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Percentiles are measurements that indicate the relative standing of a value within a dataset. A percentile tells you what percentage of observations fall below a certain value. For example, the 99th percentile (\(P_{99}\)) means that 99% of the data points fall below this value. In the context of the bone density test, it helps to identify the score below which 99% of subjects' bone density scores fall.
The process of finding a percentile in a normal distribution typically involves:

• Identifying the desired percentile
• Using a Z-table or statistical software to find the corresponding Z-score
• Converting the Z-score back to the original units

This gives you the value that separates the lower percentage of the dataset from the rest.

A Z-score (or standard score) represents the number of standard deviations a data point is from the mean. In a normal distribution, Z-scores make it easier to understand how measurements compare to the average.
The Z-score can be calculated using the formula:
\( Z = \frac{X - \tau}{\tau} \)
where X is the value, \(\mu\) (mu) is the mean, and \( \tau \) (sigma) is the standard deviation.
For example, when finding the 99th percentile in our bone density test scores, we found a Z-score of 2.33. This indicates that the score is 2.33 standard deviations above the mean of 0 for this test.

Standard deviation (\(\sigma\)) measures the spread of a set of values from the mean. It tells you how much the individual values in a dataset differ from the mean.
The formula to calculate standard deviation is:
\[ \sigma = \sqrt{\frac{1}{N-1} \sum \limits_{i=1}^{N} (X_i - \mu)^2} \]
where \(N\) is the number of observations, \( X_i \) represents each data point, and \(\mu\) is the mean.
A low standard deviation indicates that the data points are close to the mean, whereas a high standard deviation indicates that they are spread out over a wider range.
In our problem, given that \(\sigma = 1\), the scores are distributed normally with a standard deviation of 1, making our calculations straightforward.

The mean (\(\mu\)) is the average value of a dataset. In a normal distribution, the mean acts as the center point where half of the observations lie below and half lie above it.
The formula to calculate the mean is:
\[ \mu = \frac{1}{N} \sum \limits_{i=1}^{N} X_i \ \]
where \(X_i\) represents each data point and \(N\) is the number of observations.
In our bone density test example, the mean is given as 0. This simplifies our calculations because deviations are directly measured from zero. Understanding the mean is crucial for calculating the Z-score and interpreting the standard deviation in the context of a normal distribution.