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The Normal Distribution – Concepts and Fundamentals
The Normal (or Gaussian) distribution is the most common continuous probability distribution.
The function gives the probability that an event will fall between any two real number limits as
the curve approaches zero on either side of the mean. Area underneath the normal curve is
always equal to 1.
The curve itself is approximately bell shaped, and is therefore informally called the Bell Curve.
𝑓(𝑥) =
1
𝜎√2𝜋
−(𝑥−𝜇)2
𝑒 2𝜎2
For the Normal Curve, the mean, median, and mode should all be the same value (𝜇). The
Normal Curve also has the axis of symmetry 𝑥 = 𝜇. For the standard normal curve,𝜇 = 0 and
𝜎 = 1.
Notation:
For any normal curve with mean 𝜇 and standard deviation 𝜎, one can write the shorthand
abbreviation 𝑁(𝜇, 𝜎 2 ).
Empirical Rule:
Generally, for a Normal Curve, 68% of all values of 𝑥 fall between 𝜇 ± 1𝜎; 95% of all values of
𝑥 will fall between 𝜇 ± 2𝜎; and 99.7% of all values of 𝑥 will fall between 𝜇 ± 3𝜎.
The Central Limit Theorem:
One of the aspects of the normal curve that makes it so powerful is the central limit theorem,
which states that for a sufficiently large number of samples, the mean of these many samples will
be normally distributed, regardless of the distribution of the original data or events.
Middlesex Community College | Prepared by: Stephen McDonald