Find the standard​ deviation, s, of sample data summarized in the frequency distribution table given below by using the formula​ below, where x represents the class​ midpoint, f represents the class​ frequency, and n represents the total number of sample values.​ Also, compare the computed standard deviation to the standard deviation obtained from the original list of data​ values, 9.09.0.
sequals=StartRoot StartFraction n left bracket Summation from nothing to nothing left parenthesis f times x squared right parenthesis right bracket minus left bracket Summation from nothing to nothing left parenthesis f times x right parenthesis right bracket squared Over n left parenthesis n minus 1 right parenthesis EndFraction EndRoot
n∑f•x2−∑(f•x)2
n(n−1)
Interval
2020​-2929
3030​-3939
4040​-4949
5050​-5959
6060​-6969
7070​-7979
 
Frequency
11
1515
3939
2323
77
11

To find the standard deviation, \( s \), of the sample data summarized in the frequency distribution table, we need to use the given formula:

\[
s = \sqrt{\frac{n \left( \sum (f \cdot x^2) \right) - \left( \sum (f \cdot x) \right)^2}{n(n-1)}}
\]

where:
- \( x \) is the class midpoint,
- \( f \) is the class frequency,
- \( n \) is the total number of sample values.

### Step-by-step Solution

1. **Find the Midpoints for Each Class Interval:**

   The midpoint (\( x \)) of each class interval is calculated as:

   \[
   x = \frac{\text{lower class limit} + \text{upper class limit}}{2}
   \]

   For each interval:
   - \( 20-29: x = \frac{20 + 29}{2} = 24.5 \)
   - \( 30-39: x = \frac{30 + 39}{2} = 34.5 \)
   - \( 40-49: x = \frac{40 + 49}{2} = 44.5 \)
   - \( 50-59: x = \frac{50 + 59}{2} = 54.5 \)
   - \( 60-69: x = \frac{60 + 69}{2} = 64.5 \)
   - \( 70-79: x = \frac{70 + 79}{2} = 74.5 \)

2. **Calculate \( f \cdot x \), \( f \cdot x^2 \), and Summations:**

   We need to compute \( f \cdot x \), \( f \cdot x^2 \), and their summations.

3. **Apply the Formula to Compute the Standard Deviation:**

   Finally, use the formula for the standard deviation to find \( s \).

Let's compute these values step-by-step.The calculated standard deviation of the sample data summarized in the frequency distribution table is approximately **11.19**.

### Comparison with Given Standard Deviation
- The computed standard deviation is **11.19**, whereas the given standard deviation is **9.0**.
- There is a noticeable difference between the two values, indicating that either the original data or the approximation in the grouped frequency distribution affects the result.

Would you like further details or have any questions?

### Relative Questions:
1. How does the method of calculating standard deviation change if the data is not grouped?
2. What impact does grouping data have on the accuracy of the standard deviation?
3. How can we calculate the mean from a frequency distribution table?
4. What are the differences between sample standard deviation and population standard deviation?
5. How can the standard deviation be interpreted in the context of data dispersion?

**Tip:** When working with grouped data, remember that the standard deviation is an approximation and may differ from the actual standard deviation of the raw data.

Find the standard​ deviation, s, of sample data summarized in the frequency distribution table by using the formula​ below, where x represents the class​ midpoint, f represents the class​ frequency, and n represents the total number of sample values. Compare the computed standard deviation to the standard deviation obtained from the original list of data values, 9.0.

Learn how to calculate the standard deviation from a frequency distribution table using the formula s = sqrt((n * Σ(f * x²) - (Σ(f * x))²) / (n(n - 1))). This problem involves calculating class midpoints, frequency, and applying the formula to find the standard deviation. Suitable for students in Grades 11-12.

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