Question

Find the Class Marks of Classes 30-35, 35-40, and 40-45 in the data below. \[\]

Marks	Number of students
30-35	5
35−40	8
40−45	15
45−50	20
50−55	5
55−60	4
60−65	4
65−70	2
70−75	3

Answer 1

Hint: We first find lower limit and upper limit of the given class intervals 30-35, 35-40 and 40-45. We use the formula for classmark otherwise known as mid-value of a class interval with upper limit $u$ and lower limit $l$ as given by $\text{class mark}=\dfrac{l+u}{2}$.\[\]

Complete step-by-step solution:
We see that the given data in the question is a grouped data where marks are divided into continuous and groups of equal width. The groups are also called bins, categories, or class intervals, or simply classes. Here there are a total of 9 classes starting from 30 and having an equal width of 5 and increasing up to 75. The classes are 30-35, 35-40, 40-45, 45-50, 50-55, 55-60, 60-65, 65-70, and 75-70.\[\]
The numbers of students that fall into a particular class are called frequency because of how many times a particular class is frequented by the data. The frequencies are 5, 18, 15, 20, 5, 4, 4, 2, and 3. \[\]
We know that like all intervals classes have beginning points and ending points. The number at which the classes begin is called the lower limit of the interval and the number at which the classes end is called the upper limit of the interval. If we denote an interval $I$as $I=\left[ l,u \right]$ then $l$ is the lower limit of the interval and $u$ is the upper limit of the interval. \[\]
The classmark otherwise also known as the midpoint of the interval is the average value or mid-value of the lower and upper limit of the interval. The class mark of the interval $I=\left[ l,u \right]$ is given by
\[\text{class mark}=\dfrac{l+u}{2}\]
We are asked in the question to find the classmark of 30-35, 35-40, and 40-45. We see that in the class of 30-35 the lower limit $l=30$ and $u=35$. Then the class mark of 30-35 is
\[\text{class mark 30-35}=\dfrac{30+35}{2}=\dfrac{65}{2}=32.5\]
We see that in the class of 35-40the lower limit $l=35$ and $u=40$. Then the class mark of 35 -40 is
\[\text{class mark 35-40}=\dfrac{35+40}{2}=\dfrac{75}{2}=37.5\]
We see that in the class of 40-45 the lower limit $l=40$ and $u=45$. Then the class mark of 40-45 is \[\text{class mark 40-45}=\dfrac{40+45}{2}=\dfrac{85}{2}=42.5\]

Note: We note that we do not need frequency to find class marks. We can alternatively find the classmark with the formula $\text{class mark}=l+\dfrac{w}{2}$ where $w$ is the class size given by $w=u-l$. The lower and upper-class boundary is the average values of all lower limits and upper limits respectively.