The course is a set of slides with the prominent concept of Grouped frequency in Statistical Techniques in Geography.

1. Sanat Kumar Purkait
Assistant Professor & Head
Department Of Geography
Raidighi College
(University Of Calcutta)

2.  In statistics, frequency refers to the number of times a particular observation or value
occurs in a dataset. For example, if we have a dataset of the ages of a group of people,
the frequency of a particular age would be the number of people in the group who
have that age.
 Frequency can be expressed as an absolute frequency or a relative frequency.
Absolute frequency is simply the number of times a value occurs in a dataset, while
relative frequency is the proportion of times a value occurs relative to the total
number of observations in the dataset.
 Frequency distribution is a tabular representation of the frequencies of different
values in a dataset. It shows the number of observations that fall into each category or
value range, along with their corresponding frequencies. Frequency distributions are
commonly used in descriptive statistics to summarize and display the distribution of
data in a meaningful way.
 The concept of frequency is fundamental to many statistical methods, such as
hypothesis testing, probability theory, and sampling theory. It provides a way to
quantify the distribution of data and to draw conclusions about the underlying
population from a sample of observations.
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3. In statistics, frequency can be classified into the following types:
1. Absolute frequency: This refers to the number of times a particular value or observation
occurs in a dataset.
2. Relative frequency: This is the proportion of times a particular value or observation occurs in a
dataset relative to the total number of observations in the dataset.
3. Cumulative frequency: This is the sum of the frequencies of all the values up to and including a
particular value or observation in a dataset.
4. Cumulative relative frequency: This is the sum of the relative frequencies of all the values up to
and including a particular value or observation in a dataset.
5. Grouped frequency: This is the frequency of a range or interval of values in a dataset. Grouped
frequency is used when dealing with continuous data that is grouped into intervals or
categories.
6. Conditional frequency: This is the frequency of a particular value or observation in a dataset,
given that another variable or condition is true.
These different types of frequency can be used to analyze and summarize different aspects of a
dataset, depending on the research question and the type of data being analyzed.
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4. Here are examples of each classification of frequency:
1. Absolute frequency: Consider the following dataset of test scores: 80, 75, 90, 80, 85.
The absolute frequency of the score 80 is 2, because it appears twice in the dataset.
2. Relative frequency: Using the same dataset as above, the relative frequency of the
score 80 is 2/5 or 0.4, because it appears twice out of a total of five observations.
3. Cumulative frequency: Continuing with the same dataset, the cumulative frequency of
the score 80 is 2, because there are two scores of 80 or lower in the dataset (i.e., 75
and 80).
4. Cumulative relative frequency: With the same dataset, the cumulative relative
frequency of the score 80 is 0.6, because the sum of the relative frequencies of the
scores 75, 80, and 85 is 0.6 (i.e., 0.2+0.4+0.0).
5. Grouped frequency: Suppose we have a dataset of heights (in inches) of 50 students.
Rather than analyzing each individual height, we group them into intervals of 5
inches: 60-64, 65-69, 70-74, etc. The grouped frequency would then show the number
of students that fall within each interval.
6. Conditional frequency: Consider a dataset of students' grades in two subjects, Math
and Science. We can calculate the frequency of students who received an A in Math,
given that they also received an A in Science. This would be an example of conditional
frequency.
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5. 2 April 2023 SANAT K PURKAIT; Raidighi College 5
1. Class interval, or class.
2. Class frequency and total frequency.
3. Class limits- lower class limit and upper class limit.
4. Class boundaries- lower class boundary and upper class boundary.
5. Class mark or mid value, or mid point of class interval;
6. Width or size of class interval;
7. Frequency density.

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Age ( years ) Frequency ( No of Persons)
15-19 37
20-24 81
25-29 43
30-34 24
35-39 9
39-44 6
Total 200
GROUP FREQUENCY DISTRIBUTION
Now we will explain those
terms with reference to the above frequency table

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Class
interval
Frequency
(f)
Class
boundary
Mid value
(x)
Class width
(w)
Frequency
density
(f/w)
cumulative frequency
Less than More than
15-19 37 14.5-19.5 17 5 7.4 37 200
20-24 81 19.5-24.5 22 5 16.2 118 163
25-29 43 24.5-29.5 27 5 8.6 161 82
30-34 24 29.5-34.5 32 5 4.8 185 39
35-44 9 34.5-44.5 39.5 10 0.9 194 15
45-59 6 44.5-59.5 52 15 0.4 200 6
TOTAL 200
FREQUENCY TABLE

8. 2 April 2023 SANAT K PURKAIT; Raidighi College 8
When a large number of observations varying in a wide range are
available, these are usually classified in several groups according to the
size of values. Each of these groups , defined by an interval, is called class
interval, or simply class. In the previous table , column no 1 the class
interval of ages ( years) are 15-19, 20-24 etc. There are six classes in the
frequency distribution, the last class being 45-49.
When one end of a classes is not specified, the class is called an open
end classes. A frequency distribution may either one or open end classes
arises when there are relatively few observations which are apart from the
rest. In such a case, it is not considered worth-while to show several
classes with zero frequencies ( called empty class ) before reaching a class
with a very small frequency.

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The number of observations following within a class is
called its class frequency or, simply frequency. The sum of all the
class frequencies is called total frequency. In the given table, the
total frequency is 200. total frequency shows the total number of
observations considered in the frequency distribution.

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All recorded data or observations are discrete in character. For a discrete
variable, the values themselves are isolated, e.g. Records regarding the no of
workers employed in a factory will such data such as 200, 226, 512, 66 etc. ( no
fractional numbers are possible ). .Although a continuous variable can theoretically
take value ,all observations are rounded off a certain unit for convenience. For
example, if it is decided to record the height of persons to the nearest inch, we get
observations like 58, 67, 63 etc. If the decision is to record them to the nearest
tenth of an inch we may get 37.8, 67.5, 63 etc. In any case, all observations will be
rounded, either all whole numbers, or all showing upto one place of decimal, and
so on.

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When a grouped frequency distribution is constructed from these collected
data, the value of the variable are shown in several classes. For determining the
class frequencies it is necessary that these classes are mutually exclusive, i.e. Be
such that any observation is contained in only one class; for example 54-56, 57-59,
60-62 etc. ( not like 54-56, 56-58, 58-60, etc.) or 54.1-56.0, 56.1-58.0, 58.1-60.0, etc (
54.0-56.0, 56.0-58.0, 58.0-60.0 etc.). Note that classes will be written to the same
degree of precision as the original data.
In the construction of a grouped frequency distribution, the class intervals
must therefore be defined by pairs of numbers such that the upper end of one
class does not coincide with the lower end of the immediately- following class.
The two numbers used to specify the limits of a class interval for the purpose of
tallying the original observations into the various classes, are called ‘ class limits’.
The smaller of the pair is known as lower class limit and the larger as upper class
limit with reference to the particular class.

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When measurements are taken on a continuous variable, all data are
recorded nearest to a certain unit. Thus, if ages are recorded to the nearest
whole number of years, any age between 14.5 years and 15.5 years is recorded
as 15 years. Similarly, ’19 years’ denotes an age between 18.5 years and 19.5
years. Hence the class interval 15-19 actually includes all ages between 14.5 and
19.5 years. These most extreme values which would ever be included in a class
interval are called ‘class boundaries’. Class boundaries , are in fact, the real
limits of a class interval. The lower extreme point is called lower class
boundary, and the upper extreme point is called upper class boundary with
reference to any particular class.

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Class boundaries may be calculated from class limits by applying the following rule.
•Lower Class Boundary = Lower Class Limit-1/2d
•Upper Class Boundary = Upper Class Limit+ 1/2d
Where d is the common difference between the upper
class limit of any class interval and lower class limit of the
next class interval. In fact, if observations are recorded to the
nearest unit, d=1; if to the nearest tenth of a unit , d=0.1, etc.

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Note that the upper boundary of any class coincides with the lower
boundary of the next class; but the upper limit of any class is different from
the lower limit of the next class. This fact may be utilized in deciding
whether the pairs of numbers defining the class intervals in a given
frequency distribution are really class limits or class boundaries. Class
limits are used only for the construction of a grouped frequency
distribution, but in all statistical calculations and diagrams involving end
points of a classes ( e.g. Median, mode, quartiles and histogram, ogive
etc.), class boundaries are used.

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The value exactly at the middle of a class interval is called class mark or, mid
value. If lies half- way between the class limits or between the class
boundaries.
Class mark=( lower class limit + upper class limit )/2
Or,
Class mark =( lower class boundary + upper class boundary)/2
Class mark is used as a representative value of the class interval, for the
calculation of mean, standard deviation, mean deviation, etc.

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Width of a class is the difference between the lower and upper
class boundaries ( not class limits ).
Width of class = (upper class boundary – lower class boundary )

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x axis
Series 1
Series 2
Series 3
Series 4
Series 5
Series 6
Series 7

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