Compute the first quartile and third deciles from the following data :
`{:("Weekly Income (in Rs.)",58,59,60,61,62,63,64,65,66),("No. of workers",2,3,6,15,10,5,4,3,1):}`

Answer

To compute the first quartile (Q1) and third decile (D3) from the given data, we will follow these steps:

### Step 1: Organize the Data
We have the following data:
- Weekly Income (in Rs.): 58, 59, 60, 61, 62, 63, 64, 65, 66
- Number of Workers: 2, 3, 6, 15, 10, 5, 4, 3, 1

### Step 2: Create a Frequency Table
We will create a frequency table that includes the cumulative frequency.

| Weekly Income (Rs.) | No. of Workers | Cumulative Frequency |
|----------------------|----------------|----------------------|
| 58 | 2 | 2 |
| 59 | 3 | 5 |
| 60 | 6 | 11 |
| 61 | 15 | 26 |
| 62 | 10 | 36 |
| 63 | 5 | 41 |
| 64 | 4 | 45 |
| 65 | 3 | 48 |
| 66 | 1 | 49 |

### Step 3: Calculate the Total Number of Workers (N)
Total number of workers (N) = 49

### Step 4: Calculate the First Quartile (Q1)
The first quartile (Q1) is calculated using the formula:
\[ Q1 = \frac{N}{4} \]
\[ Q1 = \frac{49}{4} = 12.25 \]

Now, we look for the cumulative frequency just greater than 12.25. From the table, the cumulative frequency of 26 (at income 61) is the first one greater than 12.25.

Thus, **Q1 = 61 Rs.**

### Step 5: Calculate the Third Decile (D3)
The third decile (D3) is calculated using the formula:
\[ D3 = \frac{3N}{10} \]
\[ D3 = \frac{3 \times 49}{10} = 14.7 \]

Now, we look for the cumulative frequency just greater than 14.7. From the table, the cumulative frequency of 26 (at income 61) is the first one greater than 14.7.

Thus, **D3 = 61 Rs.**

### Step 6: Final Results
- First Quartile (Q1) = 61 Rs.
- Third Decile (D3) = 61 Rs.