To calculate the mean concentration of [tex]\( SO_2 \)[/tex] in the air, we'll follow these steps:
### Step 1: Identify the Concentration Ranges and Frequencies
We have the concentration ranges and their corresponding frequencies provided in the data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline \text{Concentration range (ppm)} & 0.00-0.04 & 0.04-0.08 & 0.08-0.12 & 0.12-0.16 & 0.16-0.20 & 0.20-0.24 \\
\hline \text{Frequency} & 4 & 9 & 9 & 2 & 4 & 2 \\
\hline
\end{array}
\][/tex]
### Step 2: Calculate Midpoints of Each Concentration Range
To find the midpoint for each range, we use the formula for the midpoint [tex]\[
\text{Midpoint} = \frac{\text{Lower Boundary} + \text{Upper Boundary}}{2}
\][/tex]
- For the range [tex]\( 0.00-0.04 \)[/tex]: [tex]\(\text{Midpoint} = \frac{0.00 + 0.04}{2} = 0.02\)[/tex]
- For the range [tex]\( 0.04-0.08 \)[/tex]: [tex]\(\text{Midpoint} = \frac{0.04 + 0.08}{2} = 0.06\)[/tex]
- For the range [tex]\( 0.08-0.12 \)[/tex]: [tex]\(\text{Midpoint} = \frac{0.08 + 0.12}{2} = 0.10\)[/tex]
- For the range [tex]\( 0.12-0.16 \)[/tex]: [tex]\(\text{Midpoint} = \frac{0.12 + 0.16}{2} = 0.14\)[/tex]
- For the range [tex]\( 0.16-0.20 \)[/tex]: [tex]\(\text{Midpoint} = \frac{0.16 + 0.20}{2} = 0.18\)[/tex]
- For the range [tex]\( 0.20-0.24 \)[/tex]: [tex]\(\text{Midpoint} = \frac{0.20 + 0.24}{2} = 0.22\)[/tex]
Thus, the midpoints are:
[tex]\[
[0.02, 0.06, 0.10, 0.14, 0.18, 0.22]
\][/tex]
### Step 3: Apply Frequency to Each Midpoint
Now, multiply each midpoint by its corresponding frequency to get the weighted values:
- [tex]\(0.02 \times 4 = 0.08\)[/tex]
- [tex]\(0.06 \times 9 = 0.54\)[/tex]
- [tex]\(0.10 \times 9 = 0.90\)[/tex]
- [tex]\(0.14 \times 2 = 0.28\)[/tex]
- [tex]\(0.18 \times 4 = 0.72\)[/tex]
- [tex]\(0.22 \times 2 = 0.44\)[/tex]
### Step 4: Sum Up the Frequencies and the Weighted Values
Sum of the frequencies:
[tex]\[ 4 + 9 + 9 + 2 + 4 + 2 = 30 \][/tex]
Sum of the weighted values:
[tex]\[ 0.08 + 0.54 + 0.90 + 0.28 + 0.72 + 0.44 = 2.96 \][/tex]
### Step 5: Calculate the Mean Concentration
The mean concentration is calculated by dividing the total of the weighted values by the sum of the frequencies:
[tex]\[
\text{Mean Concentration} = \frac{\text{Total Weighted Value}}{\text{Total Frequency}} = \frac{2.96}{30} = 0.0987
\][/tex]
Thus, the mean concentration of [tex]\( SO_2 \)[/tex] in the air is approximately:
[tex]\[
0.0987 \text{ ppm}
\][/tex]
