The ages of students in a small primary school were recorded in the table below. Age 5-6 7-8 9-10 Frequency 29 40 38 Estimate the median.

Answer Details

To estimate the median, we need to find the midpoint of the data set. In other words, we need to find the value that separates the data into two equal parts.
To do this, we can use the cumulative frequency. The cumulative frequency is the sum of the frequencies up to a certain point. In this case, we add up the frequencies starting from the youngest age group (5-6 years) and continue until we reach a cumulative frequency that is greater than the total number of students divided by 2.
Let's calculate the cumulative frequency:
5-6 years: 29 students 5-6 years + 7-8 years = 29 + 40 = 69 students 5-6 years + 7-8 years + 9-10 years = 69 + 38 = 107 students
Since the total number of students is 107 and we want to find the midpoint, which is the median, we divide 107 by 2 to get 53.5. This means that the median falls between the second and third age groups.
To estimate the median, we can use the cumulative frequency of the second age group (69) as a reference. The median lies within this age group. To find the exact estimated median, we can use the formula:
Median = Lower boundary of median group + ((Total/2) - Cumulative frequency of previous group) / Frequency of median group * Width of median group
The lower boundary of the median group is 7, the cumulative frequency of the previous group is 29, the frequency of the median group is 40, and the width of the median group is 2 (since the age group is 7-8 years).
Now, let's plug in these values into the formula:
Median = 7 + ((53.5 - 29) / 40) * 2
Calculating this, we get:
Median ≈ 7.725
Therefore, the estimated median of the ages of the students is approximately 7.725.