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The Z-score is an important measure in statistics that tells us how many standard deviations a data point is from the mean of its distribution. In simpler terms, it helps us understand where a particular score stands in relation to other data. A high positive Z-score indicates a data point that is much higher than the mean, while a high negative Z-score indicates one that is much lower.

To calculate the Z-score, you can use the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

Where:

• \(X\) is the value you want to evaluate.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation.

Using the Z-score, we can determine how rare or common a particular value is within a distribution. For example, a Z-score of 0.4 indicates the value is 0.4 standard deviations above the mean, which is a fairly common occurrence in a normal distribution.

Standard deviation (SD) is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a larger range.

Standard deviation is a crucial concept in the normal distribution because it helps define the shape of the curve. A small standard deviation results in a narrow, tall curve, indicating that most data points are clustered around the mean. Conversely, a large standard deviation means a wider, flatter curve, showing that data points are spread out over a larger range.

Mathematically, standard deviation is calculated using the formula:

\[ \sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2 } \]

Where:

• \(N\) is the number of data points.
• \(X_i\) is each individual data point.
• \(\mu\) is the mean of the data set.

In the context of our exercise, the standard deviation of $$5,000 represents how much the annual commissions of the sales representatives vary from the mean of $40,000.

A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. The normal distribution is a type of continuous probability distribution for real-valued random variables.

This distribution is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The normal distribution is often referred to as a "bell curve" because of its bell-shaped appearance.

In a normal distribution:

• The mean, median, and mode are all equal.
• The curve is symmetric with respect to the mean.
• The area under the curve represents the total probability, which is 1 or 100%.

The further a point is from the mean, the less likely it is. This allows us to calculate probabilities of certain outcomes, such as finding the proportion of data laying between two values, or above a certain value, which is precisely what the exercise involves with computing the percentages of sales representatives earning more than or between given amounts.