Frequency Distribution: Construction Of Frequency Distribution Tables, Concept Of Class Intervals, Class Mark And Class Boundary
How to Find Class Boundaries in Statistics?
In Statistics class can be defined as a Set of data or members or we can as well say that class is a group of different objects. These classes can be classified based on a few parameters. Example of a class is a set of students who are alumni with same year of graduation. In mathematical terms, it is an interval in frequency distribution. There can be different number of classes consisting of different data. At this point, let us see how to calculate class boundaries in statistics.
Class boundaries can be defined as average of upper limit of a class and lower limit of next class. Therefore, formula for finding class boundaries is given by (upper limit of a class + lower limit of next class) / 2.
We can obtain class boundaries with assistance of steps illustrated below:
Calculate upper limit of class that is last value of class.
Estimate lower limit of next class that is first value of next class.
Calculate the average of upper limit of class and lower limit of next class.
Result we obtain after step 3 is the class boundary.
Let’s take an example to understand the concept of class boundaries.
Assuming we have four classes i.e. 0 – 5, 6 – 10, 10 – 15, 15 – 20.
Then class boundaries for above classes can be obtained as illustrated below:
For class 0 – 5, upper limit = 5, lower limit of next class = 6.
Class boundary = (upper limit of a class + lower limit of next class) / 2.
= (5+6)2 = 5.5. In the same way, we can obtain class boundaries for other classes.
Class Limits, Boundaries And Intervals
Class limits are the smallest and largest observations (data, events etc) in every class. Thus, every class has two limits: a lower and upper.
Example:
| Class | Frequency |
| 200 – 299 | 12 |
| 300 – 399 | 19 |
| 400 – 499 | 6 |
| 500 – 599 | 2 |
| 600 – 699 | 11 |
| 700 – 799 | 7 |
| 800 – 899 | 3 |
| Total Frequency | 60 |
With the use of the frequency table above, what are the lower and upper class limits for the first three classes?
For the first class, 200 – 299
The lower class limit is 200
The upper class limit is 299
For the second class, 300 – 399
The lower class limit is 300
The upper class limit is 399
For the third class, 400 – 499
The lower class limit is 400
The upper class limit is 499
Class Boundaries
Class Boundaries are the midpoints between the upper class limit of a class and the lower class limit of the next class in the sequence. Thus, every class has an upper and lower class boundary.
For instance:
| Class | Frequency |
| 200 – 299 | 12 |
| 300 – 399 | 19 |
| 400 – 499 | 6 |
| 500 – 599 | 2 |
| 600 – 699 | 11 |
| 700 – 799 | 7 |
| 800 – 899 | 3 |
| Total Frequency | 60 |
With the use of the frequency table above, find out the class boundaries of the first three classes.
For the first class, 200 – 299
The lower class boundary is the midpoint between 199 and 200, that is 199.5
The upper class boundary is the midpoint between 299 and 300, that is 299.5
For the second class, 300 – 399
The lower class boundary is the midpoint between 299 and 300, that is 299.5 . The upper class boundary is the midpoint between 399 and 400, that is 399.5
For the third class, 400 – 499
The lower class boundary is the midpoint between 399 and 400, that is 399.5
The upper class boundary is the midpoint between 499 and 500, that is 499.5
Class Intervals
Class interval is the difference between the upper and lower class boundaries of every class.
For instance:
| Class | Frequency |
| 200 – 299 | 12 |
| 300 – 399 | 19 |
| 400 – 499 | 6 |
| 500 – 599 | 2 |
| 600 – 699 | 11 |
| 700 – 799 | 7 |
| 800 – 899 | 3 |
| Total Frequency | 60 |
With the use of the table above, calculate the class intervals for the first class.
For the first class, 200 – 299
The class interval = Upper class boundary – lower class boundary
Upper class boundary = 299.5
Lower class boundary = 199.5
Thus, the class interval = 299.5 – 199.5 = 100
Class limits, class boundaries, class marks.
1. Class limits.
There are two for each class.
The lower class limit of a class is the smallest data figure that can go into the class.
The upper class limit of a class is the largest data value that can go into the class. Class limits have an equivalent accuracy as the data values; the same number of decimal places as the data values.
2. Class boundaries.
They are halfway points that divide the classes.
The lower class boundary of a given class is gotten by averaging the upper limit of the previous class and the lower limit of the given class.
The upper class boundary of a given class is gotten by averaging the upper limit of the class and the lower limit of the next class.
3. Class marks.
They are the midpoints of the classes. They are calculated by averaging the limits.
Example A. DATA: 210, 222, 233, etc
| Class | Frequency | Class limits | Class boundaries | Class mark | Class size |
| 180 – 199 | 2 | 180, 199 | 179.5, 199.5 | 189.5 | 20 |
| 200 – 219 | 5 | 200, 219 | 199.5, 219.5 | 209.5 | 20 |
| 220 – 239 | 12 | 220, 239 | 219.5, 239.5 | 229.5 | 20 |
| 240 – 259 | 6 | 240 – 259 | 239.5, 259.5 | 249.5 | 20 |
| 25 |
| Class | Frequency | Class limits | Class boundaries | Class mark | Class size |
| 9.6 – 14.5 | 10 | 9.6, 14.5 | 9.55, 14.55 | 12.05 | 5 |
| 14.6 – 24.5 | 20 | 14.6, 24,5 | 14.55, 24.55 | 19.55 | 10 |
| 24.6 – 44.5 | 30 | 24.6, 44.5 | 24.55, 44.55 | 29.55 | 20 |
| 44.6 – 54.5 | 25 | 44.6, 54.5 | 44.55, 54.55 | 49.55 | 5 |
| 85 |
Open-ended classes are classes that do not have either an upper limit or a lower limit.
One instance of how to draw a frequency distribution table is with the use of tally marks to count items. A frequency distribution table is one way you can plan data to allow it make more sense. For instance, let’s say you have a list of IQ scores for a gifted classroom in a specific elementary school.
The IQ scores are: 118, 123, 124, 125, 127, 128, 129, 130, 130, 133, 136, 138, 141, 142, 149, 150, 154. This list doesn’t tell you the extent of anything. You could draw a frequency distribution table, which will give a better picture of your data than a simple list.
Frequency and Frequency Tables
The frequency of a specific data value is the number of times the data value takes place.
For instance, if four students have a score of 80 in mathematics, and then the score of 80 is said to have a frequency of 4. The frequency of a data value is frequently represented by f.
A frequency table is constructed by arranging gathered data values in ascending order of magnitude with their corresponding frequencies.
Example 5
The marks awarded for an assignment set for a Year 8 class of 20 students were as follows: 6 7 5 7 7 8 7 6 9 7 4 10 6 8 8 9 5 6 4 8
Present this information in a frequency table.
Solution:
To construct a frequency table, we move on as follows:
Step 1:
Construct a table with three columns. The first column illustrates what is being arranged in ascending order (i.e. the marks). The lowest mark is 4. Therefore, begin from 4 in the first column as illustrated below.
| Mark | Tally | Frequency |
| 4 | ||
| 5 | ||
| 6 | ||
| 7 | ||
| 8 | ||
| 9 | ||
| 10 |
Step 2:
Go through the list of marks. The first mark in the list is 6, therefore put a tally mark against 6 in the second column. The second mark in the list is 7, therefore put a tally mark against 7 in the second column. The third mark in the list is 5, therefore put a tally mark against 5 in the third column as illustrated below.
We proceed with this process until all marks in the list are tallied.
Step 3:
Count the number of tally marks for each mark and write it in third column. The completed frequency table will appear as shown below:
Generally:
We make use of the following steps to construct a frequency table:
Step 1:
Construct a table with three columns. Then in the first column, write down all of the data values in ascending order of magnitude.
Step 2:
To complete the second column, go through the list of data values and put one tally mark at the right place in the second column for every data value. When the fifth tally is reached for a mark, draw a horizontal line via the first four tally marks as illustrated for 7 in the above frequency table. We move on with this process until all data values in the list are tallied.
Step 3:
Count the number of tally marks for each data value and write it in the third column.
Class Intervals (or Groups)
When the set of data values are spread out, it is not easy to set up a frequency table for every data value as there will be a lot of rows in the table. Therefore we group the data into class intervals (or groups) to assist us organize, interpret and analyze the data.
Ideally, we ought to have between five and ten rows in a frequency table. Bear this in mind when deciding the size of the class interval (or group).
Every group begins at a data value that is a multiple of that group. For instance, if the size of the group is 5, then the groups ought to begin with 5, 10, 15, 20 etc. Likewise, if the size of the group is 10, then the groups ought to begin at 10, 20, 30, 40 etc.
The frequency of a group (or class interval) is the number of data values that fall in the range mentioned by that group (or class interval).
Example 6
The number of calls from motorists per day for roadside service was recorded for the month of November 2014. The results were as follows:
Set up a frequency table for this set of data values.
Answer:
To construct a frequency table, we move on as shown below:
Step 1:
Construct a table with three columns, and then write the data groups or class intervals in the first column. The size of each group is 40. Therefore, the groups will begin at 0, 40, 80, 120, 160 and 200 to include all of the data. Observe that in fact we require 6 groups (1 more than we first thought).
Step 2:
Move through the list of data values. For the first data value in the list, 28, place a tally mark against the group 0-39 in the second column. For the second data value in the list, 122, place a tally mark against the group 120-159 in the second column. For the third data value in the list, 217, place a tally mark against the group 200-239 in the second column.
We move on with this process until all of the data values in the set are tallied.
Step 3:
Count the number of tally marks for every group and write it in the third column. The finished frequency table is shown below:
In conclusion
Class interval means the numerical width of any class in a specific distribution.
Mathematically it is defined as the difference between the upper class limit and the lower class limit
Class Interval= Upper Class limit – Lower class limit
In statistics, the data is organized into different classes and the width of such class is known as class interval. Class intervals are in general equal in width but this may not be the case at all times.
Again, they are in general mutually exclusive. Class Intervals are very essential in drawing histograms.
For instance the following are the data of ages of a randomly chosen population of 10 people
8, 19, 58, 35, 45, 12, 6, 13, 18, 47
Then they are grouped as shown below:
| Class | Frequency |
| 0-10 | 2 |
| 10-20 | 4 |
| 20-30 | 0 |
| 30-40 | 1 |
| 40-50 | 2 |
| 50-60 | 1 |
| 60 and above | 0 |
Here, class interval is 10-0=20-10=30-20= 10
In first row, upper limit=10, lower limit= 0, class width= 10-0=10.
The grouping can be done in a different way with different class intervals as well. A frequency distribution illustrates to us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. A few graphs that can be used with frequency distributions are histograms, line charts, bar charts and pie charts. Frequency distributions are used for both qualitative and quantitative data.
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