Probability and Statistics > Basic Statistics > Sample variance and standard deviation

## How to find the sample variance and standard deviation

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The sample variance and standard deviation are ways to measure how spread out a result is. In order to find standard deviation, you have to find the variance first; Standard deviation is just the square root of variance. The variance formula can be tricky to use—especially if you are rusty on order of operations. By far the easiest way to find the variance and standard deviation is to use an online standard deviation calculator . You can also use it to check your work. Have to work the formula by hand? Read on!

## How to find the sample variance and standard deviation: Variance

Step 1: Add up the numbers in your given data set. For example, let’s say you were given data for trees in California (heights in feet):

3, 21, 98, 203, 17, 9

Add them up:

3 + 21 + 98 + 203 + 17 + 9 = 351

Step 2: Square your answer:

351 × 351 = 123,201

…and divide by the number of items. We have 6 items in our example so:

123,201 / 6 = 20,533.5

Set this number aside for a moment.

Step 3: Take your set of original numbers from Step 1, and square them individually this time:

3 × 3 + 21 × 21 + 98 × 98 + 203 × 203 + 17 × 17 + 9 × 9

Add those numbers (the squares) together:

9 + 441 + 9604 + 41209 + 289 + 81 = 51,633

Step 4: Subtract the amount in Step 2 from the amount in Step 3.

51,633 – 20,533.5 = 31,099.5

Set this number aside for a moment.

Step 5: Subtract 1 from the number of items in your data set. For our example:

6 – 1 = 5

Step 6: Divide the number in Step 4 by the number in Step 5. This gives you the **variance**:

31,099.5 / 5 = 6,219.9

### How to find the sample variance and standard deviation: Standard Deviation

Step 7: Take the square root of your answer from Step 6. This gives you the **standard deviation**:

√6,219.9 = 78.86634

*That’s it!*

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step solutions, just like this one!

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I felt that this article was ok but it could have been just alittle bit thorough on this subject since we have done a good bit of problems regarding the sample ariance and the standard deviation in statistics. There were some good points just to let you know.

You put step 3 twice. It should be step 4 on the second “step 3”

You are completely right. Thanks!

So what’s the variance??

Kim,

See Step 6.

Regards,

Stephanie

I never though it could be so sample! I truly thank Professor Stephaine Deviant MAT. Thanks