Standard Deviation Calculator

The calculator below is extremely easy to use. Simply enter a set of numbers and click on the calculate button. Please make sure you separate each number with a comma else it will not work. You will also get the result for mean, variance and population variance. Moreover the calculator will output all the steps of the calculation which is very useful for students!

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What Is Standard Deviation?

The standard deviation is a measure of spread. When comparing sets of data that have the same mean, it's useful for situations where the range is more spread out. For instance, take the two following sets of data:

14, 16, 18, 19, 20, 21
4, 8, 17, 20, 21, 38

Both of these sets of data have a mean of 18. However, it's clear that the second set of data contains a wider spread compared to the first. The first set of numbers would have a low standard deviation because the values aren't spread out as much as the first.

How To Calculate Standard Deviation?

To calculate the standard deviation, we have to follow these steps:

  1. Calculate the mean.
  2. For each number, subtract the mean and square the result.
  3. Sum up the values from step 2.
  4. Divide step 3 by the number of values.
  5. Find square root of the result.

Here is the formula for standard deviation:

standard deviation formula

Let's use another example with the number set 8, 10, 13, 15, 18, 19, 21, 24.

Step 1: Calculate the mean (finding x)
Here, we simply find the mean. We sum up the number set and divide by the number of values.
8 + 10 + 13 + 15 + 18 + 19 + 21 + 24 = 128
128 / 8 = 16
x = 16.

Step 2: Subtract the mean from each number (finding x-x^2 )
Here, we find the distance from each value to the mean (this is known as the deviations) and square each distance. This table will illustrate this step.

Value (x) Distance (x-x) Squared (x-x^2)
8 8 64
10 6 36
13 3 9
15 1 1
18 2 4
19 3 9
21 5 25
24 8 64

Step 3: Sum up the values from step 2 (finding x-x^2)
This part is simple. The E symbol in the equation means "sum".
64 + 36 + 9 + 1 + 4 + 9 + 25 + 64 = 212
x-x^2 = 212

Step 4: Divide by the number of values
Here, we divide the result from step 3 by the number of values.
212 / 8 = 26.5
This number is also known as the variance.

Step 5: Find the square root of the result
In this last step, we simply find the square root of step 4.
sqrt(26.5) = 5.148
Thus, the standard deviation for our set of data is 5.148

Quick Recap
So let's have a quick recap of how we found the standard deviation and the variance.
The set of data was 8, 10, 13, 15, 18, 19, 21, 24.
We first calculate the mean (16).
We then subtracted the mean from each result and squared it, giving us 64, 36, 9, 1, 4, 9, 25, 64.
We then summed these numbers up, totalling 212.
Next, we divided it by the number of values, giving us 26.5. This is the variance.
Lastly, we found the square root of 26.5 which gave us 5.148. This is the standard deviation.

What Is Population Standard Deviation

The above calculation is also known as the population standard deviation and only applies if we have a full set of data. For example, you could imagine that those values are the lengths of different planks of wood and you're trying to find the standard deviation among all those planks. However, what if you had 1,000 planks of wood? It would take far too long because you would need to measure each individual plank and input their lengths into a spreadsheet because it would be too slow to calculate by hand.

Because of this, there is the alternative of calculating the sample standard deviation. For example, if you didn't have time to measure 1,000 planks of wood and only measured around 20, you could use a different formula to measure the standard deviation.

As you can see, the only change is subtracting 1 from the number of values. This is known as Bessel's correction and is used to estimate the population variance from a sample when the population mean is unknown. Another good way of picturing the uses for sample standard deviation is when unknown quantities are involved. For instance, imagine trying to find the standard deviation of heights among people in a city. It would be impractical to measure every single person, so you would only need to measure a small sample and use Bessel's correction to work out the sample standard deviation instead.

Let's use the same set of numbers to work out the sample standard deviation instead.

The initial set of data was 8, 10, 13, 15, 18, 19, 21, 24.
We'll change it to 10, 13, 15, 18, 19. We will omit the 8, 21 and 24.
The steps are the same, but with the exception of dividing by n-1 instead of n.

Step 1: Calculate the mean (finding x)

10 + 13 + 15 + 18 + 19 = 75
75 / 5 = 15
x = 15.

Step 2: Subtract the mean from each number (finding x-x^2 )
The values are 25,4,0,9,16

Step 3: Sum up the values from step 2

25 + 4 + 0 + 9 + 16 = 54
x-x^2 = 54

Step 4: Divide by the number of values minus 1
54 / (5 - 1) = 13.5

Step 5: Find the square root of the result (finding the standard deviation)

sqrt(13.5) = 3.674
Thus, the sample standard deviation for our set of data is 3.674
After applying Bessel's correction to find the sample standard deviation, we are left with the following:
Variance: 13.5
Sample Standard Deviation: 3.674

To compare, here were the results for the full set of data:
Variance: 26.5
Population Standard Deviation: 5.148

When taking a sample, we lose some accuracy. However, this is natural considering we have less data to work with. The reason we take samples is that it's often cheaper and more efficient than using larger sets of data. It takes more time to collect, the calculations take longer and it's ultimately more costly. Population standard deviation is usually only used when you can be sure you've accounted for every value.

For more information check out:
http://en.wikipedia.org/wiki/Standard_deviation