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Part 1
Find the standard​ deviation, s, of sample data summarized in the frequency distribution table 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, 11.1.Standard deviationequals 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	Frequency
30-36	2
37-43	1
44-50	7
51-57	4
58-64	18
65-71	32
72-78	36

enter your response here ​(Round to one decimal place as​ needed.)

To calculate the standard deviation of the sample data summarized in the frequency distribution table, we will use the following formula:

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

Where:
- \( x \) represents the class midpoint
- \( f \) represents the class frequency
- \( n \) is the total number of sample values

Let's calculate the class midpoints, then the necessary sums, and finally the standard deviation.

### Step 1: Calculate Class Midpoints

The midpoint \( x \) for each class is given by:

\[
x = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}
\]

\[
\begin{aligned}
&\text{30-36:} & x_1 = \frac{30 + 36}{2} = 33 \\
&\text{37-43:} & x_2 = \frac{37 + 43}{2} = 40 \\
&\text{44-50:} & x_3 = \frac{44 + 50}{2} = 47 \\
&\text{51-57:} & x_4 = \frac{51 + 57}{2} = 54 \\
&\text{58-64:} & x_5 = \frac{58 + 64}{2} = 61 \\
&\text{65-71:} & x_6 = \frac{65 + 71}{2} = 68 \\
&\text{72-78:} & x_7 = \frac{72 + 78}{2} = 75 \\
\end{aligned}
\]

### Step 2: Calculate the Sums

Calculate the sum \( \sum (f \cdot x) \) and \( \sum (f \cdot x^2) \).

\[
\begin{aligned}
&\text{30-36:} & f_1 = 2 & , & x_1 = 33 & , & f_1 \cdot x_1 = 66 & , & f_1 \cdot x_1^2 = 2178 \\
&\text{37-43:} & f_2 = 1 & , & x_2 = 40 & , & f_2 \cdot x_2 = 40 & , & f_2 \cdot x_2^2 = 1600 \\
&\text{44-50:} & f_3 = 7 & , & x_3 = 47 & , & f_3 \cdot x_3 = 329 & , & f_3 \cdot x_3^2 = 15407 \\
&\text{51-57:} & f_4 = 4 & , & x_4 = 54 & , & f_4 \cdot x_4 = 216 & , & f_4 \cdot x_4^2 = 11664 \\
&\text{58-64:} & f_5 = 18 & , & x_5 = 61 & , & f_5 \cdot x_5 = 1098 & , & f_5 \cdot x_5^2 = 67038 \\
&\text{65-71:} & f_6 = 32 & , & x_6 = 68 & , & f_6 \cdot x_6 = 2176 & , & f_6 \cdot x_6^2 = 147584 \\
&\text{72-78:} & f_7 = 36 & , & x_7 = 75 & , & f_7 \cdot x_7 = 2700 & , & f_7 \cdot x_7^2 = 202500 \\
\end{aligned}
\]

Sum the values:
\[
\sum (f \cdot x) = 66 + 40 + 329 + 216 + 1098 + 2176 + 2700 = 6625
\]

\[
\sum (f \cdot x^2) = 2178 + 1600 + 15407 + 11664 + 67038 + 147584 + 202500 = 447971
\]

The total number of sample values \( n \) is:
\[
n = 2 + 1 + 7 + 4 + 18 + 32 + 36 = 100
\]

### Step 3: Plug into the Formula

\[
s = \sqrt{\frac{100 \left(447971\right) - \left(6625\right)^2}{100 \times 99}}
\]

Calculate the numerator and denominator separately:

Numerator:
\[
100 \times 447971 - 6625^2 = 44797100 - 43890625 = 906475
\]

Denominator:
\[
100 \times 99 = 9900
\]

\[
s = \sqrt{\frac{906475}{9900}} = \sqrt{91.6641} \approx 9.6
\]

### Step 4: Compare to Given Standard Deviation

The computed standard deviation \( s \) is approximately 9.6. The given standard deviation from the original list of data values is 11.1. Comparing the two, we see a difference.

If you have any questions or want more details, please let me know. Here are 5 questions you can ask next:
1. How is the formula for the standard deviation derived?
2. Can you explain the difference between sample standard deviation and population standard deviation?
3. How does changing the frequency values affect the standard deviation?
4. Can you show the step-by-step process for another frequency distribution table?
5. What are some common mistakes to avoid when calculating standard deviation from frequency tables?

**Tip:** When working with frequency distributions, always double-check your class midpoints and frequencies to ensure accuracy in calculations.

Learn how to calculate the standard deviation of sample data from a frequency distribution table using the sample standard deviation formula. Includes step-by-step calculations and comparison with given data.

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