The Mean and the Standard Deviation
Last year, golfers at Smith Ranch golf course had a mean score of 41 and a standard deviation of 2 on the first nine holes of the course, and a mean of 38 and a standard deviation of 1.8 on the last nine holes. One afternoon, Nathan scored 39 on the first nine holes and 37 on the last nine. On which half of the course did Nathan golf better, relative to the other golfers? Remember that in golf it is better to have a low score. Explain.
Math is crazy, man
To determine on which half of the course Nathan golfed better, we need to compare his scores to the mean and standard deviation of each half.
First, let's consider the first nine holes. We know that the mean score for the first nine holes is 41, and the standard deviation is 2. Nathan scored 39, which is lower than the mean.
To further determine how well Nathan did relative to the other golfers on the first nine holes, we can calculate his z-score. The z-score measures how many standard deviations away from the mean a particular value is.
The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the observed score, μ is the mean, and σ is the standard deviation.
In this case, Nathan's z-score for the first nine holes is:
z_first_nine = (39 - 41) / 2 = -1
A negative z-score indicates that Nathan's score is below the mean. However, we don't yet know how well Nathan did compared to the standard deviation.
Now let's move on to the last nine holes. The mean score for the last nine holes is 38, with a standard deviation of 1.8. Nathan scored 37, which again is lower than the mean.
Using the same formula to calculate the z-score:
z_last_nine = (37 - 38) / 1.8 ≈ -0.56
Again, a negative z-score indicates that Nathan's score is below the mean.
Comparing the z-scores for both halves, we can see that Nathan's score of -1 on the first nine holes is further below the mean than his score of -0.56 on the last nine holes. Therefore, relative to the other golfers, Nathan golfed better on the first nine holes.
In golf, it is better to have a low score. Since Nathan's scores were both below the respective means, he performed better on the first nine holes compared to the last nine holes, relative to the other golfers at Smith Ranch golf course.