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Percentiles are a useful concept in statistics to determine the position of a particular value within a distribution. By knowing in which percentile a value falls, you can understand how it compares to the rest of the data set. In a Normal distribution, percentiles are often used to showcase the proportion of data that lies below a specific point. For example, if a birth length is at the 2.5th percentile, it means that 2.5% of all recorded birth lengths are below this value.
To find the specific values at given percentiles, statisticians use z-values from the standard Normal distribution. This helps translate percentiles into usable data points. With these z-values, you can apply the formula: \( X = μ + Zσ \), where \(μ\) is the mean, \(Z\) is the z-value, and \(σ\) is the standard deviation. This formula allows you to predict the precise value of the birth length at specified percentiles like the 2.5th and 97.5th, crucial for analyzing data distributions.

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It's expressed as the number of standard deviations a particular value is from the mean. This can be extremely insightful because it allows us to understand where a specific data point lies within a distribution.
To calculate a z-score, use the formula: \( Z = \frac{(X - μ)}{σ} \), where \(X\) is the value, \(μ\) is the mean, and \(σ\) is the standard deviation. For instance, when calculating the z-score for a birth length at the 2.5th percentile, we can refer to standard Normal distribution tables, which tell us that the z-score is approximately -1.96. This indicates the length is 1.96 standard deviations below the mean.
Understanding z-scores is crucial for students and researchers as they provide insight into how extraordinary or typical a specific measurement is within a set of data.

Standard deviation is a crucial concept in statistics that measures the amount of variation or dispersion in a set of values. A smaller standard deviation indicates that the values tend to be close to the mean, while a larger standard deviation shows that values are spread out over a wider range.
In the context of the problem, the standard deviation of 0.90 inches gives us an idea of how varied the birth lengths are compared to the mean of 20.5 inches. The fact that the distribution is Normal suggests that most lengths will fall within a certain range of the mean, specifically within one or two standard deviations.
Calculating how far a certain data point is from the mean using standard deviation helps in understanding the spread of data. It's an essential aspect for interpreting z-scores and percentile ranks, offering a complete picture of how individual measurements relate to the entire data set.