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The Median of
Grouped Data
Christine Crisp
Teach A Level Maths
Statistics 1
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The Median of Grouped Data
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Statistics 1AQA
EDEXCEL
MEI/OCR
OCR
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The Median of Grouped Data
Finding the median of raw data is easy.
e.g.1 Find the median of3231172879233413
3432312823171397
The data must be put in numerical order:
The median is the middle value, which we can see is the5thvalue, so,
median = 23
The formula telling us which value we want is ,where nis the number of data items.
2
1n
If nis an even number, we average the 2middle values.
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The Median of Grouped Data
e.g.2 Find the median of
x 1 2 3 4 5f 4 7 9 6 2
There are 28observations so using we need:2
1n
th5142
128
Accumulating the frequencies:
20114Cu. f54321x
so, the 14thand 15thobservations are both 3.
The median is3.
We need to average the 14thand 15thnumbers.
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The Median of Grouped Data
With a grouped distribution, we can only estimatethe median.
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
The first2classes have a cumulative frequency of
7,
With a grouped distribution, we can only estimatethe median.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
so to reach 11, we need 4more.The first2classes have a cumulative frequency of
7,
With a grouped distribution, we can only estimatethe median.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
The first2classes have a cumulative frequency of
7,so to reach 11, we need 4more.
With a grouped distribution, we can only estimatethe median.
The 3rdclass has afrequency of 5so we need to go part-way along this class.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
The first2classes have a cumulative frequency of
7,so to reach 11, we need 4more.
With a grouped distribution, we can only estimatethe median.
The 3rdclass has afrequency of 5so we need to go part-way along this class.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
Coming up with our own method, needing 4out of thefrequency of 5, we would go 4/5thalong the class.
The class is 10wide
The 3rdclass has afrequency of 5so we need to go part-way along this class.
The first2classes have a cumulative frequency of
7,so to reach 11, we need 4more.
With a grouped distribution, we can only estimatethe median.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
Coming up with our own method, needing 4out of thefrequency of 5, we would go 4/5thalong the class.
The class is 10wide and 4/5thof 10is 8.
The first 2classes have a cumulative frequency of 7,so to reach 11, we need 4more.
With a grouped distribution, we can only estimatethe median.
The 3rdclass has afrequency of 5so we need to go part-way along this class.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
Coming up with our own method, needing 4out of thefrequency of 5, we would go 4/5thalong the class.
The class is 10wideWe would go 8along the class, which starts at 205,
and 4/5thof 10is 8.
The first 2classes have a cumulative frequency of 7,so to reach 11, we need 4more.
With a grouped distribution, we can only estimatethe median.
The 3rdclass has afrequency of 5so we need to go part-way along this class.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
Coming up with our own method, needing 4out of thefrequency of 5, we would go 4/5thalong the class.
The class is 10wide and 4/5thof 10is 8.We would go 8along the class, which starts at 205,
The first 2classes have a cumulative frequency of 7,so to reach 11, we need 4more.
With a grouped distribution, we can only estimatethe median.
The 3rdclass has afrequency of 5so we need to go part-way along this class.
205
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
Coming up with our own method, needing 4out of thefrequency of 5, we would go 4/5thalong the class.
The class is 10wide
giving 28 5.
and 4/5thof 10is 8.We would go 8along the class, which starts at 205,
This is a reasonable estimate but not quite the accepted
method which is called Linear Interpolation.
The first 2classes have a cumulative frequency of 7,so to reach 11, we need 4more.
With a grouped distribution, we can only estimatethe median.
The 3rdclass has afrequency of 5so we need to go part-way along this class.
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
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The Median of Grouped Data
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
Our reasoning was:
The median is the 11thobservation.
There are 7 in class 1 and class 2.We need to add 4 to reach the median.
We need to assume the data are evenly distributed inthe 3rdclass and it can be shown that this means themedian is found at 35along the class not 4.
You dont need to know the reason for this but Ive putan explanation at the end of the presentation.
you will get the correct answer.2
n
2
1n
If you liked our reasoning to get to the estimate, stick to
it, but in locating the median use instead of and
With a grouped distribution, we can only estimatethe median.
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The Median of Grouped Data
If you prefer to use a formula to find the estimate ofthe median, the formula is given by
where,wf
F
n
2median l.c.b.
2
n
l.c.b.
Ffw
520 510
7
5
10
105
7510
520
527 median
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
With a grouped distribution, we can only estimatethe median.
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The Median of Grouped Data
To use linear interpolation to find an estimate of themedian for a grouped frequency distribution, we
locate the class containing the median usingtotal frequency divided by 2,
Fis the cumulative frequencies up to the classcontaining the median,
fis the frequency of the class containing the median,
wis the width of the class containing the median.
( think of n/2 Fas the distance along the class to the
median ),
usew
f
Fn
2median l.c.b. where,
SUMMARY
or, use reasoning to save the need to remember
the formula.
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The Median of Grouped Data
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
The median is in the 2ndclass.Method 1: Without the formula,
The 1stclass has 7. . .
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The Median of Grouped Data
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
The median is in the 2ndclass.Method 1: Without the formula,
The 1stclass has 7. . .
There are 10in the 2ndclass . . .
so we need to go 157 = 8along the 2ndclass.
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The Median of Grouped Data
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
The median is in the 2ndclass.Method 1: Without the formula,
The 1stclass has 7. . .
There are 10in the 2ndclass . . .45
10
8
so we need to go 157 = 8along the 2ndclass.
so we want 8/10thof the class width:
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The Median of Grouped Data
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
The median is in the 2ndclass.Method 1: Without the formula,
The 1stclass has 7. . .
There are 10in the 2ndclass . . .45
10
8
The l.c.b. is 55, so the estimate of the median is 95.
so we need to go 157 = 8along the 2ndclass.
so we want 8/10thof the class width:
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The Median of Grouped Data
587Frequency, f16 - 2011 - 151 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
where,wf
Fn
2median l.c.b.
distance along class: Fn2
106 - 10
Method 2: Using the formula,
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The Median of Grouped Data
106 - 10
587Frequency, f16 - 2011 - 151 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
distance along class:
wf
Fn
2median l.c.b. where,
Fn2 15
Method 2: Using the formula,
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The Median of Grouped Data
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
distance along class:
wf
Fn
2median l.c.b. where,
Fn2 715
Method 2: Using the formula,
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The Median of Grouped Data
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
distance along class:
wf
Fn
2median l.c.b. where,
Fn2 7 8frequency of class,
15
Method 2: Using the formula,
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The Median of Grouped Data
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
Solution:
30n
2
n15
distance along class:
wf
Fn
2median l.c.b. where,
Fn2 7 810ffrequency of class,
15
Method 2: Using the formula,
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The Median of Grouped Data
Solution:
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
30n
2
n15
distance along class:
wf
Fn
2median l.c.b. where,
Fn2 7 810f
555510 wfrequency of class,
width of class,
15
Method 2: Using the formula,
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The Median of Grouped Data
Solution: 2
n15
distance along class:
wf
Fn
2median l.c.b. where,
Fn2 7 8
510
8
55 median 59
10ffrequency of class,
15
555510 wwidth of class,
58107Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
30n
Method 2: Using the formula,
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The Median of Grouped Data
In the next example and the solutions to the exercise,
Ive used the formula.
However, if you choose to use the formula, you will needto memorize it.
If you find it easy to work each problem out usingreasoning, just stick to that. Its all the formula isdoing anyway.
h d f G d
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The Median of Grouped Data
2432Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.2 Estimate the median for the following:.
11n
Solution:
The median is in the 3rdclass.
2
n55
wf
Fn
2median l.c.b.
distance along class: Fn2
50555
5
4
50510 median 111
class width= 5510515
Th M d f G d D
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The Median of Grouped Data
Exercise
61075Frequency, f
41 - 5036 - 4031 - 3521 - 30Length(cm)
Use linear interpolation to estimate the median of thefollowing:
1.
2.
162420Frequency, f
16 - 1813 - 1510 - 12Age (yrs)
Th M di f G d D
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The Median of Grouped Data
The median is in the 3rdclass.
wf
Fn
2median l.c.b.
28n 142n
536510
2535 median
distance along class:Fn
2
21214
class width= 5535540
Solutions:
61075Frequency, f
41 - 5036 - 4031 - 3521 - 30Length(cm)1.
Th M di f G d D t
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The Median of Grouped Data
Solution:
The median is in the 2ndclass.
314324
1013 median
As the data giveages, the boundariesare13 and16, not 125and155.
2.
162420Frequency, f
16 - 1813 - 1510 - 12Age (yrs)
31316 class width=
60n 302n
wf
Fn
2median l.c.b.
distance along class: Fn2
102030
Th M di f G d D t
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The Median of Grouped Data
The next 4slides show you how the linear interpolationformula is derived.
You are not expected to know the derivation so youcan skip over them unless you are interested.
SKIP
Th M di f G d D t
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The Median of Grouped Data
Well start with the example we used before.
36543Frequency, f41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)e.g. 3
There are 21observations so we want to estimate thesize of the 11thone. It lies in the 3rdclass.
Since we have 7 observations in the first 2 classes, themedian is the 4thvalue in the 3rdclass.The observations could be anywhere in the class but onaverage we expect them to be evenly spaced so we assume
that the 3rd
class looks like this:
The estimate of the median is 275.
x x x xx
215 235 255 275 295
205 305
Th M di f G d D t
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The Median of Grouped Data
205 305x x x xx
215 235 255 275 295
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)
3rdclass:
The diagram shows we want to add 7to the l.c.b.
05 1 1 1
This is 7tenths of the class width or 1010
7
The 7 tenths comes from35parts of the 5parts given bythe class frequency.
We need to express this as a formula.
Th M di f G d D t
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The Median of Grouped Data
205 305x x x xx
215 235 255 275 295
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)
3rdclass:
So, the estimate is: 527105
53520
This is 7tenths of the class width or 1010
7
The 7 tenths comes from35parts of the 5parts given bythe class frequency.
The diagram shows we want to add 7to the l.c.b.
We need to express this as a formula.
Th M di f G d D t
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The Median of Grouped Data
205 305x x x xx
215 235 255 275 295
36543Frequency, f
41 - 5031 - 4021 - 3011 - 201 - 10Length (cm)
3rdclass:
The estimate of the median is: 527105
53
520
In general, we have
wf
Fn
2median l.c.b.
We can think of n/2 Fas the distance along the classto the median.
fis the frequency and wthe width, both for the classcontaining the median.
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The following slides contain repeats ofinformation on earlier slides, shown withoutcolour, so that they can be printed and
photocopied.
For most purposes the slides can be printedas Handouts with up to 6slides per sheet.
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The Median of Grouped Data
To use linear interpolation to find an estimate of themedian for a grouped frequency distribution, we
locate the class containing the median usingtotal frequency divided by 2,
Fis the cumulative frequencies up to the classcontaining the median,
fis the frequency of the class containing the median,wis the width of the class containing the median.
( think of n/2 Fas the distance along the class to the
median ),
usew
f
Fn
2median l.c.b. where,
SUMMARY
or, use reasoning to save the need to remember theformula.
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The Median of Grouped Data
Solution:
The median is in the 2ndclass.
2
n15
distance along class:
where,
Fn2 7 8
510
855 median 59
10ffrequency of class,
15
555510 wwidth of class,
58107Frequency, f
16 - 2011 - 156 - 101 - 5Height (cm)
e.g.1 Estimate the median for the following:.
30n
wf
Fn
2median l.c.b.
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The Median of Grouped Data
2432Frequency, f16 - 2011 - 156 - 101 - 5Height (cm)
e.g.2 Estimate the median for the following:.
11n
Solution:
The median is in the 3rdclass.
2
n55
distance along class: Fn2
50555
550
510 median 111
class width= 5510515
wf
Fn
2median l.c.b.