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A percentile is a measure that indicates the value below which a given percentage of observations fall. If you are told that a 6-year-old girl who is 46.5 inches tall is in the 75th percentile, it means that her height is greater than or equal to 75% of the girls of her age. Percentiles are useful in comparing an individual's position or state relative to a larger group. They help in understanding where a particular observation stands on a scale. For instance:

• If you're in the 90th percentile, you are taller than 90% of your peers.
• If you're in the 50th percentile, you're very close to the average height, as you exceed half of your peers.

Percentiles provide a sense of distribution and are extensively used in statistics to understand relative standing in data.

A z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. Z-scores are a measure of how many standard deviations an element is from the mean. You can calculate it with the formula:\[ z = \frac{{X - \mu}}{{\sigma}} \]In this equation, \(X\) is the value you're examining, \(\mu\) is the mean of the group, and \(\sigma\) is the standard deviation.For example, if a 6-year-old girl is 46.5 inches tall, and the mean height for her age is 45 inches with a standard deviation of 2 inches, her z-score would help us understand how her height compares to the average height of her peers. A higher absolute z-score indicates a value further from the mean, whether higher or lower:

• A positive z-score indicates the value is above the mean.
• A negative z-score indicates the value is below the mean.
• A z-score of zero indicates the value is exactly at the mean.

Understanding z-scores can help you interpret data points in the context of normal distributions.

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of numbers. When data points are close to the mean, the standard deviation is smaller, indicating low variability. Conversely, a larger standard deviation indicates greater variability.In the context of our exercise, we see that the 6-year-old girl's heights are normally distributed with a mean of 45 inches and a standard deviation of 2 inches. This implies that most of the girls' heights at age 6 are within a reasonably tight range. The formula for standard deviation is:\[ \sigma = \sqrt{ \frac{ \sum_{i=1}^{n} (X_i - \mu)^2 }{n-1} } \]Here, \(X_i\) represents each data point, \(\mu\) is the mean, and \(n\) is the number of observations. Knowing the standard deviation helps to identify how spread out the data points are from the mean, which is crucial for predictions and comparisons in a normal distribution.