Sure! Let's calculate the mean, mode, and median of the given data step-by-step.
### (i) Mean of the data:
1. List the class intervals and corresponding frequencies:
[tex]\[
\begin{array}{c|c}
\text{Class Interval} & \text{Frequency (f)} \\
\hline
51-55 & 2 \\
56-60 & 10 \\
61-65 & 22 \\
66-70 & 34 \\
71-75 & 15 \\
76-80 & 10 \\
81-85 & 5 \\
91-95 & 1 \\
\end{array}
\][/tex]
2. Calculate the class midpoints ([tex]\(x_i\)[/tex]):
[tex]\[
\begin{array}{c|c}
\text{Class Interval} & \text{Midpoint (x_i)} \\
\hline
51-55 & \frac{51+55}{2} = 53 \\
56-60 & \frac{56+60}{2} = 58 \\
61-65 & \frac{61+65}{2} = 63 \\
66-70 & \frac{66+70}{2} = 68 \\
71-75 & \frac{71+75}{2} = 73 \\
76-80 & \frac{76+80}{2} = 78 \\
81-85 & \frac{81+85}{2} = 83 \\
91-95 & \frac{91+95}{2} = 93 \\
\end{array}
\][/tex]
3. Multiply each midpoint by its corresponding frequency:
[tex]\[
\begin{array}{c|c}
\text{Midpoint (x_i)} & \text{Frequency (f_i)} \\
\hline
53 & 2 \times 53 = 106 \\
58 & 10 \times 58 = 580 \\
63 & 22 \times 63 = 1386 \\
68 & 34 \times 68 = 2312 \\
73 & 15 \times 73 = 1095 \\
78 & 10 \times 78 = 780 \\
83 & 5 \times 83 = 415 \\
93 & 1 \times 93 = 93 \\
\end{array}
\][/tex]
4. Sum up the products and the total frequency:
[tex]\[
\sum f_i x_i = 106 + 580 + 1386 + 2312 + 1095 + 780 + 415 + 93 = 6767
\][/tex]
[tex]\[
\sum f_i = 2 + 10 + 22 + 34 + 15 + 10 + 5 + 1 = 100
\][/tex]
5. Calculate the mean:
[tex]\[
\text{Mean} = \frac{\sum f_i x_i}{\sum f_i} = \frac{6767}{100} = 67.67
\][/tex]
### (ii) Mode of the data:
1. Identify the modal class (the class with the highest frequency):
The class [tex]\(66-70\)[/tex] has the highest frequency of 34.
2. Find the midpoint of the modal class:
[tex]\[
\text{Midpoint of } 66-70 = \frac{66+70}{2} = 68
\][/tex]
So, the mode is 68.
### (iii) Median of the data:
1. Calculate the cumulative frequencies:
[tex]\[
\begin{array}{c|c}
\text{Class Interval} & \text{Cumulative Frequency (CF)} \\
\hline
51-55 & 2 \\
56-60 & 2 + 10 = 12 \\
61-65 & 12 + 22 = 34 \\
66-70 & 34 + 34 = 68 \\
71-75 & 68 + 15 = 83 \\
76-80 & 83 + 10 = 93 \\
81-85 & 93 + 5 = 98 \\
91-95 & 98 + 1 = 99 \\
\end{array}
\][/tex]
2. Find the median class:
The median class is the class where the cumulative frequency just surpasses [tex]\(\frac{N}{2} = \frac{100}{2} = 50\)[/tex]. Thus, the median class is [tex]\(66-70\)[/tex].
3. Find the midpoint of the median class:
[tex]\[
\text{Midpoint of } 66-70 = \frac{66+70}{2} = 68
\][/tex]
So, the median is also 68.
### Summary:
(i) Mean of the data: [tex]\( 68.35 \)[/tex]
(ii) Mode of the data: [tex]\( 68.0 \)[/tex]
(iii) Median of the data: [tex]\( 68.0 \)[/tex]
