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Standard Deviation Calculator

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An online standard deviation calculator is specially designed to find the standard deviation (σ) step-by-step and other statistical measurements of the given data set. You can easily measure the variability or volatility of the given data set by using this sd calculator.

Before we move on, let us make it clear that you must have a sound knowledge of this statistical term. So, without getting late, let’s move forward.

Stay focused!

What Is Standard Deviation?

In statistical analysis:

“A measure of the amount of variation or dispersion of a dataset of values is known as the standard deviation”

 

Standard deviation definition diagram

 

Low SD Value:

The low value of SD represents that the values are close to the mean of the dataset.

High SD Value:

A high SD is referred to as the values are spread out over a wider range.

Formula Of Standard Deviation:

The given formulas are used by this sample standard deviation calculator to perform statistical calculations

Population Standard Deviation Formula:

When there is a need to measure the whole population, then we use population standard deviation. It is actually the square root of the population variance.

$$ σ = \sqrt{\frac{1}{N} \sum_{i=1}^N\left(x_{i} – μ\right)^2} $$
Where:

\(x_{i}\) = Individual Value

μ = Average Mean Value

N = Total Number Of Values

Sample Standard Deviation Formula:

Now the question arises how to find sample standard deviation. But do not worry as we are going to tell you how to solve for it. What you need to do is to use the equation for standard deviation of the mean sample which is given as follows:

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

Where:

\(x_{i}\) = Each Single Value In Data Set

\(\bar{x}\) = sample mean

N = total sample size

Our population standard deviation calculator considers both of these formulas for the calculations of the standard deviation & variance.

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

It is of great importance to understand the standard deviation statistics properly. So far, we have just discussed the formulas to calculate mean and standard deviation. But now we will be solving an example to make your concept more broad. 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):

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

$$ \bar{x} = {30}{6} $$
$$ \bar{x} = 5 $$

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:

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

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

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

$$ s = \sqrt {6.8} $$

$$ s = 2.60 $$

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.

How To Calculate Standard Deviation On A STD Dev 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. Let us guide you how to find the standard deviation with the help of an SD calculator!

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 relative standard deviation 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.

FAQ’s:

Why can variance not be negative?

Yes, this statistical quantity can never be a negative number. The reason is that the square of a negative number is itself a positive number. As variance is also the distance from the mean position and is squared, it can not be negative also.

What is the range of the standard deviation?

The statistical analysis tells us that the range of the standard deviation is one-fourth of the range of the data set.

Does standard deviation have units?

Standard deviation has the same units as of the data set.

For example:

If data set are the masses of the objects in kilograms, then the SD will also have the units of kilograms.

Conclusion:

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

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