- 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. |

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

Class | Midpoint (x) |
Frequency (f) |
f × x |
f × x^{2} |

Notice these are all the headings you need to work out the mean,

plus an extra "*fx*^{2}" 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:
Estimate the mean and standard deviation of the masses. |

Draw a table like this:

Mass, g | Midpoint (x) | Frequency (f) | f × x | f × x^{2} |

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)

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