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Standard deviation helps us understand the spread or dispersion of a set of values. In simpler terms, it measures how much the values in a data set differ from the mean (average). A small standard deviation means the values are close to the mean, while a larger one indicates they are more spread out.
For example, in Ryan Murphy's 100-meter backstroke, the standard deviation is 0.98 seconds. This indicates that the times of all NCAA swimmers in this event usually vary by about 0.98 seconds from the average time (48.62 seconds). Likewise, for the 200-meter backstroke, the standard deviation is 2.38 seconds, indicating a wider spread of times around the mean (106.58 seconds).
Understanding standard deviation helps us see how consistent or varied the performance times are in each event, giving us a clearer picture of how Ryan's performance stands out.

The mean, commonly known as the average, is a measure that represents the central value of a data set. To find the mean, you sum all the values and then divide by the number of values.
For the 100-meter backstroke, the mean time of NCAA swimmers is 48.62 seconds. This means that when you sum all the swim times and divide by the total number of swimmers, you get 48.62 seconds. In the 200-meter backstroke, the mean time is 106.58 seconds.
Knowing the mean allows us to compare an individual’s performance against a typical performance. For instance, Ryan's times of 45.3 seconds (100-meter) and 99.32 seconds (200-meter) can be evaluated against these mean times to see how he fares compared to the average swimmer.

Performance comparison using Z-scores helps us determine how well someone performs relative to others. The Z-score tells us how many standard deviations a value (like Ryan's time) is from the mean. The formula we use is:
\text{ \frac{X - \tau}{\rho} }where \(X\) is the individual’s value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
For the 100-meter backstroke, Ryan's Z-score is calculated as: \frac{45.3 - 48.62}{0.98} = -3.39, meaning Ryan's time is 3.39 standard deviations below the mean.
In the 200-meter backstroke, his Z-score is: \frac{99.32 - 106.58}{2.38} = -3.05, indicating he is 3.05 standard deviations below the mean.
By comparing the absolute values of these Z-scores, we see that Ryan's Z-score for the 100-meter backstroke is more negative, indicating a better performance in that event compared to the 200-meter backstroke.