the number of sample from the population
• Calculate mean
• Find the difference between the each sample and mean
• Square each value
• Find the sum of the square of each value
• Divide by N-1 to get the variance of the dataset
• By taking square root of the value, you can determine the standard deviation of the dataset

Here we have example solve manually for better understanding.

Example:
Find standard deviation from mean from the sample with 6 numbers 3, 4, 9, 7, 2, 5?

Solution:

Step 1:

Calculate the mean of numbers, for this divide the sum of all numbers with the total numbers:

$$µ = {\frac{3 + 4 + 9 + 7 + 2 + 5}{6}}$$

$$µ = 30 / 6$$

$$µ = 5$$

Step 2:

Find the square of the difference of every value with mean:

$$x_1-µ = 3 – 5 = -2$$

$$x_2-µ = 4 – 5 = -1$$

$$x_3-µ = 9 – 5 = 4$$

$$x_4-µ = 7 – 5 = 2$$

$$x_5-µ = 2 – 5 = -3$$

$$x_6-µ = 5 – 5 = 0$$

Now,

$$(x_1-µ)^2 = (-2)^2 = 4$$

$$(x_2-µ)^2 = (-1)^2 = 1$$

$$(x_3-µ)^2 = (-4)^2 = 16$$

$$(x_4-µ)^2 = (2)^2 = 4$$

$$(x_5-µ)^2 = (-3)^2 = 9$$

$$(x_6-µ)^2 = (0)^2 = 0$$

Step 3:

Compute standard deviation:

$$s = \sqrt {\frac { 4+1+16+4+9+0}{ 6-1}}$$

$$s = \sqrt {\frac { 34 }{5}}$$

$$s = \sqrt {6.8}$$

$$s = 2.60$$

Step 4:

$$σ^2 = {\frac {4+1+16+4+9+0}{6-1}}$$

$$σ^2 = {\frac{34}{5}}$$

$$σ^2 = 6.8$$

Apart from this step-by-step calculation, the online standard deviation calculator is the best way to deal with S.D calculations quickly. Even the free SD calculator helps you to solve the calculations for both simple & complex calculations for standard deviations, variance. and several other measurements.

## Standard Deviation in Histograms:

The dataset is represented through a histogram, which represents the numbers in the form of bars of different heights. In a histogram, bars represent the range of the dataset. A longer bar represents the higher range of the dataset while the wider bar suggests a larger standard deviation & a narrower bar indicates a lower standard deviation. Let’s have an example:

The test marks of 600 students with a mean of 100, the histogram orientation is as:

Math test marks SD = 8.5

English test marks SD = 18.3

Physics test marks SD = 25.8

In all three subjects, the physics test has the highest standard deviation.

## How to Calculate Standard Deviation with SD Calculator:

No doubt, calculating standard deviation of a numerical dataset is not as easy task as it seems. But, the Std calculator works best for finding S.D within no time.

Inputs:

• First, select the option, either your data set value in sample or population form
• Then, enter the values for the dataset
• Lastly, hit the calculate button

Outputs:

The calculator calculate:

• Standard deviation of the dataset
• Total count (n)
• Sum (Σx)
• Mean (μ)
• Variance (σ²)
• Coefficient of Variance
• Standard Error of Mean (SE)
• Sum of Squares of the numbers
• Step-by-Step calculation
• Frequency table for the given dataset

This stdev finder uses the given dataset and displays the complete work required for your calculations.

## Wrapping it-up:

The standard deviation is referred to as the measure of the spread of numbers in a given data set from its mean value. The standard deviation helps the researcher to do the experiments when collecting the whole of the data is not possible. However, it is hard to remember the formula for doing standard deviation calculation, so the rule of thumb is to use an online standard deviation calculator that helps you to determine the standard deviation of the data set within an easy way!

## References:

From the source of Wikipedia : General understanding and basic examples

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

From the source of Investopedia: Standard Deviation vs Variance