To find out the concentration of SO₂ in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:
| Concentration of SO₂ (in ppm) | Frequency |
|---|---|
| 0.00 - 0.04 | 4 |
| 0.04 - 0.08 | 9 |
| 0.08 - 0.12 | 9 |
| 0.12 - 0.16 | 2 |
| 0.16 - 0.20 | 4 |
| 0.20 - 0.24 | 2 |
Find the mean concentration of SO₂ in the air.
To find the mean concentration, we use the Direct Method. First, we calculate the class mark () for each class interval by averaging the upper and lower limits. Then, we find the product of frequency and class mark () for each class.
The calculations are shown in the table below:
| Concentration of SO₂ (ppm) | Frequency () | Class Mark () | |
|---|---|---|---|
| 0.00 - 0.04 | 4 | 0.02 | 0.08 |
| 0.04 - 0.08 | 9 | 0.06 | 0.54 |
| 0.08 - 0.12 | 9 | 0.10 | 0.90 |
| 0.12 - 0.16 | 2 | 0.14 | 0.28 |
| 0.16 - 0.20 | 4 | 0.18 | 0.72 |
| 0.20 - 0.24 | 2 | 0.22 | 0.44 |
| Total |
Using the formula , we substitute the values: . Therefore, the mean concentration of SO₂ in the air is 0.099 ppm.
Explanation
The solution follows the Direct Method for grouped data as illustrated in the provided context (Table 13.3). The context explains that for grouped data, we find the class mark () for each interval, multiply it by the frequency (), and sum these products (). The mean is then calculated by dividing this sum by the total frequency ().
Solution Steps
Step 1: Calculate class marks () for each interval using the formula .
Step 2: Calculate the product for each class interval.
Step 3: Sum the frequencies to get and sum the products to get .
Step 4: Apply the mean formula .