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The mean is a measure of the average score, offering insight into overall performance across a dataset. In this exercise, we have two separate sets of scores for an amateur golfer, spanning the years 2005 and 2006. For each year, we first sum the scores, and then divide by the total number of scores.

For 2005, the golfer had scores of: 74, 78, 79, 77, 75, 73, 75, and 77.

• Sum of scores: 608
• Number of scores: 8
• Mean: \( \frac{608}{8} = 76 \)

Similarly, for 2006, the scores were: 71, 70, 75, 77, 85, 80, 71, and 79.

• Sum of scores: 608
• Number of scores: 8
• Mean: \( \frac{608}{8} = 76 \)

Both years exhibit the same mean score of 76, which indicates that, on average, the golfer's performance was consistent across these two years.

Standard deviation is a crucial statistical measure used to understand the variability or spread of a set of values in relation to the mean. A higher standard deviation indicates a wider spread of scores from the mean, reflecting inconsistent performance.

To calculate the standard deviation, we first compute the variance, which is the average of the squared differences between each score and the mean.

For 2005:

• Mean: 76
• Squared differences: 4, 4, 9, 1, 1, 9, 1, 1
• Variance: \( \frac{30}{8} = 3.75 \)
• Standard Deviation: \( \sqrt{3.75} \approx 1.94 \)

For 2006:

• Mean: 76
• Squared differences: 25, 36, 1, 1, 81, 16, 25, 9
• Variance: \( \frac{194}{8} = 24.25 \)
• Standard Deviation: \( \sqrt{24.25} \approx 4.92 \)

While both years have the same mean, the standard deviation is dramatically higher in 2006. This indicates that the golfer’s scores were much more varied in that year, showcasing inconsistency.

Variance analysis provides insight into the degree of fluctuation in scores over time. It is an essential concept in descriptive statistics that highlights how much scores diverge from the mean. Here, we are using variance to understand the differences in the golfer's performance between 2005 and 2006.

For 2005, the variance is 3.75. This showcases smaller discrepancies between individual scores and the mean score of 76. The golfer's performance was steady, with minimal fluctuation between high and low scores.

In contrast, the 2006 variance of 24.25 reflects greater fluctuation in scores. The large variance indicates that the golfer had rounds of both excellent and subpar performances, as evidenced by scores like 70 and 85 in 2006. This high variance points to moments of both significant improvement and inconsistency when compared to 2005.

By examining variance alongside the mean and standard deviation, it is clear that while both years shared the same average performance, 2006 featured a notable increase in performance variability.