Standard Deviation Calculator forPopulation or Sample Data Sets

The Standard Deviation Calculator on this page will instantly calculate the sum of squared differences and its square root from an entered or pasted-in data set.

This free online statistics calculator will calculate the variance and standard deviation for either a population or a sample data set.

You can either enter the numbers in the data set one at a time, or you can copy and paste an existing data set (if separated by spaces, commas, line returns, or any combination thereof), or you can enter a number and its frequency (12x4, 8x6, 9x4).

Plus, unlike other online statistics calculators, this calculator will generate and display the step-by-step process used to calculate the results.

Don't Let the Formula Scare You!

In my opinion, if the "powers that be" really wanted to entice kids into embracing mathematics, they would use kittens, puppies, bunny rabbits, etc., as math symbols, not Greek (geek) letters.

Want to scare a math-challenged student into closing the math textbook so fast that it blows out the candles in the room? Show them this:

σ2
 N (Xi - μ)2 Σ i=1
N

σ = √ σ2

That's the scary-looking formula for calculating the variance (σ2) and standard deviation (σ) of a population data set.

But wait! Keep your candles burning!

As you will see in the example further down the page, the steps to calculating variance and standard deviation are much easier than trying to decipher the Greek/geek formula.

What is Standard Deviation?

Standard Deviation is simply one of several methods for summarizing the dispersion of the values in a set of data.

Specifically, standard deviation is the square root of the variance, which attempts to summarize the variability or dispersion of values relative to the mean of the entire set.

A small standard deviation indicates the values are tightly grouped around the mean (average) of the data set.

A large standard deviation indicates the values are not tightly grouped around the mean (average) of the data set.

Population Vs Sample Statistics

Population: The data set is the total set of elements of interest for a given problem. Population parameters in the formulas on this page are denoted by σ and μ.

Sample: The data set represents only a fraction of the population as defined above. Sample parameters in the formulas on this page are denoted by s and X.

4 Simple Steps

As I stated earlier, calculating standard deviation is much easier than the formula depicts. Here are the 4 simple steps:

1. Find the mean of the data set.
2. Find the sum of the squared differences from the mean.
3. Divide the result in step #2 by n (population) or n - 1 (sample), where n is the number of items in the set.
4. Find the square root of the result in step #3.

To see just how easy the above steps are, let me walk you through an example.

Example Problem

To illustrate how easy it is to calculate variance and standard deviation, I will use the following data set:

5, 4, 7, 9, 6, 8, 7, 5, 4, 5

Note: Decimals in this example are rounded to 2 places before they are displayed on the screen.

Step #1: Calculate the mean

The mean is the average of all numbers in a data set. To calculate the mean of set of numbers, you add all of the items together and then divide that result by the number of items within the set.

 Data set: 5, 4, 7, 9, 6, 8, 7, 5, 4, 5 Data set contains 10 items Mean = (5 + 4 + 7 + 9 + 6 + 8 + 7 + 5 + 4) ÷ 10 Mean = 60 ÷ 10 Mean = 6

Step #2: Find the sum of the squared distances from the mean.

For each item in the data set we: (1) subtract the mean of the entire set from the item, (2) square the result, and then (3) sum all of the squared results, like this:

Count n X X - μ (X - μ)2
1 5 5 - 6 = -1 (-1)2 = 1
2 4 4 - 6 = -2 (-2)2 = 4
3 7 7 - 6 = 1 (1)2 = 1
4 9 9 - 6 = 3 (3)2 = 9
5 6 6 - 6 = 0 (0)2 = 0
6 8 8 - 6 = 2 (2)2 = 4
7 7 7 - 6 = 1 (1)2 = 1
8 5 5 - 6 = -1 (-1)2 = 1
9 4 4 - 6 = -2 (-2)2 = 4
10 5 5 - 6 = -1 (-1)2 = 1
n = 10 Sum = 60   Σ (X - μ)2 = 26

Step #3: Calculate the variance.

This step depends on whether or not the set you are working with represents the total data (population) or partial data (sample). Here is how you calculate the variance from the results in step #2 for either case (μ and X are both symbols for Mean, with the only real difference indicated by red text):

 Population Variance (σ2) σ2 = Σ (X - μ)2 = 26 = 2.60 n 10

 Sample Variance (s2) s2 = Σ (X - X)2 = 26 = 2.89 n - 1 9

Step #4: Calculate the square root of the variance.

This step is basically the same for both the population and the sample standard deviation calculations. The only difference is that the Variance was computed differently in Step #3 (n in denominator vs n - 1):

 Population Standard Deviation (σ) σ = √ σ2 = √ 2.60 = 1.61

 Sample Standard Deviation (s) s = √ s2 = √ 2.89 = 1.70

See how easy that was?

With that, let's use the Standard Deviation Calculator to calculate the population or sample variance and standard deviation of a set of data.

Standard Deviation Calculator

Instructions: Enter each number in the data set separated by a comma, a space, a line return, or any combination of the three (you can also paste a set of numbers into the number field as long as they are separated by one of the mentioned delimiters).

Note that you can also enter multiples of the same number by entering the number, followed by a lower case "x", followed by the multiplier (no spaces before or after the x). For example, to enter a 3 four times, you would enter 3x4.

Once you have all of the numbers entered in the number set, click the "Calculate Standard Deviation" button.

Mouse over the blue question marks for a further explanation of each entry field. More in-depth explanations can be found in the glossary of terms located beneath the Standard Deviation Calculator.

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Enter or paste numbers in data set (separated by comma, space, or return):
Total number of items in set:
Sum of items in set:
Mean:
Population variance:
Population standard deviation:
Sample variance:
Sample standard deviation:
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Standard Deviation Calculator Glossary of Terms

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Data set: Enter each element of the data set (or paste a copied data set) into this text box. Be sure each number is separated by a space, a comma, a line return, or any combination of the three. Also, if you wish to enter a frequency for each number, enter the number, followed by a lowercase x, followed by the multiplier (for 3 number twos, enter 2x3).

Total number of items in set (n or N): This is the total number of items detected in the data set field.

Sum of items in set (Σ): This is the sum of all items in the set.

Mean (μ or X): This is the average or mean of the items within the data set. This is calculated by summing the items within the set, and then dividing the sum by the number of items.

Population variance (σ2): This is the variance of the data set (also called the Mean of Squared Differences) if the items entered represent the entire (population) data set.

Population standard deviation (σ): This is the population standard deviation, which is calculated by finding the square root of the population variance.

Sample variance (s2): This is the variance of the data set (also called the Mean of Squared Differences) if the items entered represent only a portion (sample) of the data.

Sample standard deviation (s): This is the population standard deviation, which is calculated by finding the square root of the sample variance.

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