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In statistics, the mean is a measure of central tendency. It is often referred to as the average. When working with a normally distributed variable, like Kevin's commute to school, the mean is where the peak of the bell curve is centered.
The mean is calculated by summing up all individual data points and dividing the total by the number of points.

• For our problem, Kevin's travel to school has a mean of 12 minutes.
• This tells us that on average, it takes Kevin 12 minutes to commute.

In a normal distribution, the mean also equals the median and mode, serving as a critical value that represents the data set's central location.
In many real-world processes, like commute times, the mean gives us a helpful baseline for understanding typical outcomes, even though individual instances may vary.

The standard deviation is a statistical measurement that describes the spread or dispersion of a set of data. It indicates how much the individual data points deviate from the mean.
For a normal distribution:

• About 68% of data will fall within one standard deviation of the mean.
• Approximately 95% within two standard deviations.
• Almost 99.7% within three standard deviations.

In Kevin's case, the standard deviation is 2 minutes.
This means the majority of his commute times fall within 10 to 14 minutes.
A smaller standard deviation indicates data points are close to the mean, while a larger one indicates more data variability. Understanding the standard deviation helps predict how often events are outliers.

A Z-score, also known as a standard score, tells us how many standard deviations an element is from the mean.
It is a simple but powerful tool for comparing different data points within the same or different distributions.

• The Z-score formula is: \( Z = \frac{X - \mu}{\sigma} \)
• Where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

In Kevin's case, let's consider a commute time of 17 minutes:
The Z-score would be \( \frac{17 - 12}{2} = 2.5 \). This indicates 17 minutes is 2.5 standard deviations above the mean.
Z-scores are invaluable because they enable us to calculate the probability of a score occurring within our normal distribution and are essential for interpreting the probabilities in normally distributed data.

The probability table, often called the Z-table, is a chart that helps us find the probability magnitude associated with each Z-score value.
This is crucial when you want to determine the likelihood of an event occurring beyond or below certain data points in a normal distribution.

• The Z-table provides areas under the standard normal curve corresponding to specific Z-scores.
• For a Z-score of 2.5, the table indicates the cumulative probability up to that point.

Using probability tables, like the Z-table, enhances our ability to ascertain how extreme or typical a data point is relative to the distribution as a whole.
In the example of Kevin's commute time longer than 17 minutes, we subtract the cumulative probability from 1 to find the probability for more extreme times.
Probability tables simplify complex probability computations, making them easily accessible for practical application.