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Statistics
completed
Clement Radcliffe, Contributor
This week, we will complete the review
of statistics. With that in mind,
let us consider aspects of cumulative
frequency. You are asked to note the
following:
- The
frequency distribution maps, the
frequency with respect to the respective
score.
- The
cumulative frequency maps, the frequency
LESS THAN or EQUAL TO a given score.
- The
cumulative frequency, distribution
is used to construct the cumulative
distribution graph.
- This
graph may be used to determine the
number LESS THAN or GREATER THAN
a given score.
- You
may also estimate the median and
the upper and lower quartiles.
Let
us now consider the following example
which will illustrate the above.
Example
one
The
following frequency distribution table
represents the wages earned by 60
workers.
| Inches
of Rainfall |
-0.5-4.5
|
4.5-9.5
|
9.5-14.5
|
14.5
- 19.5
|
19.5
- 24.5
|
24.5-29.5
|
| Frequency |
5
|
8
|
3
|
1
|
2
|
1
|
Cumulative
Frequency |
5
|
13
|
16
|
17
|
19
|
20
|
(a)
Present the cumulative distribution
table.
(b)
Construct a cumulative frequency curve
for the data.
(c)
Using your graph, estimate:
(i)
The median
(ii)
The upper and lower quartiles and
the interquartile range
(iii)
The semi-interquartile range for the
distribution
(iv)
The number of workers who received
more than $47.50 in daily wages.
Solution
(a)
| WAGES |
40.5-45.5
|
45.5-50.5
|
50.5-55.5
|
55.5-60.5
|
60.5-65.5
|
| FREQUENCY |
5
|
12
|
28
|
11
|
4
|
Cumulative
Frequency |
5
|
17
|
45
|
56
|
60
|
N.B.
- The boundary values are used.
-
I do hope you identify the pattern
in establishing the cumulative frequency.
(b)
From
the given table, the cumulative frequency
corresponding to the highest wage
is 60.
There
are 60 workers involved.
As
with any graph, you may find the wage
corresponding to a given cumulative
frequency and vice versa.
When
the wages of the 60 workers are placed
in the order of size from the first
to the sixtieth, you must note the
following:
1st
- - 15th, 16th - - 30th, 31st - -
45th, 46th - - 60th
|
15.5th
|
30.5th
|
45.5th
|
|
LOWER
|
MEDIAN
|
UPPER
|
|
QUARTILE
|
|
QUARTILE
|
(c)
(i) The median is the 30.5th wage,
the average of the 30th and the 31st
values. From the graph, the median
is $53.
(ii)
The wages corresponding to 15.5th
and 45.5th positions are $50 and $55.50,
respectively, therefore, the lower
and upper quartiles are $50 and $55.50,
respectively.
Therefore,
the interquartile range is $55.50
- $50.00 = $5.50
(iii)
The semi-interquartile range (quartile
deviation) = ? (upper quartile - lower
quartile) = ? ($55.50 - $50) = $2.75
(iv)
From the graph, nine workers had wages
of $47.50 and below.
Therefore,
51 workers received wages above $47.50
Example
two
The
number of inches of rainfall daily
which was recorded over a period of
time is as follows:
14,
0, 9, 5, 3, 6, 7, 1, 7, 2, 8, 4, 5,
10, 18, 12, 8, 22, 27, 24.
You
are required to present the above
data and analyse it.
Solution
| Inches
of Rainfall |
0-4
|
5-9
|
10-14
|
15-19
|
20-24
|
25-29
|
| Group
Mean |
2
|
7
|
12
|
17
|
22
|
27
|
| Tally |
1111
|
1111
111
|
111
|
1
|
11
|
1
|
| Number
of days |
5
|
8
|
3
|
1
|
2
|
1
|
The
mean rainfall is: (2 x 5 + 7 x 8 +
12 x 3 + 17 x 1 + 22 x 2 + 27 x 1)/20
(a) The data is presented in a frequency
table of grouped data as follows:
(b)
The mean rainfall is: 2 x 5 + 7 x
8 + 12 x 3 + 17 x 1 + 22 x 2 + 27
x 1
20
= 190 = 9.5 inches
| Inches
of Rainfall |
-0.5-4.5
|
4.5-9.5
|
9.5-14.5
|
14.5-19.5
|
19.5-24.5
|
24.5-29.5
|
| FREQUENCY |
5
|
8
|
3
|
1
|
2
|
1
|
Cumulative
Frequency |
5
|
13
|
16
|
17
|
19
|
20
|
Histogram
illustrates
that most of the rain which fell was
up to 9.5 inches.
(c)
As there are 20 days, the median lies
between the 10th and 11th scores.
This is the class five to nine inches
of rainfall.
I
do hope that you agree that the modal
class is five to nine inches of rainfall.
Cumulative
frequency distribution
Diagram
3
From
the graph, find:
(i)
The median (7.5)
(ii)
The upper quartile (13)
(iii)
The lower quartile (4.75)
(iv)
The inter quartile range (13- 4.75
= 8.25)
Clement
Radcliffe is the principal of Glenmuir
High School in May Pen.
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