Standard Deviation Calculator

The standard deviation calculator allows you to calculate mean, variance, and standard deviation with population and sample calculation formulas.

About Standard Deviation Calculator

It's an interactive tool that runs on pre-defined algorithms and gives you the final results live, quickly, and accurately. No need to do the manual calculations. Also, it's free to use. No subscription charges or calculation limits are there.

Mostly, we can use it to know whether someone's behavior is normal or extraordinary. In addition, it is used in many fields like finance, business statistics, weather changes, and many more.

In this article, I will provide you some very important information and techniques that will help you to understand the formulas and calculations.

Standard Deviation Calculator

Most importantly, we will discuss three things in this tutorial. Which are:

  • Mean
  • Variance
  • Standard Deviation

To solve the standard deviation issues firstly, we need to figure out mean and variance. That's why we will cover these two topics here. So, you can understand all the things clearly.

Standard Deviation

First of all, let me tell you the definition.

It is a number used to tell how measurements for a group are spread out from the mean value. In other words, it is a measure of how spread out the data is.

Its symbol is "σ". Also, we can call it "Sigma"(The greek alphabet).

The standard deviation calculator uses the following formulas to perform the calculation.

Formula

The "Population Standard Deviation":

Population Standard Deviation Formula

The "Sample Standard Deviation":

Sample Standard Deviation Formula

Looks complicated right? No worries! I will help you to figure out all the things in very simple language.

Also, there is a small but very important difference between Population and Sample formula. That is "N" and "N-1" at denominator. Therefore, when we calculate sample variance, we need to divide by "N-1" (Instead of "N").

Here N is for population size and N-1 is for sample size. You can see it in the above formulas. Whereas other components are same in both equations.

Also, don't get confuse about "μ" and "x-bar(x̄)". Because these both are the same things. That is mean. x-bar is a sample mean and μ is population mean.

Don't be panic about names like population, sample, or mean. You will get all these complex things clear in the next section.

How to calculate Standard Deviation?

For easy understanding, I have divided the whole process into five simple steps.

  1. Calculate the Mean(μ)
  2. Subtract the mean from each data point
  3. Square each difference
  4. Calculate the mean of the square differences
  5. Take the square root

First of all, let's take an example to understand above all steps one by one.

Example

The following are the number of pens that five people have. So, find out the mean, the variance, and the standard deviation.

The values are: 22, 15, 35, 8, 10

In this example, we will calculate the population standard deviation.

First of all, let's have a look at the formula of standard deviation.

We can say that,

The standard deviation is equal to the square root of variance.

Standard Deviation Equation

Where,

  • σ = Standard Deviation
  • = Sum of each
  • Xi = Data points
  • μ = Mean
  • N = Number of data points

So, now you are aware of the formula and its components. Let's do the calculation using five simple steps.

1. Calculate the Mean

First of all, let me tell you the meaning of mean.

The mean is an average of total numbers.

It is defined as "μ" (The greek letter mu). Also, it's very easy to calculate the mean.

Just add up all the given values and then divide it by a total number of values.

In our case, we have five numbers. So, we can calculate the mean like this.

Mean Calculator

So, the mean value = 18.

I am sure the first step is clear now. Let's jump to the next step.

2. Subtract the mean from each data points

In this step, we will subtract each data value with mean.

Here we have,

Xi data values = 22, 15, 35, 8, 10 and Mean(μ) = 18

So, now we will find (Xi - μ).

Subtract the mean from each data points

After subtraction, we get values = 4, -3, 17, -10, -8.

3. Square each difference

In this, we will square all the values that we get from step 2. So, it will be calculated like this:

Square each difference

Hence, we get the squared values = 16, 9, 289, 100, 64.

4. Calculate the mean of the squares

In this step, we will find the mean of squared values that we get from step 3.

But before we step forward, I want to tell you something.

Some of you will raise a question that:

What will we get from step 4?

Surprisingly, we will get Variance as a final output from step 4.

Now the question is What is Variance?

It is an average of the squared differences from the mean.

Let me show you the variance formula. So, you will get more ideas.

Variance Formula
Variance Formula

As you can see, we already have found the values of (Xi - μ)2. So, our next step is to calculate the variance using these squared values.

But How?

For that, we need to calculate the mean of squared values. In short, we need to sum up all the squared values and then divide it with a total number of values. That is 5.

Therefore, the calculation will be like this:

Variance Calculator

So, as a result, we get the variance = 95.6.

Now, let's go to the final step and find the standard deviation.

5. Take the square root

In this step, we just need to calculate the square root of variance.

Square Root

Finally, we get the standard deviation value = 9.76 for population.

Sample Standard Deviation Calculation

In this section, I will tell you the process to find the sample standard deviation.

First of all, let's have a look at the sample SD formula:

Sample Standard Deviation Formula

You can see, there is just a small change in this formula as compared to the population formula. That is "N-1" with replacing of "N".

So, when we are calculating the sample standard deviation then step 1, step 2, and step 3 will be common. I mean it's same as the population calculation steps.

Only the change will be in step 4 and step 5.

Follow the steps below to find the sample standard deviation.

So, we will skip step 1, 2, and 3 and directly calculate step 4 and 5. That is find out the sample variance using squared values and then square root the variance value.

We already have found the squared values from step 3.

Which are: 16 + 9 + 289 + 100 + 64

In population, we are dividing the above values with 5. Because the number of values is 5.

But in sample, we need to divide the squared total with (N-1) = (5-1) = 4.

After division, we will get the standard variance.

And then we need to calculate the square root of the variance to get the final Sample SD result.

You can see the step 4 and 5 calculation for sample standard deviation here:

Final Sample Result

DONE!

I hope you will get all the things clear now. Also, you can easily do the manual calculation by following the five steps. But the manual calculation is only for our understanding. No need to do this all the time. Because it takes so much time to do it. So, use our calculator and save your time and effort.

How to use Standard Deviation Calculator?

It's very easy to use the calculator. Just follow these steps.

  • Firstly, it's an online tool. So, you need a device like laptop, ipad, or smartphone that can access the internet.
  • After that, open the Standard Deviation Calculator.
  • When it opens properly, then you can see the empty textarea at the top of the webpage. This textarea is for enter the numbers or values.
  • So, firstly, enter the numbers in textarea that you want to caculate its standard deviation.
  • Most importantly, enter values with space(i.e. 2 3 4 7) or comma(i.e. 2,3,4,7) seperated. In comma seperated, don't keep a single space in between numbers. Otherwise it will give you an error.
  • After entering the data click on the "Calculate" button.
  • As a result, you will get Mean, (Population + Sample) Variance, and (Population + Sample) Standard deviation at below text boxes.
  • Lastly, you can refresh the calculater using "Reset" button.

Isn't it simple? Yeah, it is. In short, you can use the standard deviation calculator and solve complex problems with few keystrokes.

Key Features

User-Friendly Interface

The calculator is useless if the user interface is weak. That's why we provide the best user-friendly interface to our online users.

Simple Design

We made a simple design. So, any technical or non-technical person can use our tool easily.

Secure & Fast

Standard deviation calculator gives you a fast and 100% secure output. We also take care of data security of the people.

Advantages

There are many advantages of this tool. Some of them are listed below.

  • Multiple Output: This calculator gives you the Mean, Variance, and Standard Deviation as output. So, it's a one-stop solution to find all the required values.
  • Handy Calculator: Our tool also works in handy devices like mobile and iPad. Beacuse we have made it mobile and iPad responsive. Therefore, you can use it anytime and anywhere.
  • Fast Results: The calculation process is very fast. You will get the results in just a few seconds. No need to wait for page loading.
  • Accurate Results: Our calculator works on coded algorithms and it is fully tested by our experts. Therefore, it gives you 100% accurate results. There is no chance for the wrong output.

Conclusion

Above all information was about standard deviation calculator. Hope it will work out and helpful for your study.