2. Page 2 of 16
TABLE OF CONTENTS
I. Q1- …………………………3-8
Cumulative frequency curve……………………………….….…3
Ways of calculating cumulative curves……………………….…3
Types of Data’s……………………………………….……….…3
Advantage/Disadvantage of cumulative curves…………….…...4
Construction of cumulative curves………………………..…….4
Estimation…………………………………………………….....5
Median………………………………………………………......7
Quartiles…………………………………………………………7
II. Q2 - ………………………………9-14
Histogram ………………………………………….…..….….…9
Bar charts…………………………………………………….... 10
Simple bar chart ………………………..……………….…..…11
Multiple bar chart ………………………………………....…...12
Component bar chart ………………………………..….......….12
Percentage component bar chart ……………………….…......13
Pie Chart…………………………………………..…………....14
III. REFERENCE ……………….………15
Citation………………………………………………………… 15
Book references………………………………………………...15
3. Page 3 of 16
Cumulative frequency Curve
As being a teacher, the cumulative frequency greatly helps in finding out the number of
observations that we get from the student’s progress report or from their attendance in the
class and so-on.
A cumulative frequency curve is a way to display cumulative information graphically.
It shows the number, percentage, or proportion of observations that are less than or equal
to particular values. Graph shows the cumulative totals of a set of values up to each of the
points on the graph. Cumulative frequency are very useful for estimating the median and
inter-quartile range of grouped data, they are also very useful for comparing
distributions. The data type can be either being any simple data or grouped discrete data
or continuous data.
Cumulative frequency calculations :-
1. The cumulative frequency can be calculated using a frequency distribution table,
which can be constructed from stem and leaf plots or directly from the data.
2. The cumulative frequency can also be calculated by adding each frequency from a
frequency distribution table to the sum of its predecessors. The last value will
always be equal to the total for all observations, since all frequencies will already
have been added to the previous total.
3. Another calculation that can be obtained using a frequency distribution table is the
relative frequency distribution. This method is defined as the percentage of
observations falling in each class interval. Relative cumulative frequency can be
found by dividing the frequency of each interval by the total number of
observations.
4. A frequency distribution table can also be used to calculate cumulative
percentage. This method of frequency distribution gives us the percentage of the
cumulative frequency, as opposed to the percentage of just the frequency.
There are 2 types of Data’s:
1-Discrete Data: it takes values only up to certain numbers
For example: The number of students in a class. [You can’t have a student]
2-Continuous data: Takes any value within a given range.
For example: A person's height: It could be any value (within the range of human
heights), not just certain fixed heights.
Q1
4. Page 4 of 16
ADVANTAGES of Cumulative frequency curves:
1- can be use to read off values both way round
2- The original information from a grouped frequency distribution can be obtained
from the C.F curves
3- Very informative when examining how values are changing within the data set.
4- -shows the running total of frequencies from the lowest interval up.
DIS-ADVANTAGES of cumulative frequency curves:
1- Difficult to compare the frequencies between each data group.
HOW CAN WE CONSTRUCT OR DRAW THE CUMULATIVE FREQUENCY
CURVES?
To draw the cumulative frequency curves we need to see if we have a grouped data or an
ungrouped data.
We know its ungrouped data when the data comes as a single row of numbers, because
each number corresponds to a single observation.
We can know if the data is grouped when the given or sample data comes in a table with
2-3 columns because each row would represent multiple observations.
THE FOLLOWING GROUPED DATA BELOW SHOWS THE TIME TAKEN FOR
MY STUDENTS TO COMPLETE THE MATHS QUIZ IN MINUTES.
Time taken (mins)
Frequency
Cumulative
Frequency
cumulative frequency will be on the y -axix
time will be on the x -axis
0<t<5 2
+
0+2= 2
5<t<10 9 2+9= 11
10<t<15 9 11+9= 20
15<t<20 8 20+8= 28
20<t<25 3 28+3= 31
25<t<30 1 31+1= 32
5. 35
30
25
20
15
10
5
Cumulative Frequency Graph
Comments on my graph: This shape of the cumulative frequency graph produces
reflects the characteristics of the time students take to complete the test and how this data
is spread or distributed within the range. This characteristic S shape is also called as an
“ogive” and it appears almost in all the cumulative frequency diagrams.
HOW CAN WE ESTIMATE VALUES USING THESE GRAPHS?
We can estimate the values using a cumulative frequency graph by drawing a straight line
that meets the cumulative frequency curve i.e. the y axis and then drawing a
corresponding line to meet on the x axis.
SHOWN BELOW:
Page 5 of 16
2
11
20
28
31
32
0
5 10 15 20 25 30
Cumulative frequency
Time ( mins)
Cumulative frequency
6. Page 6 of 16
2
Cumulative Frequency Graph
11
20
28
31
32
35
30
25
20
15
10
5
0
0<t<5 5<t<10 10<t<15 15<t<20 20<t<25 25<t<30
Here, I can easily estimate the values I want to know such as If :-
1- I want to know how many students completed the math test with in the first
20 minutes: I will draw and join graphically the corresponding lines.
Pink line shows that 29 students completed the test within 20 minutes.
2- I want to know the time by which 20 students finished their test: I will follow
the same steps applied above and mark it with other color.
Thus, the Red line shows that the 20 students finished their test in 13 minutes.
Cumulative frequency
Time ( mins)
Cumulative frequency
7. The Median Value: It is central tendency measure of a set of data; the median of
a group of numbers is the number in the middle, when the numbers are in order of
magnitude. We can find it for any given observation by the formula (n + 1)/2th value.
For example in a set of students who come for math’s tuition after school are:
4 1 5 2 5 7 8
Here, we first set the numbers in correct ascending order and then substitute the values in
our formula:
STEP 1- 1, 2, 4, 5, 5, 7, 8
STEP 2- (n + 1)/2th value [n=sum of all students]
Here, n is 7
= (7+1)/2
= 8/2 giving 4th value. Therefore, the 4th value is 5.
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Quartiles: If we divide a cumulative frequency curve into quarters:
The value at the lower quarter is subjected as the lower quartile which is
calculated by the formula: Q1= ¼ x [ n+1]th term
The value at the middle gives the median which is calculated by the formula: Q2=
½ [ n+1]th term
The value at the upper quarter is the upper quartile. And it is calculated by the
formula: Q3= ¾ x [ n+1]th term.
Below is attendance shown of the students who attended my class last month. I will use
that data to find the median and the upper and lower quartiles. And from the upper and
lower quartile range I will find the final value which is known as Inter-Quartile. This
value is calculated by subtracting the upper by lower quartile. Finding out the positions
by their formulas:
No. of students
Frequency
Cumulative frequency
From this we will
calculate the median
and the quartiles by
making the cumulative
frequency graph.
10 5 5
11 10 15
12 27 42
13 18 60
8. Page 8 of 16
14 6 66
15 16 82
16 38 120
17 9 129
Lower Quartile =Q1= ¼ x [ n+1]th term = ¼ x [ 129+1 ] = 32.5th position
Median= Q2= ½ [ n+1]th term= ½ x [ 129+1 ] = 65th position
Upper Quartile= Q3= ¾ x [ n+1]th term = ¾ x [ 129+1 ] = 97.5th position
140
120
100
80
60
40
20
0
cumulative frequency
Cumulative frequency
Graph
Cumulative
frequency Graph
From this curve, we can see the positions where the quartiles lie and thus
can approximate them as follows:
Q1= 12.1
Q2=14.2
Q3=15.9
Thus Inter-Quartile range is Q3-Q1= = 15.9 - 12.1 =3.8
9. I- HISTOGRAMS:
Histograms are compact graphs or bar charts represented in a graphical structure with
no gaps in between any point and its thickness is equal to the class interval.
Page 9 of 16
Advantages:
They are vividly strong and clearly display flow
Due to its uniformity, they are effortlessly clear to understand
They can be related to the normal cumulative curve.
Disadvantages:
Used only with continuous data type.
The data is grouped and so the values are not accurate.
Comparison becomes difficult when two or more sets of data are involved.
We can draw histograms in distinctive ways depending on the kind of data we have. For
the inclusive class interval type it is necessary to change the class intervals as class
boundaries to be used in the x-axis. Therefore, the frequency will be used in the y-axis.
NOTE:
When the class interval width of all the data is same we just change the intervals
to class boundaries by the deducting 0.5 from lower limit and adding 0.5 to upper
limit.
When the class interval is not even, we calculate frequency density and use it in
the x-axis
:
E.g.: The table below shows the money earned by the workers in a week for the past two
months.
Money Earned Weekly (wage -$)
10-19 10
20-29 5
30-39 7
40-49 2
50-59 8
60-69 6
Q2
Frequency density= Frequency ÷ Class Interval
10. Money Earned Frequency Class interval width Class Boundaries
10-19 10 9 9.5-19.5
20-29 5 9 19.5- 29.5
30-39 7 9 29.5-39.5
40-49 2 9 39.5-49.5
50-59 8 9 149.5-59.5
60-69 6 9 159.5-69.5
Frequency
9.5-19.5 19.5- 29.5 29.5-39.5 39.5-49.5 149.5-59.5 159.5-69.5
Page 10 of 16
Draw the histogram
12
10
8
6
4
2
0
frequency
II- BAR CHARTS:
They display the data in graphical terms and like histograms, they are very clear and
simple. There are two quantitative charts:
Absolute Chart: It displays original amount of data. The data will not be
converted into degrees or percentages.
Relative Chart: It is also known as proportional chart. Here the original
data is not shown on the graph because it is converted into degrees or percentages.
Advantages:
They are used in schools, large businesses, media hospitals etc.
They can be related to the normal curve.
Massive values can be correlated rapidly.
Comparison amid various groups of value is easier.
Disadvantages:
class boundaries
11. Page 11 of 16
Chances of getting cluttered due to too many bars in one graph.
They need additional information
They fail to reveal key assumptions, effects or patterns.
These are less illuminative.
There are four kinds of bar charts:
1. Simple Bar Chart:
This is an absolute chart.
It is used when we have only one variable to display on the bar graph.
It consists of a grid and vertical and horizontal columns.
E.g.: The following bar chart shows the sales department in which the workers are paid
to work for overtime they spend for a month.
No. of workers Overtime Pay (SR)
10 400
20 320
30 100
40 250
50 150
500
400
300
200
100
0
Simple Bar Chart
10 20 30 40 50
Overtime pay (SR)
No. of workers
2. Multiple Bar Chart:
This is also an absolute chart.
It is used when we have two or more variables and are displayed on the bar graph
together.
E.g.: The information given below shows that in a well known sales company, from
the last 6 years, the demand in the market of new product A is raised but for product
B has got lower.
12. Page 12 of 16
Year Demand of Product A
(units)
Demand of Product B
(units)
2008 80 50
2009 72 35
2010 65 42
2011 55 54
2012 70 65
2013 90 70
100
80
60
40
20
0
Multiple Bar Chart
2008 2009 2010 2011 2012 2013
Production Units
Year
3. Component Bar Chart:
This is an absolute chart.
It is used when we have only one bar
It is used to represent different variables into subdivisions.
E.g.: The table below shows the profit gained by 2 sales departments in 4 years. Prepare
Component Bar Chart.
Year
Dept A ($)
Dept B ($)
Total Profit ($)
2010 400 150 550
2011 300 450 750
2012 250 200 450
2013 500 450 950
units of A
Units of B
13. 1000
800
600
400
200
0
Component Bar Chart
2010 2011 2012 2013
Profit ($)
Year
Dept B
Dept A
4. Percentage Component Bar Chart:
This is a relative chart.
The data given have to be changed to percentages and the divisions are then
Page 13 of 16
expressed of the 100%.
The chart does not show anything about the original numbers.
E.g.: The following table shows the sales of 2 departments in 5 years. Prepare Percentage
Component Bar Chart.
Year Dept A ($) Dept B ($) Total
Sales($)
% of A % of B
2010 40 15 55 40/55x100=72.7 15/55x100 = 27.3
2011 30 45 75 40 60
2012 30 40 70 43 57
2013 50 45 95 52.6 47.4
14. Sales of beverages in % Pie chart
Pepsi Coke Fanta 7 up Citrus RedBull Barbican
Page 14 of 16
Percentage Component
100%
80%
60%
40%
20%
0%
Bar Chart
2010 2011 2012 2013
Sales in %
Year
% of Dept B
% of Dept A
5. Pie Chart:
This is a relative chart.
It shows relation between variables in a set of data.
It displays each segment as a proportion of 360 degrees.
E.g. Pie Chart below shows in % pie chart, how many soft beverages are consumed
by the retailer shops in a month.
27%
13%
5%
5%
17%
20%
13%
15. Page 15 of 16
REFRERENCE:
Citation:
http://www.netmba.com/statistics/histogram/
http://www.ask.com/question/difference-between-grouped-and-ungrouped-data
http://office.microsoft.com/en-001/excel-help/present-your-data-in-a-pie-chart-
HA010211848.aspx
http://www.emathzone.com/tutorials/basic-statistics/simple-bar-chart.html
http://www.cimt.plymouth.ac.uk/projects/mepres/book9/bk9_16.pdf
https://www.xtremepapers.com/revision/gcse/statistics/continuous_data_cumulati
ve_frequency_polygon.php
http://www.statcan.gc.ca/edu/power-pouvoir/ch10/5214862-eng.htm
Book:
Pledger, Keith, Alan Clegg, AS and A level Modular Mathematics, Edexcel
B.S. Everitt, The Cambridge dictionary of Statistics, 2nd Edition
SC Gupta, Kapoor VK Fundamental of Mathematical Statistics, 11th Edition,
2012.
