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The standard normal distribution is a special case of the normal distribution. It has a mean of 0 and a standard deviation of 1. This distribution is symmetric about the mean, which means the left side mirrors the right side.
When working with the standard normal distribution, we use the variable Z to represent a data point. The value of Z tells us how many standard deviations away from the mean our data point lies.
If Z is positive, the data point is above the mean. If Z is negative, it is below the mean. Most values in a standard normal distribution fall within three standard deviations of the mean. Understanding this helps in calculating probabilities for given Z values.

Cumulative probability is the probability that a random variable is less than or equal to a certain value. For the standard normal distribution, cumulative probability refers to the probability that Z is less than or equal to a certain Z value.
To find this, we use a Z-table which provides cumulative probabilities for various Z values. The table gives us the area under the curve to the left of a Z value. For example, if the Z value is 1.39, the cumulative probability is approximately 0.9177. This means there is a 91.77% chance that a randomly selected data point from the standard normal distribution will be less than 1.39.
In problems combining multiple Z values, such as finding the probability that Z is either less than -2.56 or greater than 1.39, we use cumulative probabilities for each part and then sum them up.

The Z-table is a crucial tool for finding cumulative probabilities and solving problems related to the standard normal distribution. This table lists Z values and their corresponding cumulative probability (the area under the curve to the left of the Z value).
To use the Z-table, locate the Z value in the table. For Z values not listed, approximate by using the closest available value. For example, to find the probability of Z being less than -2.56, locate -2.56 in the Z-table, which shows a cumulative probability of 0.0052.
For Z values greater than the mean, like Z = 1.39, find the cumulative probability from the Z-table (0.9177). To find the probability Z is greater than 1.39, subtract this cumulative probability from 1 (since the total area under the curve is 1):
1 - 0.9177 = 0.0823
This means there’s an 8.23% chance that Z is greater than 1.39.
The Z-table enables quick and accurate calculation of these kinds of probabilities, making it invaluable for statistics and probability exercises.