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In the world of statistics, understanding how data points relate to the mean is essential. This is where the concept of a z-score comes into play. The z-score is a measure that describes a data point's position in relation to the mean, normalized by the standard deviation. In simpler terms, it tells you how many standard deviations away a particular value is from the mean.

The z-score is calculated using the formula:
\( z = \frac{X - \mu}{\sigma} \)
where \( X \) is the value you are looking at, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

For example, let's say the mean score of a test is 50, and the standard deviation is 7. If a student scores 56 on this test, the z-score would be calculated as follows: \[ z_1 = \frac{56 - 50}{7} = \frac{6}{7} \approx 0.857 \] This means that the score of 56 is approximately 0.857 standard deviations above the mean.

The standard normal distribution is a special type of normal distribution with a mean of 0 and a standard deviation of 1. It is also known as the 'z-distribution'. This distribution is a critical tool in statistics because it allows us to perform probability calculations and make inferences about data.

When we convert raw scores into z-scores, we essentially transform the original data distribution into the standard normal distribution. This makes it easier to work with, as we can use standard normal distribution tables to find probabilities associated with different z-scores.

For instance, in our previous example, we calculated z-scores for the values 56 and 66. These z-scores (0.857 and 2.286) convert the original data into the standard normal distribution, allowing us to easily look up the probabilities for these z-scores.

Once we have our z-scores, the next step is to find the probability associated with these values using the standard normal distribution. Probability calculations involve looking up these z-scores in a z-table, which provides the area under the curve (or probability) for any given z-score.

In our example, the z-score for 56 was approximately 0.857, and the corresponding probability from the z-table is about 0.804. The z-score for 66 was 2.286, with a corresponding probability of roughly 0.989. These probabilities represent the areas under the standard normal distribution curve to the left of these z-scores.

To find the probability that the variable falls between two values (in our case, between 56 and 66), we subtract the smaller area from the larger area:

\[ P(56 \leq X < 66) = P(Z < 2.286) - P(Z < 0.857) = 0.989 - 0.804 = 0.185 \]

This tells us that there is an 18.5% chance that a randomly chosen value from this normal distribution will fall between 56 and 66.