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In the world of statistics, the Z-score is a key concept that helps us understand how far a given data point is from the mean in terms of standard deviations. Imagine you have a set of data values. The Z-score translates any individual value into a standardized format, so you can see how it compares to the overall data set.

A positive Z-score means the data point is above the mean, while a negative Z-score indicates the point is below the mean.

• The formula for calculating a Z-score is \( Z = \frac{(X - \mu)}{\sigma} \), where:
• \(X\) is the data point,
• \(\mu\) is the mean of the data set,
• \(\sigma\) is the standard deviation.

In the context of a standard normal distribution, which specifically has a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1, Z-scores are foundational for probability calculations. They indicate the number of standard deviations a data point is from the mean of the distribution.

Once you have a Z-score, the next step is calculating the probability or the likelihood of a data point being less than or greater than this Z-score in the distribution. The process of calculating probability from a Z-score involves using the properties of the standard normal distribution.

This involves:

• Understanding that the area under the curve of a probability distribution represents total probability, which is always equal to 1.
• The probability for a range is found by calculating the area under the curve for that specific interval based on our Z-score.

For instance, if you need to find the probability that a Z-score is greater than a particular value, say 1.14, you would calculate the probability that it is less, using a Z-table, and then subtract from 1.

In simpler terms, For \( P(Z > 1.14) \),

• Find \( P(Z \leq 1.14) \). Let's say this value is 0.8729.
• Then use the formula: \( P(Z > 1.14) = 1 - P(Z \leq 1.14) \).
• So, \( 1 - 0.8729 = 0.1271 \).

The Z-table, which some might call a standard normal distribution table, is an invaluable tool for finding probabilities in a standard normal distribution. It provides us with the cumulative probability from the mean to the Z-score or from the far left of the distribution (negative infinity) to the Z-score.

How do we use it?

• Look up the Z-score in the table, which gives the probability of a value being less than or equal to that Z-score.
• For example, to find \( P(Z \leq -0.36) \), directly look at the Z-table for -0.36 and find 0.3594. This value represents the area to the left under the curve for that Z-score.
• To find the probability for a value between two Z-scores, say \(-0.46 < Z < -0.09\), simply find \( P(Z \leq -0.46) \) and \( P(Z \leq -0.09) \), then subtract the smaller area from the larger to get the in-between probability.

By mastering the use of a Z-table, you can easily calculate the probability of different events occurring under a standard normal distribution, which is a foundational skill in statistical analysis.