Statistical Symmetry: A Closer Look at the Normal Distribution Curve

3 min readNov 16, 2023

A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric around its mean, meaning that the data is equally likely to fall on either side of the mean. It is a fundamental concept in statistics and probability theory, playing a crucial role in various scientific and mathematical fields. Here’s a comprehensive overview of the normal distribution:

1. Characteristics of the Normal Distribution:

a. Symmetry:

  • The normal distribution is symmetric, meaning that if you were to draw a vertical line at the mean, the two halves of the distribution would mirror each other.

b. Bell-Shaped Curve:

  • The graph of a normal distribution forms a bell-shaped curve, with the highest point at the mean.

c. Measures of Central Tendency:

  • The mean, median, and mode are all equal in a perfectly normal distribution.

d. Standard Deviation:

  • The spread or dispersion of the distribution is determined by the standard deviation. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

e. Empirical Rule:

  • The empirical rule, also known as the 68–95–99.7 rule, states that a high percentage of observations fall within certain standard deviation ranges.

f. Z-Score:

  • The Z-score represents the number of standard deviations a data point is from the mean in a standard normal distribution.

2. Probability Density Function (PDF):

  • The probability density function for the normal distribution is given by the formula: f(xμ,σ) = 1/σ2π exp(-(x-μ)² / 2σ²)
  • μ is the mean, σ is the standard deviation, and x is a particular value.

3. Standard Normal Distribution:

  • A standard normal distribution is a normal distribution with a mean (μ) of 0 and…