Find the mean, mode and median for the following data :
`{:("Class","Frequency"),(0-10," "8),(10-20," "16),(20-30," "36),(30-40," "34),(40-50," "6),("Total"," "100):}`

Answer

To find the mean, mode, and median for the given frequency distribution data, we will follow these steps:

### Step 1: Create a Frequency Distribution Table
We will organize the given data into a frequency distribution table, including cumulative frequency.

| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
|----------------|---------------|---------------------------|
| 0 - 10 | 8 | 8 |
| 10 - 20 | 16 | 24 |
| 20 - 30 | 36 | 60 |
| 30 - 40 | 34 | 94 |
| 40 - 50 | 6 | 100 |
| **Total** | **100** | |

### Step 2: Find the Median
To find the median, we need to determine the median class.

1. **Calculate n/2**:
- Total frequency (n) = 100
- n/2 = 100/2 = 50

2. **Identify the Median Class**:
- The cumulative frequency just greater than 50 is 60, which corresponds to the class interval **20 - 30**.

3. **Use the Median Formula**:
- Median \( = L + \frac{(n/2 - cf)}{f} \times H \)
- Where:
- \( L = 20 \) (lower limit of the median class)
- \( n/2 = 50 \)
- \( cf = 24 \) (cumulative frequency of the class before median class)
- \( f = 36 \) (frequency of the median class)
- \( H = 10 \) (class width)

Plugging in the values:
\[ \text{Median} = 20 + \frac{(50 - 24)}{36} \times 10 \]
\[ = 20 + \frac{26}{36} \times 10 \]
\[ = 20 + 7.22 \approx 27.22 \]

### Step 3: Find the Mode
To find the mode, we need to identify the modal class and use the mode formula.

1. **Identify the Modal Class**:
- The class with the highest frequency is **20 - 30** (frequency = 36).

2. **Use the Mode Formula**:
- Mode \( = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times H \)
- Where:
- \( L = 20 \) (lower limit of the modal class)
- \( f_1 = 36 \) (frequency of modal class)
- \( f_0 = 16 \) (frequency of the class before modal class)
- \( f_2 = 34 \) (frequency of the class after modal class)
- \( H = 10 \) (class width)

Plugging in the values:
\[ \text{Mode} = 20 + \frac{(36 - 16)}{(2 \times 36 - 16 - 34)} \times 10 \]
\[ = 20 + \frac{20}{72 - 50} \times 10 \]
\[ = 20 + \frac{20}{22} \times 10 \]
\[ = 20 + 9.09 \approx 29.09 \]

### Step 4: Find the Mean
To find the mean, we use the formula:
\[ \text{Mean} = \frac{\sum (f \cdot x)}{n} \]
Where \( x \) is the midpoint of each class.

1. **Calculate Midpoints**:
- For class 0-10: \( x = 5 \)
- For class 10-20: \( x = 15 \)
- For class 20-30: \( x = 25 \)
- For class 30-40: \( x = 35 \)
- For class 40-50: \( x = 45 \)

2. **Calculate \( f \cdot x \)**:
- \( 8 \cdot 5 = 40 \)
- \( 16 \cdot 15 = 240 \)
- \( 36 \cdot 25 = 900 \)
- \( 34 \cdot 35 = 1190 \)
- \( 6 \cdot 45 = 270 \)

3. **Sum of \( f \cdot x \)**:
\[ \sum (f \cdot x) = 40 + 240 + 900 + 1190 + 270 = 2640 \]

4. **Calculate Mean**:
\[ \text{Mean} = \frac{2640}{100} = 26.4 \]

### Final Results
- **Mean**: 26.4
- **Median**: 27.22
- **Mode**: 29.09