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This rule, also known as the empirical rule, states that for a normal distribution: approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

The PDF of a normal distribution is given by the formula:

Standard deviation measures the amount of dispersion or spread in a set of values. In a normal distribution, about 68% of values lie within one standard deviation of the mean.

The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution, regardless of the distribution of the population, as the sample size becomes larger.

The graph of a normal distribution is symmetrical and has the shape of a bell. Most data points cluster around a central peak, with frequencies tapering off equally in both directions.

A Z-score measures how many standard deviations an element is from the mean. The formula for a Z-score is:

The mean is the arithmetic average of a set of values, or the distribution's center of gravity. For a normal distribution:

The CDF is the probability that a normally distributed random variable will be less than or equal to a certain value. It is calculated as the area under the PDF curve from $-\infty$ to $x$.

In a normal distribution, the mean and standard deviation are independent of each other. Changing the mean shifts the curve left or right, while changing the standard deviation alters the width of the curve.