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A z-score table, sometimes called a standard normal distribution table, is a tool commonly used in statistics to determine the probability that a standard normal variable (z) will be less than or equal to a certain value. The table shows the cumulative probability from the mean to any value on the standard normal curve.
Understanding how to read this table is crucial for solving many statistical problems. For a given z-score, the table provides the area under the curve to the left of that score.
To use the table:

• Locate the row that matches the value of the z-score's first decimal place.
• Find the column corresponding to the second decimal digit.
• The intersection gives you the cumulative probability value from the mean to that z-score.

For example, a z-score of 1.28 corresponds to a cumulative probability of approximately 0.8997, indicating that the area to the left of 1.28 is 0.8997.

Probability calculations in the context of the standard normal distribution involve using z-scores to find the likelihood of a random variable falling within a specified range. Once you have these z-scores, you can look up their corresponding probabilities in the z-score table.
To calculate the probability of a specific range of z-scores:

• Find the cumulative probability for each z-score at the ends of the range.
• Subtract the smaller cumulative probability from the larger one.

For instance, to calculate the probability of z falling between -1 and 2, find probabilities for both z = -1 (approximately 0.1587) and z = 2 (approximately 0.9772). The probability of z being in that range is then 0.9772 - 0.1587 = 0.8185.
Such calculations help understand the likelihood of observed values occurring within a standard deviation range.

The symmetry of the normal curve is a fundamental property that simplifies many probability calculations. The normal distribution curve is symmetrical about its mean, which for a standard normal distribution is 0.
This symmetry implies:

• The areas under the curve on either side of the mean are equal, each covering 50% of the total distribution.
• If the probability of a z-score less than a certain value is known, the probability of a z-score greater than this value is the complement to 1.

For example, the area to the right of a z-score of 0 is 0.5, as is the area to the left, due to this symmetry. This feature is crucial when calculating probabilities for z-scores on either side of the mean.

Calculating the areas under the normal curve is essential for understanding probabilities relating to the standard normal distribution. Each area corresponds to the probability that the variable falls within a certain range of z-scores.
To find these areas:

• Use the z-score to calculate the respective cumulative probability from the table.
• For areas above a certain z-score, subtract the cumulative probability from 1.

For example, finding the area to the right of z = -5 delivers almost 1.00 due to the majority of the curve lying above this point.
These areas are visually represented by the plot regions under the curve, corresponding directly with the numerical probabilities provided by the z-score table. Understanding these areas helps predict the behavior of datasets fitting the model of a standard normal distribution.