Normal Distribution – Introduction and Properties
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The document provides an overview of normal distribution, describing it as a continuous frequency distribution vital in statistics and various fields. It outlines key properties such as its symmetry, equal mean-median-mode values, and application in statistical inference and hypothesis testing through z-scores. The normal distribution, also known as Gaussian distribution, was historically developed by mathematicians like Abraham de Moivre and Pierre-Simon Laplace.
Normal Distribution – Introduction and Properties
- 1. Normal Distribution - Introduction & Properties SUNDARA B. N. Assistant Professor
- 2. Normal Distribution (1) The normal distribution is a descriptive model that describes real world situations. It is defined as a continuous frequency distribution of infinite range (can take any values not just integers as in the case of binomial and Poisson distribution). This is the most important probability distribution in statistics and important tool in analysis of epidemiological data and management science.
- 3. Normal Distribution (2) The normal distribution was first discovered by Abraham de Moivre, a French mathematician he published an article on Doctrine of Chances in 1733. Later it was applied in natural and social science by Laplace in 1777. The Normal Distribution is also known as Gaussian distribution in honor of Karl Friedrich
- 4. Normal (Gaussian) Distribution Z Score (Standard Score) Z = X – μ SD (X=Observed Mean, μ=Population mean) Z indicates how many standard deviations away from the mean the point x lies. Z score is calculated to 2 decimal places. Tables Areas under the standard normal curve
- 5. Characteristics/Properties of Normal Distribution It has two parameters – mean μ and SD Mean = Median = Mode all are equal The mean of the distribution can be any numerical value (Negative, Zero or Positive) It is symmetric about mean – (Mirror image) SD determines how flat and wide the normal curve is The total area under the curve for the ND is 1 (or 100%) In a Standard Normal Distribution: The mean (μ ) = 0 and Standard deviation (σ) =1
- 6. Characteristics/Properties of Normal Distribution (2) The percentage of values in some commonly used intervals are: a) 68.3% of the values of a Normal random variable are within plus or minus ± one SD of its mean b) 95.4% of the values of a Normal random variable are within plus or minus ± two SD of its mean c) 99.7% of the values of a Normal random variable are within plus or minus ± three SD of its mean
- 7. 13.55% 2.15% 0.15% -3 -2 -1 μ 1 2 3 Diagram of Normal Distribution Curve (z distribution) 34.15%
- 8. Application/Uses of ND It’s application goes beyond describing distributions It is used by researchers and modelers. The major use of normal distribution is the role it plays in statistical inference. The z score along with the t –score, chi-square and F-statistics is important in hypothesis testing. It helps managers/management make decisions.