Distributions Commonly Used in Statistics Several types of

Distributions Commonly Used in Statistics

Several types of probability distributions are used in process of making statistical inferences about a population on the basis of a sample. The knowledge in this commonly used distributions and their properties is needed to use and interpret statistical methods correctly. We can differ between two groups of the most often used distributions: A) Distributions used for populations B) Distributions used for samples

Distibutions most often used in statistics (Summary): A) For Populations: Gaussian normal distribution Standard normal disribution Nonnormal distribution B) For Samples: t-distribution (Student´s) Chi-square distribution (Pearson´s) F- distribution (Fisher-Snedecor´s)

A) Gaussian Normal Distribution Commonly, most of biological data follow Gaussian Normal Distribution: bell-shaped curve - the majority of values is located around the mean (centre of symmetry) with progressively less observations toward the extremes.

A) Gaussian Normal Distribution The shape of the curve is determined by 2 parameters: - defines the centre of symmetry of the curve (middle of the population), indicates location of the curve on the numerical axis of studied variable - defines the spread of the curve in the inflexion point (variability of the population)

A) Gaussian Normal Distribution Within the range of: 1 - 68. 3% of all values in the population 2 - 95. 5% 3 - 99. 7%

A) Standard Normal Distribution

A) Non-normal Distribution („Irregular, Unknown“) We can define the median only (as a centre). The spread of the curve is not possible to assess.

B) Sample Distributions t – Distribution (Student) is defined for a theoretical variable t - is calculated from the mean and SD in a sample. (was published in 1908 by William Gosset under the pseudonym „Student“) is used for samples selected from the population with Gaussian Normal Distribution there are several (many) curves for samples differing in number of items (n).

B) t - Distribution (Student) p(t) Normal n=6 n=10 n= 0 t Height and spread of the curve - according to the sample size: - the smaller is the sample size the broader and lower is the curve - the bigger is the sample size the narrower and higher is the curve (in the extreme: n= the curve joins the normal distribution)

B) t - Distribution (Student) p(t) / -2 = spread of the curve Normal 1 ( =n-1) (DF) Degree of Freedom 5/3 n=6 n=10 n= 0 t Spread of the curve is defined by / -2: when n= spread = 1 (normal) when n=6 spread = 5/3

t-distribution - Use: Statistical tables of t-distribution: ® critical values in testing for difference between two means (Student t-test) ® coefficients used in calculations

B) Chi-square ( 2 - Distribution (Pearson) Asymmetrical curve – different shapes for different sample sizes ( =n-1 determines the shape): - the smaller is the sample size the higher and more asymmetrical is the curve shape - the bigger is the sample size the lower and more symmetrical is the curve shape

Chi-square distribution - Use: Statistical tables of 2 - distribution: ® critical values in testing for difference between frequencies ® coefficients used in calculations

B) F - Distribution (Fisher – Snedecor) Asymmetrical curve – different shapes for different sample sizes ( 1, 2 determine the shape): - the smaller is the sample size the lower and more asymmetrical is the curve shape - the bigger is the sample size the higher and more symmetrical is the curve shape

F-distribution - Use: Statistical tables of F- distribution: ® critical values in testing for difference between 2 variances (F-test)