Sample Distributions PDF

2/12/25, 11:55 AM OneNote Sample Distributions Friday, February 07, 2025 9:37 AM Why examine the shape of the distribution? Asymmetrical distributions: ○ Mean, median, mode d...

2/12/25, 11:55 AM OneNote Sample Distributions Friday, February 07, 2025 9:37 AM Why examine the shape of the distribution? Asymmetrical distributions: ○ Mean, median, mode differ ○ Are sample statistics reliable estimators of population parameters Asymmetirical and kutotic: ○ Do not confirm to expectations of theorestical distributions Theoretical distributions: A model for the variation in values of a variable based on the probabilities of the values” inferential statistics –comparing empirically observed distributions (your sample) with a theoretical distribution ○ If correspond, we can make inferences about future probabilities &/or the population based on our sample data ○ E.g., Normal, binomial, Poisson, log-normal, t-distribution, F-distribution The normal distribution: Theorectical distibution of frequecies of variates with several properties: ○ Symmetrical aroun mean ○ Peak is the mean ○ Asympotic (tails never reach 0) ○ Area under curve sums to 1 (reflects finite proability space) https://uvic-my.sharepoint.com/personal/lucybryden_uvic_ca/_layouts/15/Doc.aspx?sourcedoc={9faa0b91-e898-498c-a6e1-c5e15b84c2ef}&action=edit&wd=target… 1/5 2/12/25, 11:55 AM OneNote Small area of curve overlap- low proabability of drawing same value from both distributions (A & C) Normal distribution is also callef the gaussian distribution Area under the curve: % observations within range: Probabilites outside standar deviations ddecasee Area under the curve= probability of a value falling there True or fasle: Our sample distrbutions mmsut perfectlyn match the shape of the theorectical distribution (normal) to allow is to make inferences based on the theoretical expectations – FALSEL https://uvic-my.sharepoint.com/personal/lucybryden_uvic_ca/_layouts/15/Doc.aspx?sourcedoc={9faa0b91-e898-498c-a6e1-c5e15b84c2ef}&action=edit&wd=target… 2/5 2/12/25, 11:55 AM OneNote In annormal distributoin the data are symmetical around the mean- TRUE Nirmal distrbutins are defined by their locationa nd dispersion statistics- Which distribution are you more likley to radomly find in your poputltion ○ A datum >-1,52 from the mean Z-Scores–Standardizing sample variates How many standard deviations from the mean is a variate? Standarizes your sample in units of standard deviation Looking for Normality: How do we know if our sample is normally distributed? Visual inspectin- histogram, density plot, Q-Q plot Statistical test- noramlity test Caustions No universally accepted methof to assess sample normality "perfectaly" noraml samples do nto exist in real life- must make justifiable judgment call about your sample Non-normal sample? Just transform the data Q-Q plots: Quantile-quantile plot Compare observed and theoretical distribution by plotting quantiles ○ If dots call on (near) line, is good ○ Sone deviation is oky Histogramps & Q-Q plots: What are you looking for? Histogram or density: generate symmetry Difficult to interpert with small samples Q-Q plot: dots lie along/close to the line Minor deviations OK Skewness- asymmeticak distributions: Is the "tail" of the distribution "pulled" to one side? And ends fall on the same side of line https://uvic-my.sharepoint.com/personal/lucybryden_uvic_ca/_layouts/15/Doc.aspx?sourcedoc={9faa0b91-e898-498c-a6e1-c5e15b84c2ef}&action=edit&wd=target… 3/5 2/12/25, 11:55 AM OneNote Thin tails, fat tails, and kurtosis: Too much data in tails relative to noraml Bimodal: https://uvic-my.sharepoint.com/personal/lucybryden_uvic_ca/_layouts/15/Doc.aspx?sourcedoc={9faa0b91-e898-498c-a6e1-c5e15b84c2ef}&action=edit&wd=target… 4/5 2/12/25, 11:55 AM OneNote Commoly used, mostly worjk in the same way PArametric and nonParametic tests: Parametric staticstical tests must be noraml https://uvic-my.sharepoint.com/personal/lucybryden_uvic_ca/_layouts/15/Doc.aspx?sourcedoc={9faa0b91-e898-498c-a6e1-c5e15b84c2ef}&action=edit&wd=target… 5/5