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The z-score is a measure we use to understand how a value compares to the average of a set data. It tells us the number of standard deviations a particular data point is from the mean.
You can think of a z-score as a way of saying how unusual or typical a value is within a normal distribution.
For example, in the temperature problem we looked at, the z-score helps us understand how the sample mean of 98.25 compares to the usual human body temperature mean of 98.6. It's calculated using the formula:\[ Z = \frac{x̄ - μ}{σ / \sqrt{n}} \]where:

• \(xÌ„\) is the sample mean.
• \(μ\) is the population mean.
• \(σ\) is the standard deviation of the population.
• \(n\) is the sample size.

A z-score of \(-3.075\) implies that the sample mean is 3.075 standard deviations below the population mean. That's quite far off from what we'd expect!

The standard normal distribution is a special case of the normal distribution. In this standard form, it has a mean of 0 and a standard deviation of 1. It's a common reference in statistics because it lets us look up probabilities for any normal distribution by converting our values to z-scores.
When you have data with a normal distribution, like human body temperatures in our problem, you convert real-world values using the z-score formula and then make use of standard normal distribution tables or software to find probabilities.
These tables show the cumulative probability of observing a value less than your z-score. For instance, a z-score of \(-3.075\) in the standard normal distribution gives us a very low probability of 0.00105. This cumulative probability means there is a 0.105% chance that values are as low or lower.

Cumulative probability refers to the total probability from the left up to a certain point on a probability distribution curve. This is crucial when we are analyzing the likelihood of certain outcomes happening, especially when they are lower than or equal to a specific value.
In the context of our exercise, cumulative probability helps us find the probability of selecting 130 people whose average body temperature is 98.25 or lower.
From our example, when you find the cumulative probability associated with a z-score of \(-3.075\), it indicates how likely it is to randomly draw a sample with such an average. Since the probability was roughly 0.00105, this results in only about a 0.105% chance, making the particular average seem quite unlikely.
This insight is essential for statisticians and researchers when making decisions based on probability and expected outcomes.