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[edit | edit source]
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[edit | edit source]
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[edit | edit source]
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[edit | edit source]
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Contributors: Prab R. Tumpati, MD