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

## How to find the sample variance and standard deviation

Watch the video or read the steps below:

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!

**Important note: **The standard deviation formula is **slightly different **for populations and samples (a portion of the population). If you have a population, you’ll be dividing by “n” (the number of items in your data set). However, if you have a sample (which is the case for most statistics questions you’ll get in class!) you’ll need to divide by n-1.

## How to Find Standard Deviation: Example 1

Your paychecks for the last few weeks are: $600, $470, $430, $300 and $170. What is the standard deviation?

Step 1: Add up all of the numbers:

170 + 300 + 430 + 470 + 600 = 1970

Step 2: Square the total, and then divide by the number of items in the data set

1970 x 1970 = 3880900

3880900 / 5 = 776180

Step 3: Take your set of original numbers from step 1, and square them individually this time. Then add them all up:

(170 x 170) + (300 x 300) + (430 x 430) + (470 x 470) + (600 x 600) = 884700

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

884700 – 776180 = 108520

Step 5: I subtracted 1 from the number of items in my data set:

5 – 1 = 4

Step 6: Divide the number in step 4 by the number in step 5:

108520 / 4 = 27130

This is my Variance!

Step 7: Take the square root of the number from step 6 (the Variance),

√(27130) = 164.7118696390761

This is my Standard Deviation!

## How to Find Standard Deviation: Example 2

This example uses the same formula, it’s just a slightly different way of working it.

You survey households in your area to find the average rent they are paying. Find the standard deviation from the following data:

$1550, $1700, $900, $850, $1000, $950.

Step 1: Find the mean:

($1550 + $1700 + $900 + $850 + $1000 + $950)/6 = $1158.33

Step 2: Subtract the mean from each value. This gives you the differences:

$1550 – $1158.33 = $391.67

$1700 – $1158.33 = $541.67

$900 – $1158.33 = -$258.33

$850 – $1158.33 = -$308.33

$1000 – $1158.33 = $158.33

$950 – $1158.33 = $208.33

Step 3: Square the differences you found in Step 3:

$391.67^{2} = 153405.3889

$541.67^{2} = 293406.3889

-$258.33^{2} = 66734.3889

-$308.33^{2} = 95067.3889

$158.33^{2} = 25068.3889

$208.33^{2} = 43401.3889

Step 4: Add up all of the squares you found in Step 3 and divide by 5 (which is 6 – 1):

(153405.3889 + 293406.3889 + 66734.3889 + 95067.3889 + 25068.3889 + 43401.3889) / 5 = 135416.66668

Step 5: Find the square root of the number you found in Step 4 (the variance):

√135416.66668 = 367.99

The standard deviation is 367.99.

## How to Find Standard Deviation using Technology

As you can probably tell, finding a standard deviation by hand can be very lengthy for large data sets. You’ll probably want to use technology.

- How to find standard deviation on the TI 89.
- How to find standard deviation with an online standard deviation calculator.
- How to find standard deviation on the TI 83.
- How to find standard deviation in Excel.

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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