8.2 Normal Distribution

The mean (μ) is the average value of a set of data points in a normal distribution, representing the center of the distribution.

The standard deviation (σ) measures the dispersion or spread of data points around the mean in a normal distribution.

The z-score is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

A z-score of -2.5 indicates that the data value is 2.5 standard deviations below the mean.

Approximately 68% of the data falls within one standard deviation (μ ± σ) of the mean.

Approximately 95% of the data falls within two standard deviations (μ ± 2σ) of the mean.

Approximately 99.7% of the data falls within three standard deviations (μ ± 3σ) of the mean.

The empirical rule states that for a normal distribution, about 68% of data falls within one standard deviation, 95% within two, and 99.7% within three standard deviations from the mean.

The area under the normal curve represents the total probability of all outcomes, which equals 1 (or 100%).

To find the cutoff for a specific percentile, use the z-score corresponding to that percentile and apply the formula: X = μ + zσ.