Measuring spread'

1
Measuring spread.

• A Learning Object produced by
• Sidney Tyrrell, Coventry University,
• as part of her National Teaching Fellowship
  project.

2
Why measure spread?

• Data can be interpreted to give us information.

• Information is what we need for taking good
  decisions.

3
Lots of data is too much to take in

• so we summarise it

• usually by finding an average, which you
  should think of as a typical value,

• and by finding a measure of spread.

4
An average by itself

• does not give the whole picture.
• Try writing down 4 data sets each with a mean
  of 4 and a median of 4.
• Make them as different as possible.
• When you are ready click here to continue the
  presentation.

5
4 data sets with a mean and median of 4

• 4 4 4

0 4 8
0 0 4 4 14
-92 4 100
The essential difference between each set is is
the spread of the numbers.
6
A difference in spread
4
4
4
4
0
8
4
4
14
0
0
4
-92
100
7
A difference in spread
4
4
4
4
0
8
4
4
14
0
0
4
100
-92
8
A difference in spread
4
4
4
4
0
8
4
4
14
0
0
4
100
-92
9
A difference in spread
4
4
4
4
0
8
4
4
14
0
0
4
100
-92
10
A difference in spread
4
4
4
4
0
8
4
4
14
0
0
4
100
-92
11
We can measure spread using

• The range
• The interquartile range - IQR
• The standard deviation
• Click on the measure you want to find out more
  about.

12
The range

• is the difference between the largest and
  smallest values.

0
For the data set 4 4 4 the range is
For the data set 0 4 8 the range is
8 - 0 8
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The range

• is the difference between the largest and
  smallest values.

For the data set -92 4 100 the range is
100-(-92) 100 92 192
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The range a warning

• It is easy to calculate, but
• because it is found from the extreme values it
  may not be helpful,
• and may give quite the wrong idea about the data.
• Click here to return to measures of spread

15
The interquartile range

• Quartiles divide the data into quarters.
• 2 3 5 7 10 11 15 16

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The central quartile is the median

• 2 3 5 7 10 11 15 16

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Statisticians disagree about how to calculate
exactly where the quartiles lie.

• 2 3 5 7 10 11 15 16

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But agree that

• quartiles divide the data into quarters.
• 2 3 5 7 10 11 15 16

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One definition is that

• The lower quartile is the median of the lower
  half of the data set.
• 2 3 5 7 10 11 15 16

The lower quartile is 4
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With an odd number of numbers

• the lower quartile is the median of the lower
  half of the data below the median itself.
• 2 3 5 7 9 10 11 15 16

The lower quartile is 4
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The interquartile range

• is the distance between the lower
  quartile and the upper quartile.
• 2 3 5 7 10 11 15 16

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The interquartile range

• gives the spread of the middle 50 of the data
  often the bit you are most interested in.
• It is not affected by any extreme values.

23
The interquartile range

• Is used in constructing boxplots.
• The quartiles define the ends of the box.
• Click here to return to measures of spread

0
20
5
15
10
40
25
30
35
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The standard deviation, s

• is an important measure of spread for
  statisticians
• and frequently misunderstood.
• You can think of it as a typical deviation from
  the mean.
• Hence standard deviation.

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To calculate the standard deviation

• you need to start by finding the deviations or
  differences.
• From what?

From the mean.
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An example take the numbers 7,8,9

• Find the differences from the mean

• The mean is 8
• The differences are

9 8 1
8 8 0
7 8 -1
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Now you have the differences
Yes it will, whatever 3 numbers you choose!

• You need to find a standard deviation which
  suggests averaging them.
• This could be tricky as -1 0 1 0
• Will the answer always be 0?

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We get round this problem

• By squaring the differences first
• Then averaging them.

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square the differences
7 8 -1
8 8 0
9 8 1
( -1)2 1
(0)2 0
(1)2 1
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find the average

• 1

0
1


2
2 / 3
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• take the square root.

the square root of 2/3 is 0816.
this is the population standard deviation 0816.
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The population standard deviation

• is used where your data is the complete
  population being studied.

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Usually we are finding the standard deviation of
a sample a subset of the population.

• The sample standard deviation,s, is calculated
  slightly differently.
• At the stage where we find the average of the
  squared deviations we divide by one less than the
  sample size.

34
The same example the numbers 7,8,9

• Find the differences from the mean

• The mean is 8
• The differences are

1
0
-1
35
square the differences
( -1)2 1

• (1)2 1

(0)2 0
36
find the average
This time we divide by one less than the sample
size.
37
find the average

• 1

0
1


2
2 / 2 1
divide by one less than the sample size
38

• take the square root.

the square root of 1 is 1
This is the sample standard deviation.
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The variance

• Is the square of the standard deviation.
• The name reminds us it is a measure of
  variability in the data.

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Calculate the standard deviation by

• Finding the differences from the mean

• squaring the differencesnd

• finding their averageand

• then taking the square root.and

41
Finally a reminder

• Why measure spread?
• Finding the spread helps us understand our data
  better
• which should lead to better decisions and
  analysis.