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A normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetrical about the mean. This means most of the observations cluster around the central peak and probabilities for values taper off equally in both directions from the mean. Its bell-shaped curve is characterized by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)).
To determine if a set of data follows a normal distribution, you can use various methods such as graphical techniques and statistical tests. One effective graphical method is the normal quantile plot, which involves plotting the data against the quantiles of a standard normal distribution (with mean 0 and standard deviation 1). If the data points fall approximately on a straight line, the data is likely from a normal distribution.

A Z-score represents the number of standard deviations a data point is from the mean. Calculating Z-scores transforms and normalizes data, allowing comparisons across different scales or distributions.
To calculate a Z-score for a data value, use the formula: \[ z = \frac{x - \mu}{\sigma} \] where \(x\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
In the context of normal quantile plots, Z-scores correspond to specific percentiles of a normal distribution. By using tables or software, you can map percentiles to Z-scores, helping to construct the plot and analyze normality.

Earthquake depths can be analyzed using statistical methods for various purposes, such as understanding geological patterns or assessing risks. Statistically analyzing earthquake depths typically involves exploring their distribution.
In our example, we use a small sample of earthquake depth measurements: 17.3, 7.0, 7.0, 7.0, 8.1, and 6.8 (all in kilometers). These data points will be sorted, ranked, and then corresponding Z-scores will be calculated.
Specialists are often interested in whether the distribution of earthquake depths follows a normal pattern. This analysis can have implications for predicting future seismic events or understanding underlying geological processes.

Statistical analysis is a process of collecting and analyzing data to identify trends and patterns. It involves several steps, such as data collection, descriptive statistics, and inferential statistics.
In our example, we start with ranking the data and calculating the percentiles using the formula: \[ p = \frac{(i - 0.5)}{n} \] where \(i\) is the rank and \(n\) is the total number of observations.
We then convert these percentiles into Z-scores using a standard normal table. These steps ensure data standardization, allowing us to create an informative normal quantile plot.

Percentiles indicate the relative standing of a data point within a dataset. The percentile rank of a data point is a measure of the percentage of values in the dataset that are less than or equal to that particular value.
The formula to calculate the percentile for a specific rank \(i\) in a dataset of size \(n\) is: \[ p = \frac{(i - 0.5)}{n} \]
Percentiles help in comparing data points within skewed distributions or different datasets. In a normal quantile plot, percentiles are used to determine corresponding Z-scores, facilitating the transformation of individual data points to a standard scale for analysis.