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The z-score is a way to understand how far a specific value is from the mean of a distribution. In simple terms, it tells you how many standard deviations a value is away from the average.
To calculate a z-score, you can use the formula:

• \( Z = \frac{(X - \mu)}{\sigma} \)

Here, \( X \) is the value you're interested in, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
A positive z-score means the value is above the mean, and a negative z-score means it's below. Around 68% of data falls within one standard deviation, so most z-scores are between -1 and 1.
Z-scores help us use the z-table to find probabilities, showing what percent of data lies below or between certain values. This is how we understand the likelihood of an event within a given distribution.

Percentiles rank the data in a distribution, giving us an idea of how a specific value compares to the rest. For instance, being in the 99th percentile means you score higher than 99% of the other values.
In terms of height, if a boy is in the 99th percentile, he is taller than 99% of his peers. To find which height corresponds to a specific percentile, look up the z-score for that percentile and use the formula:

• \( X = \mu + Z\sigma \)

This will give you the actual value, like height, associated with the percentile. Percentiles are useful because they provide insights into data's relative standing.

Standard deviation is a measure of how spread out the numbers in a data set are. It's a key component in the normal distribution and helps to quantify the amount of variation.
When you have a small standard deviation, the data points are clustered closely around the mean. A larger standard deviation means they are spread out over a wider range of values.
In a normal distribution, approximately 68% of data points lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
In our example, with a mean height of 55.5 inches and a standard deviation of 2.7 inches, most 10-year-old boys’ heights would fall between 52.8 inches and 58.2 inches.
Understanding standard deviation helps us comprehend the diversity in data, enabling us to predict probability and variability in real-world scenarios.