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The concept of a z-score is central to understanding data within a normal distribution. A z-score measures how many standard deviations a data point is from the mean. This is useful because it allows us to determine how far or how unusual the data point is in comparison to the average.

In mathematical terms, a z-score is calculated using the formula:\[ z = \frac{(X - \text{mean})}{\text{standard deviation}} \]

• **X** is the value you are examining
• **Mean** is the average of all data points
• **Standard Deviation** indicates how spread out the values are

Understanding this can help you better interpret a wide range of data scenarios. For example, in determining salary levels, a positive z-score indicates a salary above the mean, while a negative z-score indicates a salary below the mean.

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations in a group of observations falls. For example, the 90th percentile is the value that separates the lowest 90% of the data from the highest 10%. When applying percentile to the normal distribution scenario, any particular percentile corresponds to a specific z-score. This makes understanding percentiles essential when you want to work out the position of a data point in a given dataset.

If you want to find which salary level sits at the top 10% of the earners, you would calculate the 90th percentile salary. This value means 90% of the salaries are below this value, placing the salary in the top 10% of all earners.

In the context of a normal distribution, the mean is the average of all data points or observations in your dataset. It sums up the information by providing the central measurement of that data.In our example of actuary salaries, having a mean salary of \\(98,620 means that if you added up all the actuaries' salaries and divided by the number of actuaries, you’d arrive at \\)98,620.

The mean provides a baseline for comparing other salary levels. This is crucial when calculating z-scores and defining where a salary sits in relation to the average salary.

Standard deviation is a measure of how much variation or dispersion there is from the average (mean). It represents the typical distance between each data point and the mean.

In simple terms, if the standard deviation is small, this indicates that the data points are closer to the mean, showing little variation. On the flip side, a large standard deviation indicates that data points are spread out over a wider range of values.

• For the actuaries’ salaries example, a standard deviation of \\(18,000 indicates that typically, the salaries are \\)18,000 away from the average salary.
• Larger deviations could suggest wider pay differences among actuaries.

Understanding the standard deviation helps in assessing how spread out salaries are, and thus in determining percentiles and z-scores accurately.