Cumulative Frequency Distribution

Cumulative Frequency Distribution

Cumulative Frequency In statistical investigations, sometimes we are interested in the number of observations smaller than (or greater than) a given value. In such cases, our chief concern is the accumulated frequency up to ( or above) some value of variable. This accumulated frequency is known as cumulative frequency. Cumulative Frequency corresponding to a specified value of the variable may be defined as the number of observations smaller than (or greater than) that value. The number of observations “less-than” a given value is called less-than cumulative frequency and the number of observations “greater than” a value is called more-than cumulative frequency.

Cumulative Frequency Distribution

A table showing the cumulative frequencies

against values of the variable systematically arranged in decreasing (or increasing) order is known as Cumulative Frequency Distribution. It can be derived from a grouped frequency distribution by writing down the consecutive class boundary points and noting the number of observations less than (or greater than) each class boundary point.

-:For example:-

The grouped frequency distribution shows the values of the variable in class intervals and the corresponding class frequencies:-

Value 0-------------10 10----------- 20 20-----------30 30-----------40 40----------- 50 50----------- 60 60------------70 70------------80 Total

Frequency 4 12 24 36 20 16 8 5 125

The cumulative frequency distribution

The cumulative frequency distribution shows the values of the variable (class boundaries) and the corresponding cumulative frequencies (less-than type).

Value

Cumulative frequency (“less than type”)

Less than 10

4

Less than 20

4+12=16

Less than 30

4+12+24=40

Less than 40

4+12+24+36=76

Less than 50

4+12+24+36+20=96

Less than 60

4+12+24+36+20+16=112

Less than 70

4+12+24+36+20+16+8=120

Less than 80

4+12+24+36+20+16+8+5=125

Consider the following example:Distribution of 200 individuals on the basis of age. Age (year)

Frequency

15-------------19

37

20-------------24

81

25-------------29

43

30-------------34

24

35-------------44

9

45-------------59

6

Total

200

Age (Class Boundaries)

Frequency

14.5-------------19.5

37

19.5-------------24.5

81

24.5-------------29.5

43

29.5-------------34.5

24

34.5-------------44.5

9

44.5-------------59.5

6

Total

200

The cumulative frequency distribution in case of continuous variable Age (years)

Cumulative frequency (“less than type”)

14.5

0

19.5

37

24.5

118

29.5

161

34.5

185

44.5

194

59.5

200 = total frequency

Age (years)

Cumulative frequency (“greater than type”)

14.5 19.5 24.5 29.5 34.5 44.5 59.5

200=N 163 82 39 15 6 0

Uses 1. It is used to find the Median, Quartile, deciles and percentiles or the value of the variable such that its cumulative frequency is a specified number. 2. It is also useful in finding the cumulative frequency corresponding to a given value of the variable. 3. To find the number of observations which are expected to lie between two specified value of the variable.

Problem Calculate a) The number of cases between 112 and 134 b) Number less than 112 c) Number greater than 134,from the following table:-

Class Boundary 90 - 100 100 - 110 110 - 120 120 - 130 130 - 140 140 - 150 150 - 160

Frequency 16 22 45 60 50 24 10

Class Boundary 90 100 110 112 ----120 130 134 -----140 150 160

Cumulative frequency 0 16 38 ----- y 83 143 ----- z 193 217 227 = N

y = Number of observation less than 112 z = Number of observation less than 134 To find cumulative frequency y, we have or, Or,

112 – 110 y - 38 ------------------- = ---------------------120 – 110 83 – 38 y = 47

Again, To find cumulative frequency z

or, Or

134 – 130 z - 143 ------------------- = ---------------------140 – 130 193 – 143 z = 163

Given y = 47 ,z = 163 and N = 227 , we have a) Number of cases between 112 and 134 = Number of cases below 134 – number of cases below 112 = 163 – 47 = 116 b) Number less than 112 = 47 c) Number greater than 134 = Total number of cases - Number of cases below 134 = 227 – 163 = 64 (Ans)

Consider the following example Suppose in a class there are 15 students, and their marks in statistics is given as below, 48, 52, 53, 56, 46, 59,42, 44,47,51,57,51,61,64,63 Class boundary

Frequency

40 -45

2

45 – 50

3

50 – 55

4

55 – 60

3

60 - 65

3

Class boundary

Cumulative Frequency (less than type)

40

0

45

2

50

5

55

9

60

12

65

15 = N

a) Here the number of observation between 45 and 60 is = Number of cases below 60 – number of cases below 45 = 12 – 2 = 10 b) Number less than 45 = 2 c) Number greater than 60 = Total number of cases – number less than 60 = 15 – 12 = 3 (Ans)

Cumulative distribution in case of discrete variable Number of member in a family

Frequency

1

4

2

33

3

76

4

50

5

26

6

8

7

1

Total

198

Family size

Cumulative frequency “lessthan” type

Cumulative frequency “more-than” type

1

4

198 = N

2

37

194

3

113

161

4

163

85

5

189

35

6

197

9

7

198 = N

1