Q. 10 - Elementary Statistics

Question

A quantitative data set has been grouped by using limit grouping with equal-width classes of width 5 . The class limits are whole numbers.
 a) If the class mark of the first class is 8, what are its lower and upper limits?
 b) What is the class mark of the second class?
 c) What are the lower and upper limits of the third class?
 d) Which class would contain an observation of 28 ?

Step-by-Step Solution

Answer

Ans:  

(a)   The lower limit of the first class is $6$.
The upper limit of the second class is $8$.

(b)  The class mark of the second class is 13. 

(c)  The class width is 5.
The third class interval is 16-20. 

(d)  The fifth class interval is 26-30.
 Therefore, the fifth class intervals include the observation 28. 

Step 1. Given information.

given,

      A quantitative data set has been grouped by using limit grouping with equal-width classes of width 5 . The class limits are whole numbers. 

Step 2. A set of quantitative data is collected using a standard combination of 5 equal widths.

 a) If the grade 1 grade mark is $8$, we find the lower and upper limit given below:

The lower limit of the first class is,

   $$\begin{gathered} Mark=\frac{Lower limit of the first class+ Upper limit of the first class }{2} \\ \Rightarrow Upper limit of the first class =2 Mark - Lower limit of the first class \\ =2\left(8\right)-6=10 \end{gathered}$$

Therefore,
 The lower limit of the first class is $6$.
The upper limit of the second class is $8$.

Step 3. (b) Now, find the class mark of the second class s given below:

The lower limit of the second class is,

  $$\begin{gathered} = Lower limit of the first class + Class width \\ =6+5 =11 \end{gathered}$$

The upper limit of the second Class is,

   $$\begin{gathered} = Upper limit of the first class + Class width \\ =10+5 =15 \end{gathered}$$

The class width is 5 .
 The second class interval is $11-15$
 Class mark is obtained by using the following formula:

    $$\begin{gathered} Mark=\frac{Lower limit of the second class + Upper limit of the second class }{2} \\ =\frac{11+15}{2}=13 \end{gathered}$$

Therefore, the class mark of the second class is 13.

Setp 4. (c) We find the third class intervals is given below,

The lower limit of the third class is,

  $$\begin{gathered} =Lower limit of the second class + Class width \\ =15+5=16 \end{gathered}$$

The upper limit of the third class is,

    $$\begin{gathered} = Upper limit of the second class + Class width \\ =15+5 =20 \end{gathered}$$

The class width is $5$.
The third class interval is $16-20$.

  

Step 5. (d) We find the fourth class intervals is given below:

The lower limit of the fourth class is,

   $$\begin{gathered} = Lower limit of the third class + Class width \\ =16+5 =21 \end{gathered}$$

The upper limit of the fourth class is,

    $$\begin{gathered} = Upper limit of the third class + Class width \\ =20+5 =25 \end{gathered}$$

Step 6. We find the fifth class intervals is given below:

The lower limit of the fifth class is,

  $$\begin{gathered} = Lower limit of the fourth class Class width \\ =21+5 =26 \end{gathered}$$

The upper limit of the fifth class is,

$$\begin{gathered} = Upper limit of the fourth class + Class width \\ =25+5 =30 \end{gathered}$$

The fifth class interval is $26-30$.
 Therefore, the fifth class intervals include the observation 28.