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find me variances and standard deviation using formula 

To calculate the variance and standard deviation for grouped data as seen in your example, we'll follow the same steps as in the table on "Dispersion for grouped data" and apply it to the first table you uploaded.

### The formulae:
1. **Mean (\(\bar{x}\))** for grouped dataTo calculate the variance and standard deviation for grouped data as seen in your example, we'll follow the same steps as in the table on "Dispersion for grouped data" and apply it to the first table you uploaded.

### The formulae:
1. **Mean (\(\bar{x}\))** for grouped 
   \[
   \bar{x} = \frac{\sum f_i x_i}{\sum f_i}
   \]
   where:
   - \(f_i\) = frequency of each class
   - \(x_i\) = mid-point of each class

2. **Variance (S²)** for grouped dataTo calculate the variance and standard deviation for grouped data as seen in your example, we'll follow the same steps as in the table on "Dispersion for grouped data" and apply it to the first table you uploaded.

### The formulae:
1. **Mean (\(\bar{x}\))** for grouped 
   \[
   \bar{x} = \frac{\sum f_i x_i}{\sum f_i}
   \]
   where:
   - \(f_i\) = frequency of each class
   - \(x_i\) = mid-point of each class

2. **Variance (S²)** for grouped 
   \[
   S^2 = \frac{\sum f_i (x_i - \bar{x})^2}{n - 1}
   \]
   where \(n = \sum f_i\) is the total number of observations.

3. **Standard deviation (S)** is just the square root of the variance:
   \[
   S = \sqrt{S^2}
   \]

Now, let me extract the data from your first image (table) and compute these values step by step. I'll calculate the midpoints and frequencies, then determine the mean, variance, and standard deviation using the steps outlined.Here are the results based on your grouped dataHere are the results based on your grouped 

- **Mean (\(\bar{x}\))**: 15.57
- **Variance (S²)**: 14.74
- **Standard Deviation (S)**: 3.84

Would you like further details or explanations on these calculations?

Here are some follow-up questions:
1. How would you interpret the variance and standard deviation in this context?
2. What would happen to the variance if we had more equal frequencies across the classes?
3. Can you see how the grouped data affects the precision of these calculations?
4. How would you calculate the mean and standard deviation if individual data points were available instead of grouped data?
5. What role does the class width play in the accuracy of grouped data calculations?

**Tip:** In grouped data, the larger the range of a class interval, the less precise the mean and standard deviation estimates become.

Find me variance and standard deviation using formula based on the provided frequency distribution table.

Learn how to calculate the variance and standard deviation for grouped data using a frequency distribution table. This problem demonstrates key statistical concepts like variance, mean, and standard deviation with a detailed step-by-step solution.

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