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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 makes it very useful for statistical purposes because of its simplicity.
When values are converted into a format where the mean is 0 and the standard deviation is 1, we can use these values to understand how data behaves under the normal curve.
This conversion is done using the z-score formula, which allows us to take any normal distribution and transform it into a standard normal distribution.

• Mean: always 0 in a standard normal distribution.
• Standard deviation: always 1.
• Used for simplifying probability estimates and statistical significance testing.

By transforming a normal distribution to a standard one, we can easily find probabilities using standardized tables, known as z-tables.

A z-table, or standard normal distribution table, is an essential statistical tool. It provides the area or probability beneath the bell curve of the normal distribution, up to a given z-score.
This area helps answer questions about the probability of data falling within a particular range in the distribution. The z-table reflects how much of the curve is to the left of a z-score.
To use a z-table effectively:

• Locate your z-score in the leftmost column and top row of the table, which correspond to the z-score units and the first decimal place.
• The intersecting value in the table tells you the cumulative area under the curve from the mean (z = 0) up to the specified z-score.

For positive z-scores, the table will give a number less than 0.5, representing the area under the curve from the mean to the z-score. For negative z-scores, use the symmetry of the normal distribution to find probabilities.

The normal curve is a continuous probability distribution shaped like a bell. This curve describes how data points distribute across values, and it is symmetric about the mean.
The properties of the normal curve include:

• Bell-shaped, with most data falling near the mean and fewer data points at the extremes.
• Symmetrical, meaning equal areas on both sides of the mean.
• Defined by its mean and standard deviation, though the standard normal curve standardizes these to 0 and 1, respectively.

The area under the normal curve represents probability, indicating how likely a certain value or range of values is to occur within the distribution. Because of its properties, the normal curve is fundamental in probability and statistics for understanding data behavior.

Probability is a measure of how likely an event is to occur, ranging from 0 to 1. In the context of the standard normal distribution, probabilities are associated with areas under the curve.

• An area of 0 denotes an impossible event, while an area of 1 denotes a certain event.
• Between these two extremes, we have the various probabilities of events based on their z-scores.

When using a standard normal distribution, probabilities can be found between any two z-scores by calculating the area under the curve using a z-table.
For example, to find the probability between two z-scores, calculate the area under the normal curve between these points. This involves finding the area up to each z-score and subtracting the smaller area from the larger. This result helps understand the likelihood of events within that z-score range. Understanding probabilities in this way supports better decision-making and prediction models.