Normal curve

Normal Curve

The Normal Curve, also known as the Gaussian distribution or bell curve, is a type of continuous probability distribution for a real-valued random variable. The graph of the normal distribution is characterized by its bell shape and symmetric nature about the mean, which is also its mode and median.

Definition

The normal distribution is defined by two parameters: the mean (μ) and the standard deviation (σ). The probability density function of a normal distribution with mean μ and standard deviation σ is given by:

f(x) = 1/(σ√(2π)) e^(-(x-μ)^2/2σ^2)

where e is the base of the natural logarithm, π is pi, and σ is the standard deviation.

Properties

The normal curve has several important properties:

• Symmetry: The normal curve is symmetric about its mean. This means that the left and right halves of the graph are mirror images of each other.
• Unimodality: The normal curve has a single peak at the mean, which is also the mode and median of the distribution.
• Asymptotic nature: The tails of the normal curve extend to infinity without touching the x-axis, meaning that all real numbers are within the range of possible outcomes.
• 68-95-99.7 rule: Approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. This is also known as the Empirical Rule.

Applications

The normal curve is widely used in statistics, natural sciences, social sciences, and engineering due to its desirable properties and the Central Limit Theorem, which states that the sum of a large number of independent and identically distributed random variables tends towards a normal distribution, regardless of the shape of the original distribution.

See also

• Standard Normal Distribution
• Central Limit Theorem
• Z-score
• Probability Distribution

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