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Given the following frequency table. Find the standard deviation. (10 points / 15 mins)

a. 4.56

b. 3.45

c. 5.67

d. 2.34

e. 6.78

To calculate the standard deviation from the given frequency distribution table, we can follow these steps:

### Step 1: Find the Midpoint for Each Class
The midpoint (\(x_i\)) for each class interval is found by averaging the lower and upper limits of each class.

\[
\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}
\]

### Step 2: Calculate the Deviation and Squared Deviation
For each class, subtract the midpoint from the mean (\(\overline{x}\)) and square the result to get the squared deviation.

### Step 3: Multiply the Squared Deviation by the Frequency
Multiply each squared deviation by the corresponding frequency.

### Step 4: Compute the Variance
To compute the variance, sum up all the values from Step 3 and divide by the total number of data points (total frequency).

\[
\text{Variance} = \frac{\sum f \cdot (x_i - \overline{x})^2}{N}
\]

### Step 5: Take the Square Root
The standard deviation (\(\sigma\)) is the square root of the variance.

I'll perform the calculations step by step to find the standard deviation based on the data from the image:

| Class Interval | Midpoint (\(x_i\)) | Frequency (\(f\)) |
|----------------|--------------------|-------------------|
| 21 - 30        | \(\frac{21 + 30}{2} = 25.5\)  | 14                |
| 31 - 40        | \(\frac{31 + 40}{2} = 35.5\)  | 20                |
| 41 - 50        | \(\frac{41 + 50}{2} = 45.5\)  | 25                |
| 51 - 60        | \(\frac{51 + 60}{2} = 55.5\)  | 15                |
| 61 - 70        | \(\frac{61 + 70}{2} = 65.5\)  | 7                 |

Next, I'll calculate the mean, variance, and standard deviation.The calculated standard deviation for the given frequency table is approximately **11.89**, which does not match any of the provided options. This might indicate a misunderstanding in interpreting the class intervals or options provided. 

Would you like a breakdown of each step or have further questions about the process?

Here are 5 related questions for further practice:

1. How do you calculate the mean of a frequency distribution?
2. What is the formula for variance in grouped data?
3. Why do we square deviations when calculating variance?
4. How does the choice of class interval affect the standard deviation?
5. What is the difference between population and sample standard deviation?

**Tip**: Always ensure that class intervals are interpreted correctly, as they affect the calculation of midpoints and other statistics.

Given the following frequency table. Find the standard deviation.

Learn how to find the standard deviation for a frequency distribution table with class intervals from 21 to 70. This problem involves calculating midpoints, variance, and standard deviation. Suitable for students in Grades 10-12.

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