Mean and Variance of a Normal Distribution

Mean and Variance of a Normal Distribution

Definition of Normal Distribution

  • A normal distribution is a continuous probability distribution that is symmetric about its mean.

  • The formula for a normal distribution is given by the function: f(x) = (1/σ√2π) e^−((x−μ)^2/2σ^2)

  • In the formula, ‘x’ is the variable, ‘μ’ is the mean, ‘σ’ is the standard deviation, ‘e’ is the base of natural logarithms and ‘π’ is a constant approximately equal to 3.14159.

Mean and Variance of Normal Distribution

  • The mean (μ) of a normal distribution is the central value about which data points are distributed. It is the peak of the distribution’s curve.

  • The variance (σ^2) is a measure of the dispersion of the distribution, indicating how much values deviate from the mean.

  • This specific normal distribution, with mean 0 and variance 1, is referred to as the standard normal distribution.

  • All normal distributions can be transformed into the standard normal distribution by calculating the Z-score for each value. The Z-score is calculated using the formula: Z = (X - μ) / σ.

Key Features of Normal Distribution

  • The normal distribution follows the empirical rule (also known as the 68-95-99.7 rule), which states that within a normal distribution:
    • Approximately 68% of the data falls within one standard deviation (σ) of the mean (μ).
    • Around 95% falls within two standard deviations.
    • Approximately 99.7% falls within three standard deviations.
  • The normal distribution is important in statistics due to the central limit theorem, which states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, no matter the shape of the population distribution.

Examples

  • If you have a data set that consists of the heights of a large number of adults, the data would be normally distributed with a mean at the average adult height and a standard deviation related to the average difference in height among adults.

  • The grading of standardized exam scores often assumes a normal distribution to determine passing and failing cut-offs. For instance, a Z-score of 1.0 represents a value that’s one standard deviation from the mean.