Find the median for the following distribution: Class 5−10 10−15 15−20 20−25 25−30 30−35 35−40 40−45 Frequency 5 6 15 10 5 4 2 2

Answer

To find the median for the given distribution, we will follow these steps:

### Step 1: Create a frequency table
We will list the class intervals and their corresponding frequencies.

| Class Interval | Frequency |
|----------------|-----------|
| 5−10 | 5 |
| 10−15 | 6 |
| 15−20 | 15 |
| 20−25 | 10 |
| 25−30 | 5 |
| 30−35 | 4 |
| 35−40 | 2 |
| 40−45 | 2 |

### Step 2: Calculate the cumulative frequency (CF)
We will calculate the cumulative frequency by adding the frequencies sequentially:

- CF for 5−10: 5
- CF for 10−15: 5 + 6 = 11
- CF for 15−20: 11 + 15 = 26
- CF for 20−25: 26 + 10 = 36
- CF for 25−30: 36 + 5 = 41
- CF for 30−35: 41 + 4 = 45
- CF for 35−40: 45 + 2 = 47
- CF for 40−45: 47 + 2 = 49

Now, we can summarize the cumulative frequency in a table:

| Class Interval | Frequency | Cumulative Frequency (CF) |
|----------------|-----------|----------------------------|
| 5−10 | 5 | 5 |
| 10−15 | 6 | 11 |
| 15−20 | 15 | 26 |
| 20−25 | 10 | 36 |
| 25−30 | 5 | 41 |
| 30−35 | 4 | 45 |
| 35−40 | 2 | 47 |
| 40−45 | 2 | 49 |

### Step 3: Determine n and n/2
The total number of observations (n) is the sum of the frequencies, which is 49.

Now, we calculate n/2:
\[ n/2 = 49/2 = 24.5 \]

### Step 4: Identify the median class
We look for the cumulative frequency that is greater than 24.5. The cumulative frequency of 26 (for the class interval 15−20) is the first one that exceeds 24.5. Therefore, the median class is 15−20.

### Step 5: Use the median formula
The formula for the median is:
\[ \text{Median} = l + \left( \frac{n/2 - CF}{f} \right) \times h \]

Where:
- \( l \) = lower boundary of the median class = 15
- \( n/2 \) = 24.5
- \( CF \) = cumulative frequency of the class preceding the median class = 11 (for class 10−15)
- \( f \) = frequency of the median class = 15
- \( h \) = width of the median class = 20 - 15 = 5

### Step 6: Substitute the values into the formula
Now we substitute the values into the median formula:
\[ \text{Median} = 15 + \left( \frac{24.5 - 11}{15} \right) \times 5 \]
\[ = 15 + \left( \frac{13.5}{15} \right) \times 5 \]
\[ = 15 + 0.9 \times 5 \]
\[ = 15 + 4.5 \]
\[ = 19.5 \]

### Final Answer
The median of the given distribution is **19.5**.

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