How to find Standard Deviation

Math
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By far the most common measure of variation for numerical data in statistics is the standard deviation. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation. It’s not reported nearly as often as it should be, but when it is, you often see it in parentheses, like this: ( s = 2.68).

The formula for the sample standard deviation ( s ) is

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where x i is each value is the data set, x -bar is the mean, and n is the number of values in the data set. To calculate s , do the following steps:

  1. Calculate the average of the numbers,image

  2. Subtract the mean from each number (x)
    image

  3. Square each of the differences,
    image

  4. Add up all of the results from Step 3 to get the sum of squares,
    image

  5. Divide the sum of squares (found in Step 4) by the number of numbers minus one; that is, ( n – 1).
    image

  6. Take the square root to get the result
    image

  7. which is the sample standard deviation, s . Whew!

At the end of Step 5 you have found a statistic called the sample variance, denoted by s 2. The variance is another way to measure variation in a data set; its downside is that it’s in square units. If your data are in dollars, for example, the variance would be in square dollars — which makes no sense. That’s why you proceed to Step 6. Standard deviation has the same units as the original data.

Standard deviation formula example:

Suppose you have four quiz scores: 1, 3, 5, and 7. The mean is 16 ÷ 4 = 4 points. Subtracting the mean from each number, you get (1 – 4) = –3, (3 – 4) = –1, (5 – 4) = +1, and (7 – 4) = +3. Squaring each of these results, you get 9, 1, 1, and 9. Adding these up, the sum is 20. In this example, n = 4, and therefore n – 1 = 3, so you divide 20 by 3 to get 6.67, which is the variance. The units here are “points squared,” which obviously makes no sense. Finally, you take the square root of 6.67, to get 2.58. The standard deviation for these four quiz scores is 2.58 points.

Because calculating the standard deviation involves many steps, in most cases you have a computer calculate it for you. However, knowing how to calculate the standard deviation helps you better interpret this statistic and can help you figure out when the statistic may be wrong.

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How to Find Standard Deviation

How to Find Standard Deviation
How to Find Standard Deviation765×462 135 KB

A common way to quantify the spread of a set of data is to use the [sample standard deviation]. Your calculator may have a built-in standard deviation button, which typically has an sx on it. Sometimes it’s nice to know what your calculator is doing behind the scenes.

The steps below break down the formula for a standard deviation into a process. If you’re ever asked to do a problem like this on a test, know that sometimes it’s easier to remember a step-by-step process rather than memorizing a formula.

The Process

  1. Calculate the mean of your data set.
  2. Subtract the mean from each of the data values and list the differences.
  3. Square each of the differences from the previous step and make a list of the squares.
  4. In other words, multiply each number by itself.
  5. Be careful with negatives. A negative times a negative makes a positive.
  6. Add the squares from the previous step together.
  7. Subtract one from the number of data values you started with.
  8. Divide the sum from step four by the number from step five.
  9. Take the square root of the number from the previous step. This is the standard deviation.
  10. You may need to use a basic calculator to find the square root.
  11. Be sure to use [significant figures] when rounding your final answer

A Worked Example

Suppose you’re given the data set 1, 2, 2, 4, 6. Work through each of the steps to find the standard deviation.

  1. Calculate the mean of your data set. The mean of the data is (1+2+2+4+6)/5 = 15/5 = 3.
  2. Subtract the mean from each of the data values and list the differences. Subtract 3 from each of the values 1, 2, 2, 4, 6
    1-3 = -2
    2-3 = -1
    2-3 = -1
    4-3 = 1
    6-3 = 3Your list of differences is -2, -1, -1, 1, 3
  3. Square each of the differences from the previous step and make a list of the squares.You need to square each of the numbers -2, -1, -1, 1, 3
    Your list of differences is -2, -1, -1, 1, 3
    (-2)2 = 4
    (-1)2 = 1
    (-1)2 = 1
    12 = 1
    32 = 9Your list of squares is 4, 1, 1, 1, 9
  4. Add the squares from the previous step together. You need to add 4+1+1+1+9 = 16
  5. Subtract one from the number of data values you started with. You began this process (it may seem like a while ago) with five data values. One less than this is 5-1 = 4.
  6. Divide the sum from step four by the number from step five. The sum was 16, and the number from the previous step was 4. You divide these two numbers 16/4 = 4.
  7. Take the square root of the number from the previous step. This is the standard deviation. Your standard deviation is the square root of 4, which is 2.

Tip: It’s sometimes helpful to keep everything organized in a table, like the one shown below.

We next add up all of the entries in the right column. This is the sum of the squared deviations. Next divide by one less than the number of data values. Finally, we take the square root of this quotient and we are done…