Find the mean and the mode for the following frequency distribution :
Show Hint
Always ensure that the computed values of mean, median, and mode lie within the range of the overall data set.
The mode must specifically fall within the modal class interval (35--40).
Since \(38.75\) lies within \(35-40\), this acts as a quick confirmation of your calculation accuracy!
Step 1: Understand what is being asked.
We need the mean and mode of a grouped frequency distribution. Instead of using the direct method for the mean, let us use the assumed mean (step-deviation) method, which is often faster when class marks are evenly spaced.
Step 2: Build the table with class marks, frequencies, and deviations from an assumed mean.
The class marks are $32.5, 37.5, 42.5, 47.5, 52.5$ with frequencies $3, 9, 7, 3, 2$. Choose the assumed mean $A=42.5$ (the class mark of the middle class), and find the deviation $d_i=x_i-A$ for each class:
For $x=32.5$: $d=32.5-42.5=-10$, $f=3$, $fd=-30$
For $x=37.5$: $d=37.5-42.5=-5$, $f=9$, $fd=-45$
For $x=42.5$: $d=42.5-42.5=0$, $f=7$, $fd=0$
For $x=47.5$: $d=47.5-42.5=5$, $f=3$, $fd=15$
For $x=52.5$: $d=52.5-42.5=10$, $f=2$, $fd=20$
Step 3: Sum the frequencies and the $fd$ column.
\[ \sum f_i = 3+9+7+3+2 = 24 \]
\[ \sum f_id_i = -30-45+0+15+20 = -40 \]
Step 4: Apply the assumed mean formula.
\[ \text{Mean} = A + \frac{\sum f_id_i}{\sum f_i} = 42.5 + \frac{-40}{24} = 42.5 - 1.67 = 40.83 \]
Step 5: Find the mode using an equivalent, regrouped form of the mode formula.
The modal class is $35$-$40$ (highest frequency 9), with $l=35$, $f_1=9$, $f_0=3$, $f_2=7$, $h=5$. The standard mode formula can be regrouped by splitting its denominator into two separate differences instead of combining them first:
\[ \text{Mode} = l + \frac{f_1-f_0}{(f_1-f_0)+(f_1-f_2)}\times h \]
This is algebraically the same as the usual $2f_1-f_0-f_2$ denominator, just written as a sum of two differences. Substitute the values:
\[ f_1-f_0 = 9-3=6, \quad f_1-f_2 = 9-7=2 \]
\[ \text{Mode} = 35 + \frac{6}{6+2}\times 5 = 35+\frac{6}{8}\times 5 = 35+3.75 = 38.75 \]
Final Answer:
The mean is 40.83 and the mode is 38.75, matching option (A).
\[ \boxed{\text{Mean}=40.83,\ \text{Mode}=38.75} \]
