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Part 1
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, 9.0.
sequalsStartRoot 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
Interval
30​-36
37​-43
44​-50
51​-57
58​-64
65​-71
 
Frequency
4
25
45
17
5
1
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Part 1
Standard deviationequals
  
enter your response here ​(Round to one decimal place as​ needed.)

To solve for the standard deviation \( s \) using the given formula, let's break it down step by step.

The formula for the sample standard deviation from a frequency distribution table is:

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

Where:
- \( x \) is the class midpoint.
- \( f \) is the class frequency.
- \( n \) is the total number of data points (sum of frequencies).

### Steps:

1. **Find the class midpoints \( x \):**
   - For each class interval, calculate the midpoint \( x \) as the average of the lower and upper class boundaries.

2. **Multiply \( f \) and \( x \), and compute \( f \cdot x^2 \):**
   - For each class, multiply the frequency \( f \) by the class midpoint \( x \) and the square of the midpoint \( x^2 \).

3. **Calculate the summations:**
   - Compute \( \sum{f \cdot x} \) and \( \sum{f \cdot x^2} \).

4. **Substitute values into the formula to compute the standard deviation \( s \).**

Let's calculate these values now.The computed standard deviation \( s \) is approximately 6.7 (rounded to one decimal place).

Now, comparing this with the given standard deviation of 9.0 from the original data list, we can observe that the computed value from the frequency distribution is smaller.

Would you like more details on any specific step, or further clarification?

Here are 5 related questions to expand your understanding:
1. How does grouping data into frequency distributions affect standard deviation?
2. What are the implications of a smaller standard deviation in grouped data?
3. How do you calculate variance from the standard deviation?
4. What is the difference between population and sample standard deviation?
5. Why do we square the differences when calculating standard deviation?

**Tip:** For a more accurate result, use ungrouped data, as grouping can reduce the precision of the standard deviation estimate.

Find the standard deviation, s, of sample data summarized in the frequency distribution table given below by using the formula, 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, 9.0.

Learn how to find the standard deviation of sample data from a frequency distribution table using the standard deviation formula. This problem includes steps for calculating class midpoints, summations, and comparing results with the original data. Suitable for high school students in Grades 10-12.

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