Question

Find the ‘mean’ and ‘mode’ marks of the following data:

Hint:

For grouped data, use class midpoints to calculate mean and identify the modal class (highest frequency) to find mode.

Answer 1

Step 1: Calculate Mean
We use the formula for mean: \[ \bar{x} = \dfrac{\sum fx}{\sum f} \] \begin{center} \begin{tabular}{|c|c|c|c|} \hline Class Interval & Frequency (f) & Midpoint (x) & \(fx\)
\hline 0 -- 5 & 2 & 2.5 & 5.0
5 -- 10 & 3 & 7.5 & 22.5
10 -- 15 & 8 & 12.5 & 100.0
15 -- 20 & 15 & 17.5 & 262.5
20 -- 25 & 14 & 22.5 & 315.0
25 -- 30 & 8 & 27.5 & 220.0
\hline \multicolumn{3}{|c|}{Total} & 925.0
\hline \end{tabular} \end{center} \[ \sum f = 2 + 3 + 8 + 15 + 14 + 8 = 50,\quad \sum fx = 925 \Rightarrow \bar{x} = \dfrac{925}{50} = 18.5 \] Step 2: Calculate Mode
The mode is found using the formula: \[ \text{Mode} = l + \left( \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \cdot h \] Where: \begin{itemize} \item Modal class = class with highest frequency = 15 -- 20 \item \(l = 15,\ f_1 = 15,\ f_0 = 8,\ f_2 = 14,\ h = 5\) \end{itemize} \[ \text{Mode} = 15 + \left( \dfrac{15 - 8}{2 \cdot 15 - 8 - 14} \right) \cdot 5 = 15 + \left( \dfrac{7}{30 - 22} \right) \cdot 5 = 15 + \dfrac{35}{8} = 15 + 4.375 = 19.375 \]