Standard Deviation Calculator

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What is Standard Deviation?

Standard deviation is the measure of how spread the numbers are. It is denoted by the Greek letter sigma σ. The formula is the square root of the variance. Standard deviation is also denoted by SD and it is the dispersion of the values in a data set. This dispersion can be stretched or squeezed. Higher standard deviation is an indication of wider value range. Standard deviation can be utilized in various situations just like all other concept of math and statistics. Standard deviation is mainly used to express the variability of the population and it is also commonly used to identify and calculate the margin of error.

Use of Standard Deviation

Standard deviation is commonly used in the industrial and experimental setting to test the model against real-world data. It is often used in the quality control process in different industries. Standard deviation is used to calculate the maximum and minimum values in which certain aspects of the product should fall to ascertain the quality of the product. If the value doesn’t lie in the calculated range, the process needs to be rectified to assure product quality.

Another common utilization of standard deviation is the determination of the difference in the regional climate. It is used to calculate the mean temperature of different cities and various factors are considered while measuring the standard deviation in this case such as the closeness of the city to a water body. Standard deviation can help in effectively evaluating the change in the temperature as compared to the temperature note previously and the difference of temperature among different cities.

Standard deviation is also common used in cases where employers want to identify is salary increase seem fair to all other employees, he can take the average and then calculate the variance. The analysis of the previous salary disbursement can help the employers understand the range of salaries in a specific department

Standard deviations is also used in finance to evaluate the risk associated with price fluctuation. The future is uncertain and this uncurtaining can be easily measured through using standard deviation calculator.

Standard Deviation Formula

Calculating standard deviation by hand is not recommended and you will not see statistician do it with a paper and pen in his/her hand. The calculations are complex and there is a higher chance of error. Another drawback is calculating standard deviation by hand is slow like really slow so standard deviation calculator became essential to calculate SD. If you still want to get into the tedious process of calculating standard division, here is how you can do it.

How to Calculate?

There are two major methods to calculate standard deviation. The first one is highly tedious with a paper, pen, and extra efforts to make sure that you are calculating it the right way. My personal favorite is the calculator one where I can be certain that the results are accurate with putting extra effort in the calculation of each step – just to get the same result.

Let’s assume that we have a series of number like 9, 2, 5, 4, 12, 7, 8, 11. For calculating standard deviation first we have to calculate the mean. The next step is to subtract the mean from each number and then work out the mean of those squared differences. After this step we are going to take the square root of that.

In the formula above μ (the greek letter “mu”) is the mean of all our values.

Example: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4

The mean is:


14020 = 7


μ = 7

Subtract the Mean and square the result

This is the part of the formula that says:

So what is xi ? They are the individual x values 9, 2, 5, 4, 12, 7, etc…

In other words x1 = 9, x2 = 2, x3 = 5, etc.

So it says “for each value, subtract the mean and square the result”, like this

Example (continued):

(9 – 7)2 = (2)2 = 4

(2 – 7)2 = (-5)2 = 25

(5 – 7)2 = (-2)2 = 4

(4 – 7)2 = (-3)2 = 9

(12 – 7)2 = (5)2 = 25

(7 – 7)2 = (0)2 = 0

(8 – 7)2 = (1)2 = 1… etc …

And we get these results:

4, 25, 4, 9, 25, 0, 1, 16, 4, 16, 0, 9, 25, 4, 9, 9, 4, 1, 4, 9

To work out the mean, add up all the values then divide by how many.

First add up all the values from the previous step.

But how do we say “add them all up” in mathematics? We use “Sigma”: Σ

The handy Sigma Notation says to sum up as many terms as we want:
Sigma Notation

We want to add up all the values from 1 to N, where N=20 in our case because there are 20 values:

Example (continued):

Which means: Sum all values from (x1-7)2 to (xN-7)2

We already calculated (x1-7)2=4 etc. in the previous step, so just sum them up:

= 4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9 = 178

But that isn’t the mean yet, we need to divide by how many, which is done by multiplying by 1/N (the same as dividing by N):

Example (continued):

Mean of squared differences = (1/20) × 178 = 8.9

(Note: this value is called the “Variance”)

Take the square root of that:

Example (concluded):

σ = √(8.9) = 2.983

Why Standard Deviation Calculator?

After solving the example equation, you can understand how tiring it gets and why this method is not used by the statisticians anymore. Standard deviation calculator does all the work for you, you just have to add the values and BOOM! You will get the result easily and it will be accurate too. Rather than depending on paper pen, it is far better to use the calculator which is actually designed to consider your mathematical needs.