Question

1. table 1.12 shows the distance covered by 20 randomly selected runners in kilometers.

table 1.12
\

$$\begin{array}{|c|c|c|c|} \\hline \\text{class limit} & \\text{frequency} & \\text{class boundary} & \\text{lcf} \\\\ \\hline 6-10 & 2 & 5.5-10.5 & 2 \\\\ \\hline 11-15 & 1 & 10.5-15.5 & 3 \\\\ \\hline 16-20 & 4 & 15.5-20.5 & 7 \\\\ \\hline 21-25 & 6 & 20.5-25.5 & 13 \\\\ \\hline 26-30 & 5 & 25.5-30.5 & 18 \\\\ \\hline 31-35 & 2 & 30.5-35.5 & 20 \\\\ \\hline \\end{array}$$

\text{lcb} =
\text{cf} =
\text{f} =
\text{n} =
\text{i} =

\frac{n}{2} = \frac{20}{2} = 10

Explanation:

⚡ Using what you learned: Median · 🆕 New Concept Discovered: Grouped Data Median
Finding the middle value when data is grouped into intervals.

Step 1: Identify the Median Class

To find the median of grouped data, we first locate the median class. This is the class interval containing the middle value, which is at position:

$$ \frac{n}{2} = \frac{20}{2} = 10 $$

Looking at the cumulative frequency (\(cf\)) column:

• The first class (\(6\text{--}10\)) contains up to the 2nd value.
• The second class (\(11\text{--}15\)) contains up to the 3rd value.
• The third class (\(16\text{--}20\)) contains up to the 7th value.
• The fourth class (\(21\text{--}25\)) contains up to the 13th value.

Since the 10th value falls into the cumulative frequency of 13, the median class is \(21\text{--}25\).

Step 2: Determine the Required Parameters

Using the median class \(21\text{--}25\) (with class boundary \(20.5\text{--}25.5\)), we extract the values for each variable:

• \(LCB\) (Lower Class Boundary of the median class):

$$ LCB = 20.5 $$

• \(cf\) (Cumulative frequency of the class before the median class):

$$ cf = 7 $$

• \(f\) (Frequency of the median class):

$$ f = 6 $$

• \(n\) (Total number of frequencies):

$$ n = 20 $$

• \(i\) (Class width/interval size):

$$ i = 25.5 - 20.5 = 5 $$

Answer:

• \(LCB\) = \(20.5\)
• \(cf\) = \(7\)
• \(f\) = \(6\)
• \(n\) = \(20\)
• \(i\) = \(5\)