The standard deviation for grouped and ungrouped data calculation (step by step illustration)

The standard deviation for grouped and ungrouped data calculation (step by step illustration)

Non-Grouped Data
Non-grouped data is just a list of values. The standard deviation is given by the formula:

Standard Deviation

s means ‘standard deviation’.
S means ‘the sum of’.
x bar means ‘the mean’

black calculator near ballpoint pen on white printed paper
Photo by Pixabay on

Find the standard deviation of 4, 9, 11, 12, 17, 5, 8, 12, 14
First work out the mean: 10.222

Now, subtract the mean individually from each of the numbers given and square the result.

This is equivalent to the (x – x bar)² step. x refers to the values given in the question.

(x – x bar )238.71.490.603.1645.927.34.943.1614.3

Now add up these results (this is the ‘sigma’ in the formula): 139.55
Divide by n. n is the number of values, so in this case, is 9. This gives us: 15.51
And finally, square root this: 3.94

The standard deviation can usually be calculated much more easily with a calculator and this may be acceptable in some exams. On my calculator, you go into the standard deviation mode (mode ‘.’).

Then type in the first value, press ‘data’, type in the second value, press ‘data’.

Do this until you have typed in all the values, then press the standard deviation button (it will probably have a lower case sigma on it). Check your calculator’s manual to see how to calculate it on yours.

NB: If you have a set of numbers (e.g. 1, 5, 2, 7, 3, 5 and 3), if each number is increased by the same amount (e.g. to 3, 7, 4, 9, 5, 7 and 5), the standard deviation will be the same and the mean will have increased by the amount each of the numbers was increased by (2 in this case).

This is because the standard deviation measures the spread of the data. Increasing each of the numbers by 2 does not make the numbers any more spread out, it just shifts them all along.

Grouped Data
When dealing with grouped data, such as the following:


the formula for standard deviation becomes:
Standard Deviation
Try working out the standard deviation of the above data. You should get an answer of 1.32 .
You may be given the data in the form of groups, such as:

3.5 – 4.59
4.5 – 5.514
5.5 – 6.522
6.5 – 7.511
7.5 – 8.517

In such a circumstance, x is the midpoint of groups.

arithmetic mean

arithmetic mean is the average of all values in a set of distribution.

it is determined by adding up all values and dividing by the sum of observations added.

arithmetic mean is used to assess the distribution value whether was high or low

advantages of the arithmetic mean

  • it is easy to calculate and the majority of people understand it
  • it is used to check the values if high or low
  • it can be used for further calculation; for example arithmetic mean is used to calculate standard deviation

disadvantages of arithmetic mean

  • arithemetic mean has the big weakness of being pulled toward an outlier (extreme score)
  • it need high mathematical knowledge to calculate arithematic mean for grouped data


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