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Find the standard deviation and variance for the above data.

To calculate the variance and standard deviation for the given data, we'll follow these steps:

1. **Determine the midpoints of each class**.
   Each pollution level range has a midpoint, which is calculated as:
   \[
   \text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}
   \]
   
   For the given dataTo calculate the variance and standard deviation for the given data, we'll follow these steps:

1. **Determine the midpoints of each class**.
   Each pollution level range has a midpoint, which is calculated as:
   \[
   \text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}
   \]
   
   For the given 
   - For the range \(100-200\): Midpoint = \(\frac{100 + 200}{2} = 150\)
   - For the range \(200-300\): Midpoint = \(\frac{200 + 300}{2} = 250\)
   - For the range \(300-400\): Midpoint = \(\frac{300 + 400}{2} = 350\)
   - For the range \(400-500\): Midpoint = \(\frac{400 + 500}{2} = 450\)

2. **Use the frequency (number of cities) for each class**:
   - 4 cities for \(100-200\)
   - 12 cities for \(200-300\)
   - 3 cities for \(300-400\)
   - 6 cities for \(400-500\)

3. **Calculate the mean (\(\mu\))**:
   \[
   \mu = \frac{\sum (f \cdot x)}{\sum f}
   \]
   where \(f\) is the frequency (number of cities) and \(x\) is the midpoint.

4. **Calculate the variance**:
   \[
   \sigma^2 = \frac{\sum f \cdot (x - \mu)^2}{\sum f}
   \]
   where \(\sigma^2\) is the variance, and \(x - \mu\) is the deviation from the mean.

5. **Calculate the standard deviation**:
   \[
   \sigma = \sqrt{\sigma^2}
   \]

Let's proceed to compute the values.The calculations yield the following results for the given dataThe calculations yield the following results for the given 

- **Mean (\(\mu\))**: 294.0
- **Variance (\(\sigma^2\))**: 10,464.0
- **Standard Deviation (\(\sigma\))**: 102.29 (rounded to two decimal places)

Would you like any further details or explanations? Here are some related questions:

1. How is the midpoint chosen for a class interval?
2. Why is the mean important when calculating variance and standard deviation?
3. What do the variance and standard deviation tell us about the distribution of pollution levels?
4. Can we use this method for any type of frequency distribution?
5. How would adding another class interval affect the results?

**Tip**: The standard deviation measures how spread out the values are around the mean—larger standard deviations indicate greater variability in the data.

Learn how to calculate the variance and standard deviation for pollution levels across cities. This guide includes steps for determining midpoints, calculating the mean, and understanding variance and standard deviation. Suitable for high school students in Grades 9-12.

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