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.
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:
|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.
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: 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.
As I stated earlier, calculating standard deviation is much easier than the formula depicts. Here are the 4 simple steps:
To see just how easy the above steps are, let me walk you through an example.
To illustrate how easy it is to calculate variance and standard deviation, I will use the following data set:
Note: Decimals in this example are rounded to 2 places before they are displayed on the screen.
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|
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|
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|
|Sample Variance (s2)|
|s2 =||Σ (X - X)2||=||26||= 2.89|
|n - 1||9|
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.
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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.