Frequency Distribution

Gaurav Sikri
Frequency Distribution
• The Initial step in Statistics is to collect relevant data related to specific purpose
According to Connor,
“ Frequency distribution is simply a table in which the data are grouped into different classes and the
number of cases which fall in each class are recorded”
• Raw Data:
• The original collection of data for the predetermined purpose in a systematic manner is known as
Raw Data.
• The data is unprocessed and is to be used for making wise decisions after application of statistical
methods.
• Array:
• Arrangement of data in ascending or descending order is known as an array.
• It helps us to study certain characteristics of the data and in proper application of statistical methods.
• Frequency:
• Number of times a value occurs is known as frequency.
• Suppose there are 20 nursing students who scored 50 marks.
• So variable is marks and frequency is 20. Frequency can also be within a range, e.g. 25 students
scoring between 50 and 60 marks.
Types of Frequency Distribution

Data
Distribution

Ungrouped Grouped

Individual Discrete
Continuous
Distribution or Distribution or
Distribution
Individual series Discrete series
Types of Frequency Distribution: Ungrouped Frequency Distribution
• A form of ungrouped frequency distribution is also known as individual frequency distribution

• A Statistical series in which all the observations are listed out and have a frequency of one only .

• The value of variable are mentioned individually and are independent of the preceding or
succeeding values.

• If any value is repeated, it is mentioned as many times as it repeats.

Types of Frequency Distribution: Ungrouped Frequency Distribution
• Each value is treated as separate value and variable is normally represented as “X” or “x”

• There will be only one column for the variable

and
• No separate column for frequency
• because frequency of each variable value is one only.

• The individual distribution can be presented in an array

i.e. either in
Ascending order
or
Descending order.
• It can be used only with small samples.
• Marks:
• Random: 45, 48, 64, 46, 40, 64, 70, 56, 61, 59, 60, 52
• Ascending order Array: 40, 45, 46, 48, 52, 56, 59, 60, 61, 64, 70
• Note: The gap between individual values may not be equal.
Types of Frequency Distribution: Grouped Frequency Distribution
• In this distribution,
• The variable is accompanied by the number of times it occurs in the given data
• It is known as frequency of occurrence.

• In this distribution,
• A group is formed
• which consists of variable and the frequency of occurrence of that variable.

• Whenever there is large sized sample or population,

• It is advisable to have grouped form of the frequency distribution.
Types of Frequency Distribution: Grouped Frequency Distribution
• It is of two types;
1. Discrete frequency distribution 2. Continuous frequency distribution
Discrete frequency distribution
According to Boddington,
• In a discrete distribution
• Individual values of the variable differ from each other by a definite amount.
• Value of the variable cannot be expressed in the form of fractions.

• Variable is a complete unit and is capable of expression as such only, for example,

• number of children number of patients number of students number of trees number of eggs.

Number of Children 0 1 2 3 4 5
Number of Families 4 5 4 2 1 1
Types of Frequency Distribution: Continuous Frequency Distribution
• When measurements of the variable are expressed in the form of
2
• Range consisting of a lower limit and an upper limit, 5 7
• It is known as class interval. 4
5 9

• Any frequency distribution expressed in this way of 6

5 11
• Lower limit and Upper limit
8
• is known as Continuous frequency distribution. 5 13

10
5 15
• The variables can be expressed in the form of fractions.
•
Types of Frequency Distribution: Continuous Frequency Distribution
• For example,
• Blood pressure, Weight, Temperature, Pulse rate is measured

• Each measurement convey a definite meaning and is different in significant and use
• For large size of sample, such type of distribution is more suitable.

Weight (Kgs.) No. of Persons

10-20 15
20-30 12 10-20, 20-30, 30-40 etc. are known as classes.
30-40 10
40-50 5 • The values on the left side of the class 10, 20, 30 are lower limits
50-60 8
• The values on the right side of the class 20, 30, 40 are higher limits
Types of frequency
• Simple Frequency:
• It refers to the number of observations that fall into each class interval.
• It also refers to the number of repetitions of items of various class intervals in the population.
• Sum of all the observations gives us total frequency for all the classes.
• Relative frequency:
• It is a particular type of presentation of frequency in which the number of observations falling into
each class interval is expressed as a percentage of the total number of observations.
• Its use is restricted to particular studies.
Types of frequency
• Cumulative frequency:
• It refers to progressive total of frequency of individual class interval.
• It means it is running total for each class.
• Thus a cumulative frequency distribution
• is the sum of the class and all the class below it in a frequency distribution.
Cumulative frequency (Less Than Type)
• The total frequency or sum of frequency of all the classes less than the upper limit of the class
is called the cumulative frequency of that class,
• It is obtained by adding successively the frequency of all the previous classes,
• including the class against which it is to be mentioned.
• Cumulative frequency (C.F):
Cumulative Frequency Simple Frequency
Variable Frequency Classes Frequency
Below 10 15 = a 0-10 15=F1 a = F1
Below 20 24 = b 10-20 9=(24-15)=F2 b – a = F2
Below 30 45 = c 20-30 21=(45-21)=F3 c – b= F3
Below 40 50 = d 30-40 5=(50-45)=F4 d – c = F4
Below 50 75 = e 40-50 25=(75-50)=F5 e – d =F5
In this way, cumulative frequency can be calculated as well as converted to simple frequency for the purpose of
application of different statistical method and procedures

Imp: In Less than type, the last value of the cumulative frequency will be equal to the sum of frequency
Cumulative frequency (More Than Type)
• Cumulative frequency of more than type is also described as greater than type of a particular value of the
variable and
• It is obtained
• by adding the frequencies of all values greater than that value or lower limit of the class,
• starting from the frequency of that particular value
• It will represent the
• Total frequencies of all the classes more than or equal to the class value
or Cumulative Frequency Simple Frequency
• Lower limit of the class to which it relates Variable Frequency Classes Frequency
More than 10 75 = a 10-20 15=(75-60)=F1 a- b = F1
More than 20 60 = b 20-30 20=(60-40)=F2 b – c = F2
More than 30 40 = c 30-40 22=(40-18)=F3 c – d= F3
More than 40 18 = d 40-50 6=(18-12)=F4 d – e = F4
More than 50 12 = e 50-60 12=(12-0)=F5 e =F5

Imp: In More than type, the first value of the cumulative frequency will be equal to the sum of frequency
Class Limits
• Each class interval figure is known as limits of class interval.
• A class is within two limits and the class limits are the lowest and highest value that has been included in
a class.
• These limits are known as ‘lower limit’ and ‘upper limit’
• In a class of 100-120, value 100 is lower limit and 120 is upper limit.
Class Interval:
• These are measurements in which some variable is measured and mentioned in continuous group.
• Difference between class limits or between upper limit and lower limit of the class is known as
• class interval.

• In a class of 100-120, class interval is 20 i.e., 120-100 = 20

Class interval types : Exclusive and Inclusive
• Exclusive class Interval:
• It is a type of class interval
• In which values equal to or more than the upper limit of that class are excluded.

• It means values equal to or more than lower limit

21 is included here
• and values less than upper limit are included in the class.
• For example
Classes 10-20 20-30 30-40 40-50 50-60
Frequency 5 9 12 10 4 = 40

• This method ensures continuity of the data

40 is included here
• because upper limit of one class is the lower limit of succeeding class.

There are no gaps and is continuous.

Class interval types : Exclusive and Inclusive
• Inclusive class Intervals: In this class interval,
• The value equal to upper limit of the concerned class
• is included in that class interval

• In such a case upper limit of the class is less than the lower limit of the succeeding class interval.

• Values upto or more than lower limit and equal to or below upper limit are included in that class.

• Overlapping of the class interval is avoided. For example,

Classes 10-19 20-29 30-39 40-49 50-59
Frequency 4 8 9 11 5 = 37

• There is no continuity
• Because of the gap between upper limit of a class and lower limit of the succeeding class,
• e.g. gap between 20 and 19, 30 and 29
Conversion of Inclusive to Exclusive type
• Some statistical operations can be performed only on exclusive type of classes. So, it becomes essential
to convert inclusive into exclusive classes.

• Correction factor is counted.

𝐿𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑒𝑑𝑖𝑛𝑔 𝑐𝑙𝑎𝑠𝑠 − (𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑝𝑟𝑒𝑐𝑒𝑑𝑖𝑛𝑔 𝑐𝑙𝑎𝑠𝑠)

𝐶. 𝐹. =
2

𝐿2 − 𝑈1
𝐶. 𝐹. =
2
where 𝐿2 is the lower limit of succeeding class
𝑈1 is the upper limit of preceding class

• Now we subtract correction factor from the lower limit of each class
• and then add correction factor in the upper limit of each class.
• It converts inclusive class into exclusive class.
Conversion of Inclusive to Exclusive type
• The new class obtained shall possess all the characteristics of exclusive class.
• There will be change in class limits only.

• Classes: 10-19 20-29 30-39 40-49 50-59

• Converting it into exclusive classes

20−19 1
• Correction Factor (CF) 𝐶. 𝐹. = = = 0.5 (As 𝐿2 = 20, 𝑈1 = 19)
2 2
Exclusive Classes
• Now gap between upper limit and lower limit of the next class has been abolished by conversion.

Lower Limit - CF Upper Limit + CF Classes

10 – 0.5 = 9.5 19 + 0.5 = 19.5 9.5 – 19.5
20 – 0.5 = 19.5 29.5 + 0.5 = 29.5 19.5 – 29.5
30 – 0.5 = 29.5 39 + 0.5 = 39.5 29.5 – 39.5
40 – 0.5 = 39.5 49 + 0.5 = 49.5 39.5 – 49.5
50 – 0.5 = 49.5 59 + 0.5 = 59.5 49.5 – 59.5

• It is to be noted that the

mid value of the previous i.e. inclusive classes and newly obtained exclusive classes shall remain
same.

• There will be complete continuity or no gap in these exclusive classes.

Class Mid Value
• It is known as ‘mid points’ or ‘class mark’.
• It is a value lying half way between upper and lower limit of a class interval. Open End Class New Class
• It means mid value is the average of the two limits of the class interval. Below 20 15 – 20
𝐿𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 +𝑈𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝐿+𝑈 20 – 25 20 – 25
• Mid Value = =
2 2 25 – 30 25 – 30
• Class intervals are represented through mid values. 30 and above 30 – 35
• Statistical operations are performed after calculating mid values of the classes.
OPEN END CLASSES
• If the value of lower limit or upper limit or both limits of some classes are not given or kept open,
Then the intervals are known as open end classes.

• To calculate the mid values in such open end classes,

• Lower limit is estimated on the basis of succeeding class interval,
• Upper limit is determined on the basis of preceding class.
Formation of Frequency Distribution Table
• Example: From the marks obtained by six nursing students, form an array (Individual series)
Marks: 25 18 42 38 35 40

• Solution: Array: 18, 25, 35, 38, 40, 42

• Example: Following is the height measurement of students. Form a discrete frequency distribution
• Height (cms): 100, 120, 105, 106, 110, 120, 105, 112, 110, 111, 105, 112, 106, 105, 110, 111, 120, 100,
106, 110, 105, 106, 112, 106, 105
Height (cm) Tally Bars Frequency
• Solution: Frequency distribution-discrete form 100 || 2
105 |||| | 6
106 |||| 5
110 |||| 4
111 || 2
112 ||| 3
120 ||| 3
Total = 25
Formation of Frequency Distribution Table
• Example: Present the following pulse rate measurements of the patients in the form of frequency
distribution, by taking class magnitude of 4
Pulse rate: 64, 63, 60, 69, 72, 75, 68, 76, 79, 72, 68, 64, 75, 76, 64, 62, 70, 74, 77, 69, 65, 78, 79, 60, 63,
72, 72, 73, 75, 77
• Solution: Frequency distribution table. Class magnitude = 4

Pulse rate Tally bars Patients (F)

60 – 64 |||| 5
64 – 68 |||| 4
68 – 72 |||| 5
72 – 76 |||| |||| 9
76 – 80 |||| || 7
Total = 30

• The above classes are in exclusive form

• In which pulse rate equal to upper limit of the class has not been included in that class.
• Example: Present the following data in cumulative form
Classes: 60-64 64-68 68-72 72-76 76-80
Frequency: 5 4 5 9 7 = 30

• Solution:
• Cumulative frequency (Less than form): Cumulative frequency (More than form):

Pulse rate F Pulse rate Patients Pulse rate F Pulse rate Patients
60 – 64 5 Less than 64 5 60 – 64 5 More than 60 30
64 – 68 4 Less than 68 5+4=9 64 – 68 4 More than 64 30–5 = 25
68 – 72 5 Less than 72 5+4+5 = 14 68 – 72 5 More than 68 30–5–4 = 21
72 – 76 9 Less than 76 5+4+5+9 =23 72 – 76 9 More than 72 30–5–4–5 =16
76 – 80 7 Less than 80 5+4+5+9+7=30 76 – 80 7 More than 76 30–5–4–5–9 =7
Calculate the mid values from the following classes:
Classes: 10–20 20–30 30–40 40–50 50–60
Frequency: 7 9 4 5 2

• Solution: Calculation of Mid Values:

Classes F Mid Values
10-20 7 (10+20)/2 = 15
20-30 9 (20+30)/2 = 25
30-40 4 (30+40)/2 = 35
40-50 5 (40+50)/2 = 45
50-60 2 (50+60)/2 = 55
Determine classes from the following mid values
• MV: 15 25 35 45 55 65
MV Classes Classes
• Solution: Determine classes from mid values:
15 15-5 15+5 10-20
25 25-5 25+5 20-30
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑚𝑖𝑑 𝑣𝑎𝑙𝑢𝑒 10
𝑑= = =5 35 35-5 35+5 30-40
2 2
45 45-5 45+5 40-50
55 55-5 55+5 50-60
• Classes = MV ± d
65 65-5 65+5 60-70

The following points should be considered

• Class interval should be reasonable, preferably 2, 5, 10, 25….50 etc
• No. of classes should be manageable
• As far as possible, class intervals should be equal.
• Avoid open end classes.
• Selection of class intervals and number of classes should be as per purpose of the statistical study.