Standard deviation worksheet for calculating mean and standard deviation of golf scores for John and Joe.

Problem Overview:

We are tasked with calculating the mean and standard deviation for the golf scores of John and Joe over 6 competitions. We will then compare their performances based on these statistics.

Data:

- John's Scores: 74, 73, 74, 73, 71, 73
- Joe's Scores: 68, 74, 70, 67, 80, 81

Steps to Solve:

#### Step 1: Calculate the Mean
The mean ($\bar{x}$) is calculated using the formula:
\[
\bar{x} = \frac{\sum x}{n}
\]
where $\sum x$ is the sum of all scores and $n$ is the number of scores.

##### For John:
\[
\text{Sum of John's scores} = 74 + 73 + 74 + 73 + 71 + 73 = 438
\]
\[
\text{Mean for John} = \frac{438}{6} = 73
\]

##### For Joe:
\[
\text{Sum of Joe's scores} = 68 + 74 + 70 + 67 + 80 + 81 = 440
\]
\[
\text{Mean for Joe} = \frac{440}{6} \approx 73.33
\]

#### Step 2: Calculate the Standard Deviation
The standard deviation (S.D.) is calculated using the formula:
\[
\text{S.D.} = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}
\]
where $x$ is each score, $\bar{x}$ is the mean, and $n$ is the number of scores.

##### For John:
1. Calculate $(x - \bar{x})$ for each score:
\[
\begin{aligned}
&74 - 73 = 1, \\
&73 - 73 = 0, \\
&74 - 73 = 1, \\
&73 - 73 = 0, \\
&71 - 73 = -2, \\
&73 - 73 = 0.
\end{aligned}
\]

2. Square each difference:
\[
\begin{aligned}
&1^2 = 1, \\
&0^2 = 0, \\
&1^2 = 1, \\
&0^2 = 0, \\
&(-2)^2 = 4, \\
&0^2 = 0.
\end{aligned}
\]

3. Sum the squared differences:
\[
\sum (x - \bar{x})^2 = 1 + 0 + 1 + 0 + 4 + 0 = 6
\]

4. Calculate the standard deviation:
\[
\text{S.D. for John} = \sqrt{\frac{6}{6-1}} = \sqrt{\frac{6}{5}} = \sqrt{1.2} \approx 1.095
\]

##### For Joe:
1. Calculate $(x - \bar{x})$ for each score:
\[
\begin{aligned}
&68 - 73.33 = -5.33, \\
&74 - 73.33 = 0.67, \\
&70 - 73.33 = -3.33, \\
&67 - 73.33 = -6.33, \\
&80 - 73.33 = 6.67, \\
&81 - 73.33 = 7.67.
\end{aligned}
\]

2. Square each difference:
\[
\begin{aligned}
&(-5.33)^2 = 28.4089, \\
&(0.67)^2 = 0.4489, \\
&(-3.33)^2 = 11.0889, \\
&(-6.33)^2 = 39.9689, \\
&(6.67)^2 = 44.4889, \\
&(7.67)^2 = 58.8289.
\end{aligned}
\]

3. Sum the squared differences:
\[
\sum (x - \bar{x})^2 = 28.4089 + 0.4489 + 11.0889 + 39.9689 + 44.4889 + 58.8289 = 183.2344
\]

4. Calculate the standard deviation:
\[
\text{S.D. for Joe} = \sqrt{\frac{183.2344}{6-1}} = \sqrt{\frac{183.2344}{5}} = \sqrt{36.64688} \approx 6.054
\]

Final Results:

- John:
- Mean: $73$
- Standard Deviation: $\approx 1.095$

- Joe:
- Mean: $\approx 73.33$
- Standard Deviation: $\approx 6.054$

Comparison:

- Mean: Joe has a slightly higher mean score (73.33) compared to John (73), indicating that Joe's average performance is marginally better.
- Standard Deviation: Joe has a much higher standard deviation (6.054) compared to John (1.095). This suggests that Joe's scores are more variable, meaning he has more inconsistent performance, while John's scores are more consistent.

Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{John:} & \text{Mean} = 73, \text{S.D.} \approx 1.095 \\
\text{Joe:} & \text{Mean} \approx 73.33, \text{S.D.} \approx 6.054 \\
\end{array}
}
\]