Class Mark

In statistics and data analysis, the class mark refers to the midpoint or center value of a class interval in a frequency distribution or histogram. When you have grouped data into intervals or classes, the class mark is the value that represents the center of each interval. It is calculated by adding the lower class limit to the upper-class limit and then dividing by 2. In this article, we will discuss the topic of class marks in detail including its definition, formula and various use cases of class marks as well.

What is a Class Mark?

Class Mark refers to the middle value of each class interval. It’s calculated based on the upper limit and lower limit of the class interval which is termed as a class size also.

In simple words, It’s a middle value or midpoint of the given class, in the frequency distribution. Suppose, we talk about the class marks in statistics. They can be used in various places like calculating mean and drawing line graphs etc. But it’s much easier to find it by using the formula; (Upper Limit value must be added with the Lower Limit value) and this is all divided by 2.

Class Mark Definition

Class mark is the average of upper and lower limit of class interval.

In simple word, we can say class mark is the midpoint of a class interval in a frequency distribution.

Note: Class Mark is arithmatic mean of upper and lower limits of class interval.

Class Mark Formula

The difference between the average upper limit and the average lower limit of the class interval is also known as the class mark. So, for that finding sum of class boundaries we have to use the formula given below;

Class Mark = (Upper limit + Lower limit)/2

Where,

• Class Interval: Class interval represents the size of each group in a frequency distribution. It consists of intervals with specified ranges, like 10-20, 20-30, etc..
• Upper Limit: Upper limit refers to the highest value within a class interval or range in statistical analysis. It helps define the boundary of each interval.
• Lower Limit: It is the class interval’s lowest value. In that specific class, there could not be an item that was smaller than the lower limit. Suppose, the class mark of 10-20. Here, the upper limit is 10.

How to Calculate Class Mark?

To calculate the class interval we have to follow some of the steps;

• Step 1: First, Clarify the Upper limit and the lower limit of the class interval.
• Step 2: After that, Add upper limit and lower limit together.
• Step 3: Lastly, Divide the sum by 2 to find the average, of the class mark.

Class Mark in Frequency Distribution

Class mark in a frequency distribution denotes the average value or midway of each class interval and it’s a process of histogram as well.

Even though, It is computed by summing the class interval’s upper and lower bounds, then dividing the total by two. Whereas, the numbers at the x-axis or horizontal axes can quickly determine the middle of each class.

Age Group	Frequence (fi)	Class Mark (xi)
0-9	40	(9 + 0)/2 = 4.5
10-19	92	(19 + 10)/2 = 14.5
20-29	66	(29 + 20)/2 = 24.5
30-39	38	(39 + 30)/2 = 34.5
40-49	20	(49 + 40)/2 = 44.5
50-59	14	(59 + 50)/2 = 54.5
60-69	9	(69 + 60)/2 = 64.5
70-79	3	(79 + 70)/2 = 74.5

Types of Class Mark

Classmarks can be categorized into two main types which are given below;

• Simple Class Marks
• Exclusive Class Marks
• Inclusive Class Marks

Simple Class Marks

These class marks represent the midpoint of each class interval and are calculated by taking the average of the lower and upper class limits.

For example, if a class interval is from 10 to 20, the simple class mark would be (10 + 20) / 2 = 15.

Exclusive Class Marks

Exclusive class marks are used when the class intervals do not include both endpoints. In this case, the class mark is calculated similarly to simple class marks, but with consideration for the exclusion of one or both endpoints.

For example, if a class interval is from 10 to 20 but excludes 20, the exclusive class mark would be (10 + 19) / 2 = 14.5.

Inclusive Class Marks

Inclusive class marks are used when the class intervals include both endpoints. These class marks represent the midpoint of the inclusive range.

For example, if a class interval is from 10 to 20 and includes both 10 and 20, the inclusive class mark would be (10 + 20 + 1) / 2 = 15.5.

Following table shows the difference between these class mark.

Class Interval	Lower Limit	Upper Limit	Simple Class Mark	Exclusive Class Mark	Inclusive Class Mark
10 – 20	10	20	(10 + 20)/2 = 15	(10 + 19)/2 = 14.5	(10 + 20 + 1)/2 = 15.5
20 – 30	20	30	(20 + 30)/2 = 25	(20 + 29)/2 = 24.5	(10 + 20 + 1)/2 = 25.5
30 – 40	30	40	(30 + 40)/2 = 35	(30 + 39)/2 = 34.5	(10 + 20 + 1)/2 = 35.5
40 – 50	40	50	(40 + 50)/2 = 45	(40 + 49)/2 = 44.5	(10 + 20 + 1)/2 = 45.5
50 – 60	50	60	(50 + 60)/2 = 55	(50 + 59)/2 = 54.5	(10 + 20 + 1)/2 = 55.5

Application of Class Mark

Class marks are used

• To Calculate the Mean
• To make Graphs and Charts

Class Mark in Mean Calculation

In a frequency distribution, the class mark is the average value or midpoint of each class interval.

It is employed to depict the data’s central trend within each interval. As we have read above how we can find the class marks by putting formula. So, After finding the class mark, It is true that the class mark is necessary to determine the mean, or average, in a frequency distribution. The total FiXi value is obtained by multiplying each class’s frequency by the matching class mark and then adding these products.

So, If we want to find the mean. First, we have to find the total frequency(Fi) and the total of FiXi values, then put the formula of the mean. Therefore, To find the mean we can divide this total by the sum of all frequencies.

Mean of any frequency distribution is given as:

x̄ = Σfixi/ Σfi

• x̄ is mean,
• Σfixi is sum of product of f_i and x_i, and
• Σfi is sum of all frequencies.

Let’s consider an example of calculation of mean using class mark.

Example: Find’s mean of the table given below:

Time (In-minutes)	No.of machines (Frequency)
20-25	03
25-30	07
30-35	11
35-40	08
40-45	05
45-50	02

Solution:

Time (in minutes)	(Frequency) Fi	(Class Mark) Xi	Fi × Xi
20-25	3	(20 + 25)/2 = 22.5	3 × 22.5 = 67.5
25-30	7	(25 + 30)/2 = 27.5	7 × 27.5 = 192.5
30-35	11	(30 + 35)/2 = 32.5	11 × 32.5 = 357.5
35-40	8	(35 + 40)/2 = 37.5	8 × 37.5 = 300
40-45	5	(40 + 45)/2 = 42.5	5 × 42.5 = 212.5
45-50	2	(45 + 50)/2 = 47.5	2 × 47.5 = 95
	Σfi = 63		Σfixi = 1225

Let’s calculate the mean now; x̄ = Σfixi/ Σfi

• Σfixi = 1225, and
• Σfi = 63

⇒ x̄ = 1225/63

⇒ x̄ = 34.027

Therefore, the means of the given class mark is 34.027.

Read More about Mean in Statistics.

Class Mark in Graphs

Class marks are frequently shown on the x-axis, or horizontal axis, in graphs and charts to indicate the middle or average value of each class period in histograms. In histograms, frequency polygons, and frequency curves, class marks play a crucial role in representing the data within each class interval. Let’s describe all the terms in short:

• Frequency Polygon: Plotting the class frequency on a line graph against the class midpoint creates a frequency polygon. Whereas, By connecting the tops of the rectangles in the histogram’s midpoints, it can be obtained. It is created by joining the class marks with line segments.
• Frequency Curve: A smoothed curve that shows the data’s frequency distribution is called a frequency curve. It is created by drawing a smooth free-hand curve.

For Example: Draw a histogram of a frequency polygon and a frequency curve of the following data:

Marks	0-10	10-20	20-30	30-40	40-50	50-60
No. of students	5	12	15	22	14	4

Histogram and frequency polygon both for data given above can be illustrated as follows:

Conclusion

In conclusion, class marks are average values of class intervals which helps up plot histogram, frequency polygons and help us analysis the data. It uses upper and lower limit and take it’s average to find the value of class mark for each interval. In this article, we explored the concept of class mark in detail including examples, formula, types as well as applications.

Read More,

• Mean, Median and Mode
• Mean, Median and Mode of Grouped Data
• Graphical Representation of Data

Sample Problems on Class Mark

Example 1: Find the class mark for class interval 10-25.

Solution:

Given:

• Upper Limit = 25
• Lower Limit = 10

As we know, Class Mark = (Upper limit + Lower limit)/2

⇒ Class Mark = (25 + 10)/ 2

⇒ Class Mark = 35/2

⇒ Class Mark = 17.5

Example 2: In the class interval 40-49, find the class mark.

Solution:

Given: Class interval is 40-49.

• Upper Limit = 49
• Lower Limit = 40

As we know, Class Mark = (Upper limit + Lower limit)/2

⇒ Class Mark = (49 + 40)/ 2

⇒ Class Mark = 89/2

⇒ Class Mark = 44.5

Example 3: Find the class marks of the following frequency distribution table:

Class Interval	Frequency
0-10	2
10-20	5
20-30	8
30-40	4
40-50	1

Solution:

Let’s find out the class mark of each class interval.

Class Interval	Frequency	Class Mark
0-10	2	(0+10)/2 = 5
10-20	5	(10+20)/2 = 15
20-30	8	(20+30)/2 = 25
30-40	4	(30+40)/2 = 35
40-50	1	(40+50)/2 = 45

Practice Problems on Class Mark

Q1: In a frequency distribution, the class intervals are 5 – 9, 10 – 14, 15 – 19, and so on. Calculate the class marks for these intervals.

Q2: In a frequency distribution, the class intervals are 50-59, 60-69, 70-79, 80-89, and so on. Calculate the class marks for these intervals.

Q3: For the following frequency distribution table, calculate the class marks:

Class Interval	Frequency
5 – 10	15
11 – 15	20
16 – 20	25
21 – 25	18

Frequently Asked Questions on Class Mark

What is Class Mark?

The class mark is generated by averaging the upper and lower boundaries of the interval, is the midpoint or average value inside each class interval in a frequency distribution.

What is the Formula of Class Mark?

The formula to find class mark is given as, Class mark = (upper limit + lower limit)/2, or (sum of class boundaries)/2.

What is the Class Mark of Class 100-190?

The class mark of the class interval 100-190 is 145.

What is the Class Mark of Class Interval 90-120?

The Class Mark of Class Interval 90-120 is 105.