# Statistical Probability Distributions Notes

### Osmosis High-Yield Notes

This Osmosis High-Yield Note provides an overview of Statistical Probability Distributions essentials. All Osmosis Notes are clearly laid-out and contain striking images, tables, and diagrams to help visual learners understand complex topics quickly and efficiently. Find more information about Statistical Probability Distributions:

Normal distribution and z-scores

Paired t-test

Standard error of the mean (Central limit theorem)

Hypothesis testing: One-tailed and two-tailed tests

NOTES NOTES STATISTICAL PROBABILITY DISTRIBUTIONS NORMAL DISTRIBUTION & Z-SCORES osms.it/normal-distributions-z_scores NORMAL DISTRIBUTION ▪ Data grouped around central value, no left/ right bias, in “bell curve” shape ▪ Probability distribution for normal random variable x ▪ f(x) = ▪ ▪ ▪ ▪ ▪ 1 σ 2π e 1 −( )[( x− µ )/σ )2 2 µ = mean of normal random variable x σ = standard deviation π = 3.1416 … e = 2.71828 … Normal distribution: μ = 0, σ = 1 Z-SCORES ▪ Standardized score ▪ Uses data set mean, standard deviation to determine measurement location ▫ Represents deviation from mean ▪ Expressed in standard deviations ▪ Sample z-score for measurement x ▪ z= x−u σ ▪ µ = population mean ▪ σ = standard deviation OSMOSIS.ORG 71
STANDARD ERROR OF THE MEAN osms.it/standard-error-of-mean ▪ AKA SEM, standard deviation ▪ σx = σ ▪ σ = standard deviation ▪ n = sample size n PAIRED T-TESTS osms.it/paired-t-test ▪ Statistical hypothesis test (parametric) ▪ Determines if two groups are statistically different (compares two groups’ means) ▪ Groups can occur naturally (e.g. smokers compared to non-smokers)/groups can be created experimentally (e.g. control group compared to treatment group) ▪ t = difference between means variance/sample size ▪ = sample mean - population mean sample standard error of the mean ▪ ▪ ▪ ▪ ▪ x1 − x2 s12 s2 2 + n1 n2 x1 = mean of sample 1 x2 = mean of sample 2 n1 = sample size of sample 1 n2 = sample size of sample 2 ∑ (x − x ) 1 ▪ s12 = variance of sample 1 = ▪ s22 = variance of sample 2 = n1 − 1 ∑ (x 2 2 1 − x2 ) 2 n1 − 1 ONE-TAILED & TWO-TAILED TESTS osms.it/one-tailed-two-tailed-tests ▪ Tails: ends of probability curve ▪ Alternative (research) hypothesis proposes groups under investigation are different in some way/relationship between them exists 72 OSMOSIS.ORG ONE-TAILED TESTS ▪ Alternative hypothesis is directional (i.e. speciﬁes direction of difference/relationship) ▫ Extreme values of one of distribution tails are of interest given difference type/ expected relationship (solid theoretical
Chapter 11 Biostatistics & Epidemiology: Statistical Probability Distributions basis required for one-tailed test) ▫ Alternative hypothesis predicts relationship between groups either positive/negative (e.g. Group A will score higher on particular test than Group B) TWO-TAILED TESTS ▪ Alternative hypothesis is non-directional (i.e. non-speciﬁed direction of difference/ relationship) ▫ Extreme values on either tail of sampling distribution support null hypothesis rejection (e.g. Group A scores will be different than Group B) OSMOSIS.ORG 73

### Osmosis High-Yield Notes

This Osmosis High-Yield Note provides an overview of Statistical Probability Distributions essentials. All Osmosis Notes are clearly laid-out and contain striking images, tables, and diagrams to help visual learners understand complex topics quickly and efficiently. Find more information about Statistical Probability Distributions by visiting the associated Learn Page.