The following distribution gives the daily income of 50 workers of a factory:
`{:("Daily income", 200-250,250-300,300-350,350-400,400-450,450-500),("Number of workers",10,5,11,8,6,10):}`
Convert the distribution to a less than type cumulative frequency distribution and draw its ogive. Hence obtain the median daily income.

Answer

To convert the given distribution of daily income into a less than type cumulative frequency distribution and to find the median, we will follow these steps:

### Step 1: Create a Cumulative Frequency Table

We will first create a table to organize the data. The table will have the following columns: Daily Income Range, Number of Workers (Frequency), and Less Than Type Cumulative Frequency.

| Daily Income Range | Number of Workers (Frequency) | Less Than Type Cumulative Frequency |
|---------------------|-------------------------------|-------------------------------------|
| 200 - 250 | 10 | 10 |
| 250 - 300 | 5 | 15 |
| 300 - 350 | 11 | 26 |
| 350 - 400 | 8 | 34 |
| 400 - 450 | 6 | 40 |
| 450 - 500 | 10 | 50 |

### Step 2: Calculate Less Than Type Cumulative Frequencies

- For the range 200 - 250: Cumulative frequency = 10
- For the range 250 - 300: Cumulative frequency = 10 + 5 = 15
- For the range 300 - 350: Cumulative frequency = 15 + 11 = 26
- For the range 350 - 400: Cumulative frequency = 26 + 8 = 34
- For the range 400 - 450: Cumulative frequency = 34 + 6 = 40
- For the range 450 - 500: Cumulative frequency = 40 + 10 = 50

### Step 3: Plot the Ogive

To draw the ogive, we will plot the cumulative frequencies against the upper limits of the income ranges.

- Points to plot:
- (250, 10)
- (300, 15)
- (350, 26)
- (400, 34)
- (450, 40)
- (500, 50)

### Step 4: Draw the Ogive

1. Draw a graph with the x-axis representing Daily Income and the y-axis representing Cumulative Frequency.
2. Mark the points obtained from the table.
3. Connect the points with a smooth curve.

### Step 5: Find the Median

To find the median, we need to locate \( n/2 \) where \( n \) is the total number of workers.

- Total number of workers, \( n = 50 \)
- Therefore, \( n/2 = 25 \)

Now, we look at the cumulative frequency to find the median class:
- The cumulative frequency just greater than 25 is 26, which corresponds to the income range 300 - 350.

**Median Class: 300 - 350**

### Step 6: Apply the Median Formula

Using the median formula:

\[ \text{Median} = L + \left( \frac{\frac{n}{2} - CF}{f} \right) \times h \]

Where:
- \( L \) = lower limit of the median class = 300
- \( n \) = total frequency = 50
- \( CF \) = cumulative frequency of the class preceding the median class = 15 (for the class 250 - 300)
- \( f \) = frequency of the median class = 11
- \( h \) = class width = 50 (from 300 to 350)

Substituting the values:

\[ \text{Median} = 300 + \left( \frac{25 - 15}{11} \right) \times 50 \]
\[ = 300 + \left( \frac{10}{11} \right) \times 50 \]
\[ = 300 + \frac{500}{11} \]
\[ = 300 + 45.45 \approx 345.45 \]

### Final Result

The median daily income is approximately **345.45**.

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