Find out median of the following series: Items: 20-30, 30-40, 40-50, 50-60, 60-70, 70-80, 80-90 Frequency: 5, 10, 16, 18, 12, 10, 8

Steps to Solve

1. Calculate the cumulative frequencies

To find the median, we first need to find the cumulative frequencies for each class interval.

• 20-30: 5
• 30-40: 5 + 10 = 15
• 40-50: 15 + 16 = 31
• 50-60: 31 + 18 = 49
• 60-70: 49 + 12 = 61
• 70-80: 61 + 10 = 71
• 80-90: 71 + 8 = 79

2. Determine the median class

The median is the middle value of the data set. Since the total frequency (N) is $5 + 10 + 16 + 18 + 12 + 10 + 8 = 79$. The median position is $N/2 = 79/2 = 39.5$ The median class is the class interval where the cumulative frequency just exceeds 39.5, which is the 50-60 class.

3. Apply the median formula

The formula for the median of grouped data is: $Median = L + \frac{(N/2 - cf)}{f} \times h$ Where:

• $L$ is the lower boundary of the median class = 50
• $N$ is the total frequency = 79
• $cf$ is the cumulative frequency of the class preceding the median class = 31
• $f$ is the frequency of the median class = 18
• $h$ is the class width = 10

Now we substitute the values into the formula: $Median = 50 + \frac{(79/2 - 31)}{18} \times 10$ $Median = 50 + \frac{(39.5 - 31)}{18} \times 10$ $Median = 50 + \frac{8.5}{18} \times 10$ $Median = 50 + 0.4722 \times 10$ $Median = 50 + 4.722$ $Median = 54.722 \approx 54.7$