Cumulative Frequency graphs

It is very easy to understand data graphically view rather than mathematical view. So, for representing large amount of data we use different graphical methods in Probability and Statistics. Here we discuss a graphical method which is related with frequency and when we create this graphical plot, then this plot is called as a cumulative frequency graph. With the help of this cumulative frequency graph, we can easily analyze Median of data, percentage of data and we can easily observe that which data is bigger and which data is smaller. So, this cumulative frequency graph is useful for observing data. Now we discuss whole method which is useful for creating cumulative frequency graph.
We use following steps for creating cumulative frequency graph-
Step 1: First of all, we analyze frequency table, means it is in ascending order or in descending order and if there is no order, then we order it in ascending way because with the help of ascending order, we can easily evaluate cumulative frequency graph.
Step 2: After a perfect order we can add frequency one by one for evaluating the cumulative total like we have frequencies,
A - 4
B - 5
C - 3
D – 2
Now we add one by means,
A – 4,
B – 4 + 5 = 9,
C – 9 + 3 = 12,
D – 12 + 2 = 14.
So, cumulative total for above frequencies is 14
Step 3 : After first two steps, we will plot the graph between value and cumulative frequencies where cumulative total makes an important role because each and every sum, where we evaluate cumulative total property is called as cumulative frequency for that value like cumulative frequencies for above frequencies is:
Value cumulative frequency,
A.         4
B.          9
C.         12
D.         14
Generally in cumulative frequency graph, we make value as a horizontal line and cumulative frequency as a vertical line.
Suppose we have following frequency table and we have to make cumulative frequency graph of given frequency table -
Marks frequency
21-30   11
41-50   15
11-20    2
31-40    19
71-80    13
91-100  40
81-90    6
51-60    42
61-70    31
Step 1: First of all, we analyze frequency table and it is not in any proper order. So, we make a proper.
Ascending order of given frequency table -
Marks frequency
11-20   2
21-30   11
31-40   19
41-50   15
51-60   42
61-70   31
71-80   13
81-90   6
91-100 40
Step 2: after a perfect order we can add frequency one by one for evaluating the cumulative total
Marks frequency cumulative total
11-20       2        2
21-30      11       2 + 11 = 13
31-40      19       13 + 19 = 32
41-50      15       32 + 15 = 47
51-60      42       47 + 42 = 89
61-70      31       89 + 31 = 120
71-80      14       120 + 14 = 134
81-90       6        134 + 6 = 140
91-100     40       140 + 40 = 180
So, cumulative total of above frequency table is 180.
Step 3: This each and every sum which we calculated in above step is called as cumulative frequency.
For each value -
Marks frequency cumulative total cumulative frequency
11-20       2                 2                         2
21-30       11             2 + 11 = 13          13
31-40       19             13 + 19 = 32        32
41-50       15             32 + 15 = 47        47
51-60       42             47 + 42 = 89        89
61-70       31             89 + 31 = 120      120
71-80       14             120 + 14 = 134    134
81-90         6             134 + 6 = 140      140
91-100       40           140 + 40 = 180    180
Now we make cumulative frequency plot graph between marks and cumulative frequency of marks, where we assume horizontal line as marks and vertical line as a cumulative frequency.
With the help of these cumulative frequency graphs we can easily evaluate median of this cumulative frequency table like in above cumulative frequency table -
Median of cumulative frequency is 180 / 2 = 90 because it is a Mean Point of this cumulative frequency table; so, marks which are related with this cumulative frequency are called as median of this cumulative frequency table.
We can easily analyze many mathematical questions from above cumulative frequency table, like how many students have scored more than 40 marks and how many students who have marks greater than 90 marks. The answer is -
Number of students, who are greater than 40 is
= [cumulative total] - [student, who have less than 40 marks]
= 180 – (2 + 11 + 19)
= 180 – 32
= 148
So, there are 148 students, who have greater than 40 marks.
Now we calculate number of students, who are greater have marks greater than 90.
In cumulative frequency table, there are 40 such students, which have more than 90 marks.
So, with the help of this cumulative frequency plot, we easily evaluate many mathematical problems like mean, median of data, observation which is related with data.