Taking of a measurement and the process of counting yield numbers that contain information. The objective of a person applying the tools of statistics to these numbers is to determine the nature of this information. This task is made much easier if the numbers are organized and summarized. Even quite small data sets are difficult to understand without some summarization. Statistical quantities such as the mean and variance can be extremely helpful in summarizing data but first we discuss tabular and graphical summaries. There are several ways to present a statistical data like; Frequency table Simple bar diagrams Multiple Bar Diagrams Histogram Frequency Polygon etc. Steam and Leaf plots Pie Charts A frequency distribution is a tabular arrangement of data in which various items are arranged into classes or groups and the number of items falling in each class is stated. The number of observations falling in a particular class is referred to as class frequency and is denoted by "f". In frequency distribution all the values falling in a class are assumed to be equal to the midpoint of that class. Data presented in the form of a frequency distribution is also called grouped data. A frequency distribution table contains a condensed summary of the original data. There are two types of frequency distribution i) Simple Frequency distribution ) ii) Grouped Frequency distribution.

1. Lecture No 03
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MUHAMMAD ALFAHHAD
FARWA BUTT
“Presentation of data”

2.  Taking of a measurement and the process of counting yield numbers that contain information. The
objective of a person applying the tools of statistics to these numbers is to determine the nature of this
information.
 This task is made much easier if the numbers are organized and summarized.
 Even quite small data sets are difficult to understand without some summarization. Statistical quantities
such as the mean and variance can be extremely helpful in summarizing data but first we discuss
tabular and graphical summaries.
There are several ways to present a statistical data like;
 Frequency table
 Simple bar diagrams
 Multiple Bar Diagrams
 Histogram
 Frequency Polygon etc.
 Steam and Leaf plots
 Pie Charts
Presentation of Data
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3. The Ordered Array
 A first step in organizing data is the preparation of an ordered array. An ordered array is a listing of the
values of a collection (either population or sample) in order of magnitude from the smallest value to the
largest value.
 An ordered array enables one to determine quickly the value of the smallest measurement, the value of the
largest measurement,
 The following table shows number of fishes (frequency) against each magnitude of oxygen intake.
Table: Arrangement in a Class of 1 Magnitude
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4. Frequency Distribution
 A frequency distribution is a tabular arrangement of data in which various items are arranged into classes
or groups and the number of items falling in each class is stated.
 The number of observations falling in a particular class is referred to as class frequency and is denoted
by "f".
 In frequency distribution all the values falling in a class are assumed to be equal to the midpoint of that
class.
 Data presented in the form of a frequency distribution is also called grouped data. A frequency
distribution table contains a condensed summary of the original data.
There are two types of frequency distribution i) Simple Frequency distribution ) ii) Grouped Frequency
distribution
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5. i) Simple Frequency distribution
In simple frequency distribution the observations are not divided into groups or classes but here individual
values are shown.
ii) Grouped Frequency distribution
In grouped frequency distribution the observations are divided into groups or classes with the corresponding
respective frequencies.
The following two tables shows Grouped frequency distribution of Class of 2 magnitude and Class of 3
magnitude.
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6. Class limits/ Class intervals
 The class limits are defined as the values of the variables which are used to separate two classes. The
smaller number is called lower class limit and larger number is called upper class limit.
 Sometimes classes limits/class boundaries are taken as 20--25, 25--30 etc. In such a case, these class
limits means " 20 but less than 25", "25 but less than 30" etc.
 Class Intervals should be neither two low nor too high, generally 5 to 20 classes are ideal.
Class Boundaries
 Class boundaries are the limits up to which the two limit of class may be extended to fill up the gap
between the classes.
 The lower value of class boundary is called lower class boundary and the upper value of class boundary
is called upper class boundary.
 The main object to defined class boundaries is to removes the difficulty, if any, in knowing the class to
which a particular value should be assigned.
Terms associated with the frequency distribution
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7. Terms associated with the frequency distribution
Class Size/ width of class/ Class interval
Class interval is basically the length of a class. It is obtained by
1) The difference between the upper class boundary and the lower class boundary. (Not the difference
between class limits).
2) The difference between either two successive lower class limits or two successive upper class limits.
3) The difference between two successive midpoints.
Mid Value
The class mark or the midpoint is that value which divides a class into two equal parts. It is obtained by
dividing the sum of lower and upper class limits or class boundaries of a class by 2.
Class Frequency
 The number of observation falls in a class is known as class frequency. The sum of frequency of all the
classes of frequency distribution is called the total frequency/ cumulative frequency.
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8. Construction of a Frequency Distribution
1. Decide the number of classes
 No hard and fast rule for deciding on the no of classes. Statistical experience tells us that no less than 5
and no more than 20 classes are generally used.
 The number of classes is determine by the formula i.e. K=1+3.3 log(n), where “K” denotes the number
of classes and “n” denotes the total number of observations.
2. Determine the range of variation of the data.
The difference between the largest and smallest values in the data is called the range of the data. i.e.
R = largest observation - smallest observation
where R denote the range of the data.
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9. 3. Determine the approximate size of class interval
The size of the class interval is determine by dividing the range of the data by the number of classes i.e.
h= R/K; where h denotes the size of the class interval.
In case of fractional results the next higher whole number is usually taken as the size of the class interval.
4. Decide where to locate the class limits
The lower class limit of the first class is started just below the smallest value in the data and then add class
interval to get lower class limit of the next class, repeat this process until the lower class limit of the last
class is achieved.
5. Distribute the data into appropriate classes
Take an observation and marked a vertical bar "I"(Tally) against the class it belongs.
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10. Example: The following data is the final plant height (cm) of thirty plants of wheat. Construct a frequency
distribution.
87, 91, 89, 88, 89, 91, 87, 92, 90, 98, 95 , 97, 96, 100, 101, 96, 98, 99, 98, 100, 102, 99 , 101, 105, 103
107, 105, 106, 107, 112
Solution:
Step- 1: Calculate the Range
R = Largest observation - Smallest observation
= 112 - 87 = 25
Step- 2: Number of classes
The number of classes is determine by the formula
K = 1+3.3 log (n) = 1+3.3 log(30)= 1+3.3(1.4771)= 5.87 = 6
Step-3: Size of class interval
The size of the class interval h= R/K
h = 25/6 = 4.17 = 5
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11. Step- 4: Choose the lowest value
Minimum Value = 87, so start the class interval from 86.
Step-5: Calculate the mid point
Average of lower and upper class limits of any consecutive classes.
Step- 6: Convert the class limits to class boundaries
Step-7: Assigned the observations to the Classes
Starting from first observation and assigned the observation to the classes they belong. Tally mark is
made in the tally column against this class.
2
h
midpiont
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12. Class
Limits
Class
Boundaries
Mid-
Points
Entries Tally f c.f.
86---90 85.5----90.5 88 87,89,88,89,87,90 IIII I 6 6
91---95 90.5----95.5 93 91,91,92,95 IIII 4 10
96---100 95.5----100.5 98 98,97,96,100,96,98,99,98, 100,99 IIII IIII 10 20
101----105 100.5----105.5 103 101,102,101,105,103,105 IIII I 6 26
106----110 105.5----110.5 108 107,106,107 III 3 29
Table: Frequency distribution of the height of plants.
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