We can compare numbers from different groups by standardizing the variable. One way of standardizing is by calculating a z-score:
\[z = \frac{x - \mu}{\sigma}\]
where \(x\) is the number you are converting to a z-score, \(\mu\) is the mean of the group, and \(\sigma\) is the standard deviation. The standard deviation measures how far spread out the numbers are from the mean. We can convert two different athletic performance numbers to z-scores in order to compare them to see who is more extraordinary.
1.
Two athletes want to compare their success in their respective
events. One athlete runs the 100 meter dash for their high school while
another athlete runs the 800 meter dash. Race times for the 100 meter
dash have a mean of 11.24 seconds and a standard deviation of 0.7
seconds. Race times for the 800 meter dash have a mean of 118 seconds
with a standard deviation of 3.7 seconds.
The athlete who runs the 100 meter dash ran their race with a time of 10.52 seconds. The athlete who runs the 800 meter dash ran their race with a time of 109 seconds. Which athlete performed better, relative to other times in their races?
We can convert each athlete’s race time to a z-score so that we can compare them directly to one another.
The 100 meter dash has a mean time, \(\mu\), of 11.24 seconds and a standard deviation, \(\sigma\) of 0.7 seconds. The z-score for a race time, \(x\), of 10.52 seconds is
\[z = \frac{x - \mu}{\sigma}\]
\[z = \frac{10.52 - 11.24}{0.7}\]
\[z = \frac{-0.72}{0.7}\]
\[z \approx -1.03\]
The 800 meter dash has a mean time, \(\mu\), of 118 seconds and a standard deviation, \(\sigma\) of 3.7 seconds. The z-score for a race time, \(x\), of 109 seconds is
\[z = \frac{x - \mu}{\sigma}\]
\[z = \frac{109 - 118}{3.7}\]
\[z = \frac{-9}{3.7}\]
\[z \approx -2.43\]
The athlete that ran the 800 meter dash has a lower z-score, indicating that their time was faster when compared to the mean and standard deviation of athletes in this race.
1. Two athletes want to compare their performance in their respective fields. One athlete threw a shot put 53.2 feet. Another athlete threw their javelin 69.2 meters. Assume athletes in this competition threw the shot put for an average of 49.8 feet with a standard deviation of 2 feet, and athletes threw the javelin an average 64.9 meters with a standard deviation of 3 meters. Which athlete performed better with respect to the mean and standard deviation of their events?