In statistics, graph plays an important role. With the help of these graphs, we can easily understand the data. So in this article, we will learn how to represent the cumulative frequency distribution graphically.

### Cumulative Frequency

The frequency is the number of times the event occurs in the given situation and the cumulative frequencies are the sum of all the previous frequencies up to the current frequency. Or in other words, the cumulative frequency of a class is the frequency calculated by adding the frequencies of all the classes preceding the given class. For example:

Interval | Frequency |

0-10 | 2 |

10-20 | 4 |

20-30 | 5 |

In the above table, we have intervals and frequencies now we are going to find the cumulative frequency by adding all the previous frequencies up to the current frequency.

Interval | Frequency | Cumulative frequency |

0-10 | 2 | 2 |

10-20 | 4 | 6 |

20-30 | 5 | 11 |

So, here the cumulative frequency of interval(0-10) is 2 because this is the very first frequency. Similarly, the cumulative frequency of interval(10-20) is 6 because it is the sum of 2(previous frequency) + 4(current frequency), and the cumulative frequency of interval(20-30) is 11 because it is the sum of 2 + 4(previous frequency) + 5(current frequency). Such type of table is known as a cumulative frequency table.

### Cumulative Frequency Curve

Let us considered a grouped frequency distribution be given to us. Take a graph paper and mark the upper-class limits along the x-axis and the corresponding cumulative frequencies along the y-axis. Join these points successively by smooth curves, we will get a curve, this curve is known as cumulative frequency curve. Or in other words, the graphical representation of cumulative frequency distribution is known as cumulative frequency curve. It is also known as ogive, and it is the most efficient way to represent data. The cumulative frequency curve is of two types:

**(1) Less than cumulative frequency curve**

As we know that the cumulative frequency curves are created using the cumulative frequencies so, in less than the cumulative frequency curve, the frequencies of all the preceding class or interval are added to the current class or interval frequency. You can create more than cumulative frequency by adding the frequency of first-class to the frequency second-class and so on. For example:

Interval | Frequency |

5-10 | 2 |

10-15 | 4 |

15-20 | 5 |

In the above table, we have intervals and frequencies now we are going to find less than the cumulative frequency by adding all the previous frequencies up to the current frequency.

Interval | Frequency | Cumulative frequency | Upper Limit |

5-10 | 2 | 2 | 10 |

10-15 | 4 | 2 + 4 = 6 | 15 |

15-20 | 5 | 2 + 4 + 5 = 11 | 20 |

**How to draw less than cumulative frequency curve:**

In this case, we use the upper limit of the classes to draw the curve. Now, the step-by-step process of plotting a less than cumulative frequency curve:

- Take a graph paper and mark the upper-class limits along the x-axis and the corresponding cumulative frequencies along the y-axis.
- Join these points successively by line segments, we will get a polygon, known as a cumulative frequency polygon.
- Join these points successively by a smooth curve, we will get a curve, known as cumulative frequency graph.
- Take a point A (0, N/2) on the y-axis and draw AP || x-axis, cutting the above curve at a point P. Draw PM ⊥ to the x-axis, cutting the x-axis at M.
- Then, the median length of OM.

**(2) More than cumulative frequency curve**

As we know that the cumulative frequency curves are created using the cumulative frequencies so, in more than cumulative frequency curve, the frequencies of succeeding class or interval are added to the current class or interval frequency. You can create more than cumulative frequency by subtracting the frequency of the second-class from the first class and so on. For example:

Interval | Frequency |

5-10 | 20 |

10-15 | 4 |

15-20 | 5 |

In the above table, we have intervals and frequencies now we are going to find more than the cumulative frequency:

Interval | Frequency | Cumulative frequency | Lower Limit |

5-10 | 20 | 20 | 5 |

10-15 | 4 | 20 – 4 = 16 | 10 |

15-20 | 5 | 16 – 5 = 11 | 15 |

**How to draw more than cumulative frequency curve:**

In this case, we use the lower limit of the classes to draw the curve. Now, the step-by-step process of plotting a more than Cumulative Frequency curve:

- Take a graph paper and mark the lower class limits along the x-axis and the corresponding cumulative frequencies along the y-axis.
- Join these points successively by line segments, we will get a polygon, known as a cumulative frequency polygon.
- Join these points successively by a smooth curve, we will get a curve, known as cumulative frequency graph.
- We assume that P be the point of intersection of less than’ and ‘more than curves. Draw PM ⊥ to the y-axis, cutting x-axis at M.
- Then, median = length of OM.

### Sample Problem

**Question 1. Following is the age distribution of group students. Now, draw the cumulative frequency curve of less than type and find the median value.**

Age (in years) | Frequency |
---|---|

4-5 | 36 |

5-6 | 42 |

6-7 | 52 |

7-8 | 60 |

8-9 | 68 |

9-10 | 84 |

10-11 | 96 |

11-12 | 82 |

12-13 | 66 |

13-14 | 48 |

14-15 | 50 |

15-16 | 16 |

**Solution:**

For the given table, we have to prepare the more than series as shown below:

Age (in years)

c.f.

Less than 5

36

Less than 6

78

Less than 7

130

Less than 8

190

Less than 9

258

Less than 10

342

Less than 11

438

Less than 12

520

Less than 13

586

Less than 14

634

Less than 15

684

Less than 16

700

On a graph paper, take the scale

Along the x-axis: 5 small div. = 1.

Along the y-axis: 1 small div. = 10.

And, plot all the points A(5, 36), B(6, 78), C(7, 130), D(8, 190), E(9, 258), F(10, 342), G(11, 438),

H(12, 520), I(13, 586), J(14, 634), K(15, 684) and L(16, 700).

Join these points successively with a freehand, we will get the cumulative frequency curve or an ogive.

Here, N = 700 ⇒ N/2 = 350

Take a point P(0, 350) on the y-axis and draw PQ|| x-axis, meeting the curve at Q. Draw QM 1 x-axis, intersecting x-axis at M. Then, OM = 10 units.

Hence, median = 10.

**Question 2. For the given frequency distribution, draw a cumulative frequency graph of more than type and find the median value.**

Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
---|---|---|---|---|---|---|---|

Frequency | 5 | 15 | 20 | 23 | 17 | 11 | 9 |

**Solution:**

For the given table, we have to prepare the more than series as shown below:

More than 60

9

More than 50

20

More than 40

37

More than 30

60

More than 20

80

More than 10

95

More than 5

100

Scale:Along the x-axis, 10 small div. = 5.Along the y-axis, 1 small div.= 1.

Plot all the points A(5, 100), B(10, 95), C(20, 80), D(30, 60), E(40, 37), F(50, 20) and G(60, 9).

Join AB, BC, CD, DE, EF and FG with a freehand, and we will get the required curve, as shown in below figure.

Here, N = 100

⇒ N/2 = 50

From P(0, 50) draw PQ || x-axis, meeting the curve at Q. Draw QM ⊥ OZ, meeting x-axis at M. Clearly, OM = 35 units

Hence, median = 35.

**Question 3. The following table gives the production yield of rice of 100 farms of a village:**

Production (kg/hectare) | 40-45 | 45-50 | 50-55 | 55-60 | 60-65 | 65-70 |
---|---|---|---|---|---|---|

Number of farms | 4 | 6 | 16 | 20 | 30 | 24 |

**Draw a cumulative frequency graph of more than type.**

**Solution:**

For the given table, we have to prepare the more than series as shown below:

More than 65

24

More than 60

54

More than 55

74

More than 50

90

More than 45

96

More than 40

100

Scale: Along the x-axis, 1 small div.= 1

Along the y-axis, 1 small div. = 1

On a graph paper, plot all the points A(40, 100), B(45, 96), C(50, 90), D(55, 74), E(60, 54) and F(65, 24).

Join AB, BC, CD, DE and EF with a free hand, and we will get a More Than Ogive.

**Question 4. During the medical checkup of 35 students of a college their weights were recorded as follows:**

Weight(in kg) | 38-40 | 40-42 | 42-44 | 44-46 | 46-48 | 48-50 | 50-52 |
---|---|---|---|---|---|---|---|

No. of Students | 3 | 2 | 4 | 5 | 14 | 4 | 3 |

**Draw a less than and a more than type ogive from the given data. Hence, find the median weight from the graph.**

**Solution:**

(i) Less than Series:For the given table, we have to prepare the less than series as shown below:

Weight (in kg)

Number of Students

Less than 40

3

Less than 42

5

Less than 44

9

Less than 46

14

Less than 48

28

Less than 50

32

Less than 52

35

Scale:Along the x-axis, 5 small div. = 1 kg.Along the y-axis, 10 small div.= 5 kg.

Plot all the points A(40, 3), B(42, 5), C(44, 9), D(46, 14), E(48, 28), F(50, 32) and G(52, 35).

Join AB, BC, CD, DE, EF and FG with a free hand to get the curve ‘Less Than Series’.

(ii) More than Series:For the given table, we have to prepare the more than series as shown below:

Weight (in kg)

Number of Students

More than 38

35

More than 40

32

More than 42

30

More than 44

26

More than 46

21

More than 48

7

More than 50

3

Now plot the points on the same graph: P(38,35), Q(40, 32), R(42, 30), S(44, 26), T(46, 21), U(48, 7) and V(50,3)

and join PQ, QR, RS, ST, TU and UV with a free hand to get ‘More Than Series’.

The two curves intersect at the point L. Draw LM ⊥ OX.

Hence, median weight = OM = 46.5 kg.