91Ó°ÊÓ

Z-scores are a way to describe the position of a data point within a distribution, helping us understand how far and in what direction the point lies from the mean. They are standardized scores, meaning each score indicates the number of standard deviations a point is away from the mean. In the case of a standard normal distribution, the mean is 0 and the standard deviation is 1. Therefore, a z-score tells us directly how far a value is from the mean in terms of standard deviations.

For instance, a z-score of 1.04 means the data point is 1.04 standard deviations above the mean. Similarly, a z-score of -2.20 indicates the point is 2.20 standard deviations below the mean.

Calculating and understanding z-scores is fundamental when working with standard normal distributions as they transform data to a comparable standardized form, integrated into further statistical analysis.

Probability calculation in the context of z-scores involves determining the likelihood of a particular range of values occurring within a standard normal distribution. The process of finding probabilities for specific z-ranges helps us answer the question: "What is the chance that a randomly selected score falls within a particular interval?"

To calculate probabilities between two z-scores, we follow a few simple steps:

• First, obtain the cumulative probability for the higher z-score.
• Next, find the cumulative probability for the lower z-score.
• Finally, subtract the cumulative probability of the lower z-score from the higher one. This resulting value represents the probability of a randomly picked z-value falling within that interval.

In our example, we want to find the probability between z = -2.20 and z = 1.04. We subtract the cumulative probability at z = -2.20 from that at z = 1.04. This probability can be visualized as the shaded area under the curve between these two points.

A Z-table, also known as the standard normal distribution table, is a mathematical table used to determine the percentage of values below a given z-score in a standard normal distribution. The table is a powerful tool that assists in converting z-scores into probabilities and vice versa.

The Z-table lists z-scores in its first column and provides the cumulative probabilities up to those z-scores in the adjacent columns. It is divided into two sections based on positive and negative z-scores due to the symmetrical property of the standard normal distribution.

When using a Z-table:

• Identify the correct z-score from your data.
• Locate this z-score in the Z-table.
• Read across to find the cumulative probability.

For example, if you need to find the probability for z = 1.04, locate 1.0 in the row and use 0.04 in the column, then find the intersection point for the cumulative probability.