# Standard Deviation

 From: Alex Watson Date: 21 April 1999 Subject: Standard Deviation of Grouped Data Please help me understand how to find the standard deviation of grouped data.

### Maths Help suggests:

We suggest you set out your calculation using a table with headings:

 Class Midpoint (x) Frequency (f) f × x f × x2

Notice these are all the headings you need to work out the mean,
plus an extra "fx2" column.

Work out the mean using:

(Click here to see an example of working out the mean of a grouped distribution)

... and the variance using this formula:

Square root the variance to calculate the standard deviation.

Here is a worked example:

100 parcels were weighed. Their masses were as follows:

 Mass (g) 0- 250- 500- 750- 1000-1250 Frequency 7 18 33 26 16

Estimate the mean and standard deviation of the masses.

Draw a table like this:

 Mass, g Midpoint (x) Frequency (f) f × x f × x2 0-250 125 7 875 765625 250-500 375 18 6750 2531250 500-750 625 33 20625 12890625 750-1000 875 26 22750 19906250 1000-1500 1250 16 20000 25000000 Totals: 100 71000 61093750

The estimate of the mean is:

mean = 71000 ÷ 100 = 710 grams.

The estimate of the variance is:

var = 61093750 ÷ 100 - (710)2 = 106837.5

...and the standard deviation is the square root of the variance:

standard deviation = sqrt(106837.5) = 326.86 grams (2dp)