Given below are the number of family members in 50 families of a locality :
`{:("Number of members",2,3,4,5,6,7,8),("Number of families",5,11,3,8,6,9,8):}`
For this frequency distribution, calculate the mean and median.
Using empirical formula, calculate its mode.

Answer

To solve the problem, we will follow these steps:

### Step 1: Organize the Data
We have the following data for the number of family members and the corresponding number of families:

| Number of Members (X) | Number of Families (F) |
|-----------------------|------------------------|
| 2 | 5 |
| 3 | 11 |
| 4 | 3 |
| 5 | 8 |
| 6 | 6 |
| 7 | 9 |
| 8 | 8 |

### Step 2: Calculate the Mean
To calculate the mean, we need to find the sum of the product of the number of members and the number of families (XF) and then divide it by the total number of families (ΣF).

1. Calculate XF for each row:
- For X = 2: XF = 2 * 5 = 10
- For X = 3: XF = 3 * 11 = 33
- For X = 4: XF = 4 * 3 = 12
- For X = 5: XF = 5 * 8 = 40
- For X = 6: XF = 6 * 6 = 36
- For X = 7: XF = 7 * 9 = 63
- For X = 8: XF = 8 * 8 = 64

2. Now, sum up all the XF values:
- ΣXF = 10 + 33 + 12 + 40 + 36 + 63 + 64 = 258

3. The total number of families (ΣF) is given as 50.

4. Now, calculate the mean:
\[ \text{Mean} = \frac{\Sigma XF}{\Sigma F} = \frac{258}{50} = 5.16 \]

### Step 3: Calculate the Median
To find the median, we need to determine the cumulative frequency (CF) and find the median class.

1. Calculate the cumulative frequency:
- CF for 2: 5
- CF for 3: 5 + 11 = 16
- CF for 4: 16 + 3 = 19
- CF for 5: 19 + 8 = 27
- CF for 6: 27 + 6 = 33
- CF for 7: 33 + 9 = 42
- CF for 8: 42 + 8 = 50

The cumulative frequency table is:

| Number of Members (X) | Number of Families (F) | Cumulative Frequency (CF) |
|-----------------------|------------------------|---------------------------|
| 2 | 5 | 5 |
| 3 | 11 | 16 |
| 4 | 3 | 19 |
| 5 | 8 | 27 |
| 6 | 6 | 33 |
| 7 | 9 | 42 |
| 8 | 8 | 50 |

2. The median position is given by:
\[ \text{Median Position} = \frac{N + 1}{2} = \frac{50 + 1}{2} = 25.5 \]

3. From the CF, we see that the median class is the one where the cumulative frequency first exceeds 25.5, which is 27 (for X = 5).

4. Therefore, the median is:
\[ \text{Median} = 5 \]

### Step 4: Calculate the Mode using the Empirical Formula
The empirical formula for mode is given by:
\[ \text{Mode} = 3 \times \text{Mean} - 2 \times \text{Median} \]

1. Substitute the values:
\[ \text{Mode} = 3 \times 5.16 - 2 \times 5 \]
\[ = 15.48 - 10 = 5.48 \]

### Final Results
- Mean = 5.16
- Median = 5
- Mode = 5.48