Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most of the observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions.

The mean is the average of all data points, and the standard deviation measures the dispersion or spread of the data points around the mean.

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

A 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 represents the number of standard deviations a data point is from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.

The area under the normal curve represents the probability of a random variable falling within a particular range of values.

Z-scores can be used to determine percentiles, which indicate the percentage of data points that fall below a particular value in a normal distribution.

To find the x-value, use the formula: X = μ + (z * σ), where μ is the mean, z is the z-score, and σ is the standard deviation.

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1.

A z-score of -1.43 indicates that the value is 1.43 standard deviations below the mean.