Frequency Distribution

 What is a frequency distribution?

The frequency of a value is the number of times it occurs in a dataset.

A frequency distribution is the pattern of frequencies of a variable. It’s the number of times each possible value of a variable occurs in a dataset.

A frequency distribution refers to data classified on the basis of some variable that can be measured such as prices, wages, age, and number of units produced or consumed.

Types of frequency distributions

There are four types of frequency distributions:

Ungrouped frequency distributions: 

The number of observations of each value of a variable.

You can use this type of frequency distribution for categorical variables.

Example

121	104	175	201	156	185	216	8	177	175
191	227	215	118	247	199	182	215	252	150
256	116	158	205	209	244	101	173	243	171

 

Grouped frequency distributions:

The number of observations of each class interval of a variable. Class intervals are ordered groupings of a variable’s values.

You can use this type of frequency distribution for quantitative variables.

Class	Frequency (f)
100-120	4
120-140	1
140-160	3
160-180	5
180-200	4
200-220	7
220-240	1
240-260	5
	N= 30

Relative frequency distributions:

 The proportion of observations of each value or class interval of a variable.

You can use this type of frequency distribution for any type of variable when you’re more interested in comparing frequencies than the actual number of observations.

Cumulative frequency distributions: 

The sum of the frequencies is less than or equal to each value or class interval of a variable.

You can use this type of frequency distribution for ordinal or quantitative variables when you want to understand how often observations fall below certain values.

A frequency distribution may be either continuous or discrete (also called discontinuous).

 

Example

Prepare a frequency distribution table by using the given data (consider the class interval is 20)

121	104	175	201	156	185	216	8	177	175
191	227	215	118	247	199	182	215	252	150
256	116	158	205	209	244	101	173	243	171

 

Number of data= 30

Class	Tally	Frequency (f)	Cumulative frequency (fc)
100-120	////	4	4
120-140	/	1	5
140-160	///	3	8
160-180	////	5	13
180-200	////	4	17
200-220	////    //	7	24
220-240	/	1	25
240-260	////	5	30
		N= 30

 

 

Class Intervals

The span of a class, that is the difference between the upper limit and the lower limit is known as class interval.

Upper Limit – Lower Limit = Class Interval

Close End Class

Class	Class Interval
20 - 40	20
40 - 60	20
60 - 80	20

Open End Classes

Class

Below - 400

400 - 600

600 and Above

 

Class Limits

The Class Limits are the lowest and highest values that can be included in the class

 

Class
10	-	20
20	-	30
30	-	40
Lower Limits	-	Upper Limits
10	-	19
20	-	29
30	-	39

 Methods of Classifying data according to Class Interval

1.      Exclusive – Upper limit of one class is the lower limit of another class

2.      Inclusive

  Example,

Exclusive	Inclusive
20 - 40	20 - 39
40 - 60	40 - 59
60 - 80	60 - 79

Class Frequency

The Number of Observations Corresponding to the particular class is known as the Frequency of that class or the Class Frequency.

 

Class	Tally	Frequency (f)
10-20	////	5
20-40	/	1
40-60	///	3

 

Class Mid-Point

It is the value lying halfway between the lower limit and the upper limit of a class interval.

 

Mid-Point of Class/ Correction Factor = (Upper Limit + Lower Limit) / 2

 

Principles of Classification / Determining Class

·         The Number of Class should be between 5 to 25.

·         As far as possible one should avoid odd values of class intervals such as 3,7,9,26,39 etc.

·         Class intervals should be 5 or multiples of 5. Such as 10.15,20,35, etc.

·         The lower limit should be either 0 or 5 or multiple of 5.

·         All classes should be of the same size

The formula,

i = R / K

  = R / 1 + 3.322 log N

Where,

i = Interval

R= Range

   = Highest Value of data – Lowest Value of data

 

k = 1 + 3.322 log N

N= Number of Observation/ Population

Log = The ordinary logarithm to the base of 10

Example

The profit of 30 Companies (in Million) for 1 year are given below

20	22	35	42	37	42	48	53	49	65
39	48	67	18	16	23	37	35	49	63
65	55	45	58	57	69	25	29	58	65

 

Classify the above data by taking a suitable class interval

 

Answer

Here N= 30

 

i = R / K

  = (Highest Value of data – Lowest Value of data) / 1 + 3.322 log N

= (69-16)/ 1 + (3.322 log 30)

= (69-16)/ 1 + (3.322 x 1.4771)

= 53/ 1+ 4.91

= 53/ 5.91

= 8.97

= 9

As odd number should be avoided so i = 10

Profit (in Million)	Tally	No of Companies
15-25	////	5
25-35	//	2
35-45	////  //	7
45-55	////  /	6
55-65	////	5
65-75	////	5
	N=	30