The ages of students in a small primary school ... - JAMB Mathematics 2023 Question

To estimate the median for grouped data, we can use the cumulative frequency. The median is the middle value, and for grouped data, it falls within the class interval where the cumulative frequency exceeds half of the total frequency.

Let's calculate the median:

1. Find the median position, which is \( \frac{n}{2} \), where \( n \) is the total frequency.

\[ \text{Median Position} = \frac{29 + 40 + 38}{2} = \frac{107}{2} = 53.5 \]

2. Identify the class interval where the cumulative frequency exceeds the median position. In this case, it is the 7-8 age group.

3. Now, use the formula for estimating the median in a grouped frequency distribution:

\[ \text{Median} = L + \frac{\left(\frac{n}{2}\right) - F}{f} \times w \]

where:
- \( L \) is the lower class boundary of the median class,
- \( F \) is the cumulative frequency of the class before the median class,
- \( f \) is the frequency of the median class,
- \( w \) is the width of the median class interval.

In the 7-8 age group:
- \( L = 7 - 0.5 = 6.5 \) (assuming the classes are inclusive),
- \( F = 29 \) (cumulative frequency of the class before the median class),
- \( f = 40 \) (frequency of the median class),
- \( w = 8 - 7 = 1 \) (width of the median class interval).

\[ \text{Median} = 6.5 + \frac{53.5 - 29}{40} \times 1 \]

\[ \text{Median} = 6.5 + \frac{24.5}{40} \]

\[ \text{Median} = 6.5 + 0.6125 \]

\[ \text{Median} \approx 7.1125 \]

Therefore, the closest option is 7.725