Example: Find the median of the following distribution. Class & Frequency 40-44 & 7 45-49 & 10 50-54 & 22 55-59 & 15 60-64 & 12 65-69 & 6 70-74 & 3

## Step1: Calculate the cumulative frequencies.
### We need to add a cumulative frequency column to the table. The cumulative frequency for each class is the sum of the frequencies of all classes up to and including that class.

## Step2: Determine the median class.
### The median is the middle value of the dataset. First, calculate the total frequency, $$N$$ . Then, find the class that contains the $$\frac{N}{2}$$ -th value. This class is the median class.

## Step3: Apply the median formula.
### The formula for the median of grouped data is:
### $$Median = L + \frac{\frac{N}{2} - CF}{f} \times h$$ , where:
### - $$L$$ is the lower boundary of the median class.
### - $$N$$ is the total frequency.
### - $$CF$$ is the cumulative frequency of the class preceding the median class.
### - $$f$$ is the frequency of the median class.
### - $$h$$ is the class width.

## Step4: Calculate the values for the formula.
### From the table, we can identify the values needed for the median formula.

## Step5: Substitute the values into the formula and calculate the median.
### Substitute the values obtained in the previous step into the median formula and perform the calculation.

**Detailed Solution:**

## Step1: Calculate the cumulative frequencies.
### The cumulative frequencies are calculated as follows:
### - 40-44: 7
### - 45-49: 7 + 10 = 17
### - 50-54: 17 + 22 = 39
### - 55-59: 39 + 15 = 54
### - 60-64: 54 + 12 = 66
### - 65-69: 66 + 6 = 72
### - 70-74: 72 + 3 = 75

## Step2: Determine the median class.
### The total frequency, $$N$$ , is 75.
### $$\frac{N}{2} = \frac{75}{2} = 37.5$$
### The median class is the class that contains the 37.5-th value, which is the 50-54 class (since its cumulative frequency is 39, which is the first cumulative frequency greater than 37.5).

## Step3: Apply the median formula.
### The median formula is: $$Median = L + \frac{\frac{N}{2} - CF}{f} \times h$$

## Step4: Calculate the values for the formula.
### - $$L$$ (lower boundary of the median class 50-54) = 49.5
### - $$N$$ (total frequency) = 75
### - $$CF$$ (cumulative frequency of the class preceding the median class) = 17
### - $$f$$ (frequency of the median class) = 22
### - $$h$$ (class width) = 5

## Step5: Substitute the values into the formula and calculate the median.
### $$Median = 49.5 + \frac{37.5 - 17}{22} \times 5$$
### $$Median = 49.5 + \frac{20.5}{22} \times 5$$
### $$Median = 49.5 + 0.9318 \times 5$$
### $$Median = 49.5 + 4.659$$
### $$Median = 54.159$$
### $$Median \approx 54.16$$