How To Calculate Standard Deviation using Standard Deviation Formula

Standard deviation is a mathematical term and most students find the formula complicated therefore today we are here going to give you stepwise guide of how to calculate the standard deviation and other factors related to standard deviation in this article. As standard deviation is one of the hardest things so it can not be calculated by hands there is still no statistician who can calculate it on hands.

The only reason behind it is that standard deviation is quite a complex task and there is a high risk to create a mistake and apart from this making calculation by hand is a very slow procedure. Therefore, mostly all statisticians focus on the spreadsheets or either the computer programs for calculating a standard deviation. Now, you might be thinking why we are discussing this topic or what is the need for learning standard calculations. The only reason behind all this stuff is to make you understand the concept of calculating the standard deviation with fewer chances of mistake. We all know that the symbol of the standard deviation is sigma. Now let us come to the explanation of the formula of standard deviation.

Standard deviation

Let us come to explore about standard deviation in depth, if we talk about the value of standard deviation then by the formula itself we can say that it is directly proportional to the mean value or else we can say that if the mean value is low then the value for standard deviation will also be less or if the value of mean for data set is high then the value of standard deviation will also be high. There is not only one situation where we can opt for standard deviation there are various circumstances at which you can freely use standard deviation formula for lots of equations and questions.

Standard deviation formula

This concept of standard deviation can also be used for measuring statistical data which is also known as the expression for variability in the population, in this situation we can also use another name of standard deviation or it is mostly referred as the standard error of mean or the standard error of estimation in respect of the mean. You can also calculate population standard deviation by the help of standard deviation and it will also support you for calculating sample standard deviation. Now, we are going to get the standard definition of the standard deviation of which is also defined with respect to the population standard deviation. When we are in need of calculating the entire population then it is used, we can simply define it as the square root of variance for any data set in which the population members will be taken as the sample, in this condition we are going to use the below-provided formula for calculating a standard deviation.

Formula For Standard Deviation

Formula For Standard Deviation

In this above-provided equation we are seeing a sign like reverse Z which is known as the sign of summation. I hope that you are already aware of this sign and if not then, first of all, come to know about this. So, while you calculate the standard deviation simply put up the given values in the above formula and you will get the result.

How To Calculate Standard Deviation

So, we are here going to explain the formula of standard deviation and will also tell you how to calculate the standard deviation by using this formula.

Step 1: First of all you need to calculate the arithmetic mean of the number or set of numbers which you are having. By calculating it we will get the value of x-bar within the equation. The very common and easy formula of a calculating average mean is

Take out the sum of all given numbers and divide the result by the total number of a particular data set.

This is the most common and old method of calculating mean manually but instead of it you can also calculate the mean by using spreadsheet software which can give you the result in just a single click for any size of data large, medium or small. Although you can also calculate standard deviation also by using spreadsheet software but the formula of calculation is very big and it is difficult to memorize that.

Step 2: Next, after calculating average mean you are required to perform the (x-x)² function.

With the help of the computer, it is very easy to perform this function just by dragging and dropping the formula and after that just copy and paste those functions as there is no requirement of writing each and everything.

If you are interested in calculating this function manually by your hands then the best way of doing this is to prepare a table by the data set you are having like, for example, we are having 5,12,16,21,28 data. The average mean of this set is 16.4 now, out this mean in the formula of standard deviation as shown below.

And the result after calculating (x-x)² will be following.

For calculating this function we are squaring each and every value and then we will find the square root of the result as squaring is because there will be no negative value.

Step 3: As we have found the result of xx bar now we will get the value which we are simply going to add. Now, substitute these values in the given formula.

As there is total 5 digit in the set so we are going to going to do 5-1 which is 4, next substitute it in the formula.

Next, you have to find out the square root of the given result 305.2/4 it will be 76.3 and a square root of it will be 8.73

So, at last, we have calculated the standard deviation for our data.

Sample Standard Deviation Formula

Sample standard deviation Formula

Lots of different problems can arise while making any solution and out of them, one can be a problem which is not easy to sample with each and every member for the entire population by using the above equation. In this condition, we will be in need of doing some modification within equation which will help us in calculating the random population of a sample which you are studying.

Sample Standard Deviation

  1. First, you should be aware of the sample standard deviation, it is also known as the true standard deviation for the given population which is based on the small sample from the entire population. Now we are going to calculate sample standard deviation.
  2. First of all, you have to calculate the mean by adding all individual data and then dividing all of them by the total number.
  3. After this, you have to subtract mean of each individual measurement and then take the square of a result. You need to continue the same process for all numbers.
  4. Next, add the result comes out from the above calculation. After it you have to find the value of n where n is the total counting of data numbers.
  5. Now, divide the result came from the 3rd step with the 4th step. By this, you will get the value for s².
  6. At last, calculate the sample standard deviation by taking a square root of the last result.

Sample standard deviation Formula


If we compare the result from both the sets that are from standard deviation and the sample standard deviation then we are going to see a lot of variation among the result of both of them. You will see a wide percentage difference among both of them therefore standard deviation is known as the accurate one as it takes the accurate measurements.

Why Take Sample?

The main question now arises is that if the answer of sample standard deviation is not accurate then why we use it? The main reason behind it is because calculating sample standard deviation is easy and cheap also. Therefore we use the sample by assembling the population so that we can get them on results easily without any complicated procedure.

Standard Deviation ( σ ) Calculator

Standard Deviation ( σ ) Calculator

At this point another main thing which we are going to learn is the standard deviation calculation. We all know that the symbol of standard deviation is sigma and till now we have learned a lot about the standard deviation but here is the basic problem which we all feel is that how to find standard deviation and to calculate it we are having this standard deviation calculator which will really go to help you in calculating the values of standard deviation. In the previous time the main target of calculating standard deviation was just to support the input and then getting the final result or solution for the calculations you have performed but from then till now a lot of changes has arrived under these calculators which let you perform different tasks. Now, we are going to discuss about the modern standard deviation calculator as at this stage we are not only stuck to the calculation of final result apart from this it will also going to let you calculate other factors like mean, mean statistics, mean average and will also allow you in making various inputs by using lots of methods with this modern calculator for making the calculation of standard deviation as it will not only give you the final result along with it you are also going to get a stepwise procedure for letting you understand more better.

After this it is easy to get know that this modern calculator will not only help you in finding mean value, standard deviation and variance apart from this you will get the entire calculation in a stepwise format. If you wanted to do calculation for standard deviation manually without help of any kind of standard deviation calculator to get mean or variance value then we are going to get lots of different variations in the result which will vary widely from person to person but by help of calculator you can get almost exact result every time whenever you calculate the same data set at all steps. If you are going to calculate standard deviation by using the calculator there is a great need of understanding the value of simplicity for the given data set which you are going to require at the time of calculation. If you calculate the standard deviation in the text file and CSV format input then there will be a detailed stepwise calculation for all questions which will be calculated by this input method with help of a calculator. Therefore, for calculating standard deviation you will be going to need a proper formula which is provided here.

Standard Deviation

The next most important thing which comes is how we will calculate the standard deviation and to get a perfect error less calculation there is the need of a perfect formula that is why below we are giving you main formula needed to calculate sample standard deviation formula, mean formula, sample variance formula etc. Let us have a look over them,

Standard Deviation S2

At this point we are having lots of different calculators regardless of standard deviation calculator, these are quite helpful to calculate other precious values like as of variance, population standard deviation, sample standard deviation and many more. So, now let us come to get some most useful calculators which will let you calculate different values. These are as follows:

  1. Variance Calculator
  2. Population standard deviation calculator
  3. Sample standard deviation calculator
  4. A coefficient of variance calculator
  5. Variance and standard deviation calculator
  6. Sample variance calculator
  7. Relative standard deviation calculator
  8. Sample mean calculator
  9. Variance and standard deviation calculator
  10. Standard deviation calculator excel
  11. Mean standard deviation calculator

Standard Deviation S2

Mean Median Mode Definition

Here, we are going to know about other important definitions like mean, median, and mode.


It is also called as the average value of provided data set in terms of mathematics we can also be called it as arithmetic mean. To calculate the value of mean we can use the old method in which first of all we add all the given numbers of data set and then count them and divide the result after addition with the total number of digits. After doing all this stuff you will later get your result. Let us make it more clear by having an example:

4, 7, 9, 2

4+7+9+2 = 22

After it, we are going to divide it with the total number of digits we are having that is 4

22/4 = 5.5

So, this value 5.5 is the mean value.


It is only known as the middle digit of the entire data set you are having, let us take an example that we are having 4,5,7,3,2,1,9 as a data set. So, in this value, we are having 3 as the median. Now, we are having another question in our mind that if there is the value which is in even number then what will be the median. To get the answer to this question we are having another set of data like 3, 6, 9, 2, 4, 8. So, in this situation, we are going to take 9 and 2 as the middle terms and then we will have the average of these both digits like ( 9+2 ) / 2 = 5.5

So, the median for this data set is 5.5


A mode is one of the easiest things to find as in it we only have to count most repetitive or re-occurring digit in the provided data set. Like we have the below-provided data set

2, 5, 3, 2, 6, 4, 2, 6, 2, 8, 9

Here, the most re-occurring digit is 2 which comes for the four-time therefore, the mode value of this set is 2.

How To Calculate Median in Excel or Mean

In this topic, we are going to learn about the excel formula for calculating the value of mean and here we are learning all the details of calculating.

How to Calculate median in Excel or mean

To calculate this at first we have to open the excel spreadsheet wherein the section of a column you just have to enter the score as we can see in the above picture. Later you add all of the data just put your cursor on the place at where you are in need of calculating mean and then click the mouse button. Next, later on by clicking at the formula tab then simply select the fx or insert function after it you are going to see a dialogue box appeared at your screen and after that go through the option of average from the statistical category and at last click on the OK button.

Afterward, if you are also in need of calculating median value then for it click at the option of median instead of selecting mean and you will finally get your result.

Calculate Standard Deviation in Excel

We have already calculated a mean and median value in the excel sheet and now we are going to learn the calculation of standard deviation in the Excel sheet which is one of the most important parts of this article. So, as before we are going to open the excel spreadsheet and then we will place the cursor at the desired location where we want to get the value of standard deviation and then click on the mouse button. Then, reach the tab of formula and then, at last, select the fx or insert function and a dialogue box will appear in front of you.

Calculate Standard Deviation in Excel

After it from the statistical category we are going to select STDEV.S option and then we will make a range selection and click OK which is available at the right side of the bottom screen. Then select the cell range and drag the mouse cursor then at last click OK and you will get the value of your standard deviation.

How to Calculate Variance in Excel

To find the statistical variance value in the excel sheet we have to simply utilize the function VAR which is used for variance calculation and an inbuilt feature of excel. We will have the set of data from 1 till the N values so that we can calculate variance value for the given set of data. You are in need of selecting an empty cell in your spreadsheet and in this cell you have to insert value in VAR like as of value 1, value 2 till value N although you also have the option of choosing syntax VAR where you need not enter manual values for the VAR ranges as in it you can simply mention VAR (A2: A18 ) or the number of rows you are having. Then, later on, select the option of insert function and then click at the “function library” then select the option of insert function and at last click on the VAR function. Next, click with the cursor of the mouse at a particular place and click on OK from dialogue box by this you will get the variance value by its help.

How to Calculate Variance in Excel

Standard Deviation Graph

If we are having lots have values for the standard deviation then at those instances we need to create a graph for entire values which will make everything more clear and memorable. There can be lots of circumstances where you can need to make a graph for the values of standard deviation but there is a basic problem which you can face is how to create this graph. It is very easy for creating a graph by using various tools which are available in the excel. Above we have also provided you the sample graph for the standard deviation which will help you in getting an idea for drawing the graph of standard deviation.

Standard Deviation Graph

Standard Deviation Percentage

Are you in need of calculating the percentage value for your standard deviation then for this you are in need of finding mean an average for the provided set of data without even calculating the values for mean and average deviation and after this you have to divide the value with average deviation and then multiply the result by 100 and at last you will get the value for standard deviation percentage.

Standard Deviation Percentage

Standard Deviation Example

Standard Deviation Example

As of now, we have assembled a lot of information about the standard deviation and how to calculate it. In these last topics, we are going to make the calculations more easy for keeping your concept more clear by using an example.

So, let us take the following set of data

9, 2, 4, 5, 7, 3

Step 1: At first, we are going to find the mean value for the given set of data.

         ( 9 + 2 + 4 + 5 + 7 + 3 ) / 6 = 5

Step 2: In this step we are going to subtract the value of mean for the set of data and then we are going to take the square of each and every value like shown below.

* ( 9 – 5 )2 = 16

* ( 2 – 5 )2 = 9

* ( 4 – 5 )2 = 1

* ( 5 – 5 )2 = 0

* ( 7 – 5 )2 = 4

* ( 3 – 5 )2 = 4

Step 3: Next, we are going to simply find the value of mean for these squared values like as follows.

     ( 16 + 9 + 1 + 0 + 4 + 4 ) / 6 = 5.6

This above value will be known as the variance or you can say it as sample variance.

Step 4: To calculate the value of standard deviation we are going to take the square root of the above value like this.

5.6 = 2.36


So, today in this article we come to learn about standard deviation and its use after that we learned how to calculate the standard deviation with a stepwise guide. It becomes easy to calculate any data set when you have each and every minute information with you for calculating. Next, we come to know about sample standard deviation which is denoted by s, it is easy to calculate the sample standard deviation but it shows vast variation in the result as there is no precise data. The formula for both of them is stated below.

So finally, we have come to learn about almost each and every information related to the subjects like as of sample variance, deviation, mode, mean, median and all other things which can help you in gaining some extra knowledge. The excel spreadsheet is a program which is used mostly in the windows and there are almost all people who wanted to calculate the standard deviation for making the calculations easier. So, after reading this entire article you will come to learn almost all the details with the basic concept and will solve all your problems within seconds.