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In statistics, the z-score is a measure that describes a value's position relative to the mean of a group of values. It shows how many standard deviations a data point is from the mean.
The z-score formula is:
\[ Z = \frac{X - \bar{X}}{s} \]
Here, \(X\) is the value being considered, \(\bar{X}\) is the mean of the dataset, and \(s\) is the standard deviation.
Calculating the z-score helps us understand how unusual or typical a particular value is within a set.
For Roberto and Zandra, their z-scores let us compare how well they performed compared to other competitors of the same gender in the triathlon.

Standard deviation is a measure of the amount of variation or dispersion in a set of values.
It tells us how spread out the values are and is commonly used to gauge the variability in a dataset.
For example, if the finishing times in a race have a high standard deviation, it indicates that the times vary widely from the mean.
On the other hand, a low standard deviation means the times are closely clustered around the mean.
The formula for standard deviation is:
\[ s = \sqrt{\frac{\sum (X_i - \bar{X})^2} {n-1}} \]
Where \(X_i\) is each individual value, \(\bar{X}\) is the mean, and \(n\) is the number of values.
Understanding standard deviation is essential when evaluating the performance of participants in competitions or any other scenario involving statistical data.

The mean finishing time is the average amount of time it takes for participants to complete a race.
To calculate the mean, sum up all the finishing times and divide by the number of participants.
The formula for the mean is:
\[ \bar{X} = \frac{\sum X_i}{n} \]
Where \(X_i\) represents each participant's finishing time and \(n\) is the total number of participants.
In the context of Roberto and Zandra's triathlon, we are given the mean finishing times for all men (69.4 minutes) and all women (84.7 minutes).
Comparing an individual's finishing time to the mean gives us insight into how well they performed relative to other participants.
For a deeper analysis, measuring the z-score helps us see how far above or below the mean a specific finishing time is, taking into account the variability.