Standard Deviation (Edexcel GCSE Statistics)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
Standard Deviation
What is the standard deviation of a data set?
The standard deviation of a data set is a measure of dispersion (i.e. a measure of spread)
It measures how the data is spread out relative to the mean
If the standard deviation is small then most data values are close to the mean
If the standard deviation is large then many data values will be further away from the mean
If the data has units (seconds, cm, etc.), then the standard deviation has the same units as the values in the data set.
The Greek letter (lower case sigma) is often used for standard deviation
How do I calculate the standard deviation for a data set?
There are two different formulas you can use to calculate the standard deviation
Usually the second formula will be the quickest one to use
But make sure you know how to use both of them
In this formula:
is the number of values in the data set
is the mean of the data set
is 'any value' in the data set
In this formula:
is the number of values in the data set
is the sum of all the data values
is the sum of the squares of all the data values
Sometimes a question will give you the values of and for a data set
In that case definitely use this formula!
Both formulas are on the exam formula sheet
So you don't need to remember them
You just need to know how to use them
Examiner Tips and Tricks
Your calculator may be able to calculate the standard deviation for a list of data values
Worked Example
For the following set of data values
6 9 2 11 5
(a) Calculate the mean.
Add up the values and divide by the number of values (5)
mean = 6.6
(b) Calculate the standard deviation using .
It is easiest to set up a table to work out the different values
total |
Now we have all the values to put into the formula
standard deviation = 3.14 (3 s.f.)
(c) Calculate the standard deviation using .
It is easiest to set up a table to work out the different values
total |
---|
Now we have all the values to put into the formula
standard deviation = 3.14 (3 s.f.)
Standard Deviation from a Table
How do I find the standard deviation for data in a table?
A data set may be presented in a table of data values and associated frequencies
In this case the formulas to use are different
These formulas are not on the exam formula sheet
So you need to remember them
But note that they are closely related to the basic formulas
Usually the second formula will be the quickest one to use
But make sure you know how to use both of them
In this formula:
is the mean of the data set
is 'any value' in the data set
is the frequency associated with a particular data value
is the sum of all the frequencies (this is the same as the total number of data values in the data set)
In this formula:
is 'any value' in the data set
is the frequency associated with a particular data value
is the sum of for all the data values in the set
is the sum of for all the data values in the set
is the sum of all the frequencies (this is the same as the total number of data values in the data set)
Sometimes a question will give you the values of and for a data set
In that case definitely use this formula!
Examiner Tips and Tricks
Your calculator may be able to calculate the standard deviation for a list of data values and their associated frequencies
Worked Example
Kira collected data about the numbers of pet rabbits owned by the members of her local house rabbits association. This data is shown in the following table:
Number of rabbits | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
Frequency | 2 | 6 | 4 | 6 | 2 |
Work out the standard deviation of this data set.
Method 1: using
It is easiest to set up a table to work out the different values
number, | |||
---|---|---|---|
total |
So , and
That gives us everything we need to put into the formula
standard deviation = 1.18 (3 s.f.)
Method 2: using
It is easiest to set up a table to work out the different values
number, | |||||
---|---|---|---|---|---|
total |
Now that we have the sum of the f and fx columns we can work out the mean
The sum of the fx column is the sum of all the data values
And the sum of the f column is the total number of data values
Now we can complete the rest of the table
number, | |||||
---|---|---|---|---|---|
total |
So and
That gives us everything we need to put into the formula
standard deviation = 1.18 (3 s.f.)
Standard Deviation for Grouped Data
How do I find the standard deviation for grouped data?
For grouped data we no longer have access to the original data values
Therefore we can only find an estimate for the standard deviation
To calculate an estimate for the standard deviation for a set of grouped data:
Use the same formulas as used for data in a table
See the 'Standard Deviation from a Table' spec point
But use the midpoints of the class intervals as the data values
i.e. as the values for in the formulas
The mean will also be an estimate where it appears in a formula
Examiner Tips and Tricks
Your calculator may be able to calculate an estimate for the standard deviation from a list of midpoints and their associated frequencies
Worked Example
Kira collected data about how long the pet rabbits, owned by the members of her local house rabbits association, took to eat their lunch. This data is shown in the following table:
Time, t (minutes) | 0 ≤ t < 3 | 3 ≤ t < 6 | 6 ≤ t < 9 | 9 ≤ t < 12 |
---|---|---|---|---|
Frequency | 1 | 5 | 8 | 6 |
Work out an estimate for the standard deviation of this data set.
Method 1: using
It is easiest to set up a table to work out the different values
Remember to use the class interval midpoints as the x values
midpoint, | |||
---|---|---|---|
total |
So , and
That gives us everything we need to put into the formula
standard deviation = 2.59 (3 s.f.)
Method 2: using
It is easiest to set up a table to work out the different values
Remember to use the class interval midpoints as the x values
midpoint, | |||||
---|---|---|---|---|---|
total |
Now that we have the sum of the f and fx columns we can work out the estimated mean
The sum of the fx column is the estimated sum of all the data values
And the sum of the f column is the total number of data values
Now we can complete the rest of the table
midpoint, | |||||
---|---|---|---|---|---|
total |
So and
That gives us everything we need to put into the formula
standard deviation = 2.59 (3 s.f.)
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