The document includes an extensive table of contents with page numbers to all the important formulas, concepts, etc. The notes include all lectures for the exam, except for the overview lecture.
Statistics 2022 Exam I
Contents
Statistical techniques..............................................................................................................................4
Measuring variables...............................................................................................................................4
Numerical...........................................................................................................................................4
Categorical..........................................................................................................................................4
Expressing difference in metric..............................................................................................................5
1. Cumulative distribution..................................................................................................................5
2. difference between centres relative to distribution (effect size D (delta Δ))..................................5
Calculating (arithmetic) mean................................................................................................................5
Dispersion (variation + variance)............................................................................................................5
Standard deviation (standardized deviation metric)..............................................................................6
Varpool & Spool............................................................................................................................................7
Median & quartiles.................................................................................................................................7
Total variation....................................................................................................................................9
Eta2 = proportion explained variation in y explained by x.......................................................................9
Interpretation of Eta2..............................................................................................................................9
Statistical regression.............................................................................................................................10
Linear regression..................................................................................................................................10
Formula straight line............................................................................................................................10
Deviation observation, explained part, residual part........................................................................11
Testing the model: R2............................................................................................................................11
Mean regression and linear regression: result splitting variation.................................................11
A linear regression model is adequate if:.....................................................................................11
Sample vs population...........................................................................................................................12
Theory of estimates: population to sample..........................................................................................12
Student interval: σ unknown................................................................................................................13
Estimations...........................................................................................................................................14
Point estimate & interval estimate...................................................................................................14
Confidence interval..............................................................................................................................14
If sigma is known:.............................................................................................................................14
If sigma is unknown:.........................................................................................................................15
Interpretation of IC...........................................................................................................................15
Hypotheses...........................................................................................................................................16
1
, By Z. Floren
Testing hypotheses:..............................................................................................................................16
Exceedance probability (p-value).....................................................................................................17
Critical value.....................................................................................................................................17
Confidence interval (CI)....................................................................................................................18
Comparing two groups.........................................................................................................................19
Test for two paired samples.............................................................................................................19
Tests for two independent samples.................................................................................................19
Which test to use:.............................................................................................................................19
2 groups of paired measurement.........................................................................................................20
Paired sample T-test........................................................................................................................20
Wilcoxon Rank Sign test...................................................................................................................21
Sign test............................................................................................................................................21
2 independent groups..........................................................................................................................21
Independent-samples T test.............................................................................................................22
Levene’s test – T-test with or without assumption...........................................................................22
Mann-Whitney U-test: u-value.........................................................................................................23
Test for comparison 2 proportions.......................................................................................................24
Type I and II error.................................................................................................................................24
Sample-effect size.................................................................................................................................24
More than 2 groups..............................................................................................................................25
Between group variance: S2between......................................................................................................25
Within group variance: S2within...........................................................................................................25
Anova (for independent samples)........................................................................................................25
ANOVA-table....................................................................................................................................25
One-way ANOVA (F-value)................................................................................................................26
For two groups (compared to T-test)................................................................................................26
Eta2 in ANOVA, estimating effect population η2................................................................................26
ANOVA conditions................................................................................................................................27
Planned comparison 2 combinations (contrasts, cj).............................................................................27
Orthogonal contrasts (uncorrelated contrasts)................................................................................28
Contrasts in SPSS..............................................................................................................................28
Polynomial contrasts (check trends).................................................................................................28
Post-hoc comparisons..........................................................................................................................29
Repetition one-way ANOVA.................................................................................................................29
ANOVA flowchart.................................................................................................................................29
Control for other variable(s): 2-way ANOVA.........................................................................................30
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, By Z. Floren
Practical 1.............................................................................................................................................31
Cumulative quantile plot:.................................................................................................................31
Maximum cumulative percentage difference Delta.........................................................................31
Arithmetic mean without calculator.................................................................................................31
Measures of dispersion....................................................................................................................31
Practical 2.............................................................................................................................................32
ANOVA table.....................................................................................................................................32
Eta2 computed from ANOVA table................................................................................................32
Deriving R2 from the ANOVA table................................................................................................32
Practical 3.............................................................................................................................................32
Similarity and difference between a student (df) distribution and the standard normal distribution
..........................................................................................................................................................32
Confidence interval directed & undirected hypothesis....................................................................32
Practical 4.............................................................................................................................................33
Choosing which test to perform.......................................................................................................33
Reading results of Wilcoxon Signed Rank test and the sign test in SPSS..........................................34
Hypothesis tests for two independent samples from two populations............................................34
Practical 5.............................................................................................................................................35
Mean squares ss and df....................................................................................................................35
Estimation value for the population η2 value....................................................................................35
One-way ANOVA using contrasts......................................................................................................35
T-value contrast............................................................................................................................36
Difference between ANOVA and 2 independent sample t-test........................................................36
Difference in distribution between the groups -> homogeneity test (of Levene based on the mean)
..........................................................................................................................................................36
Relationship t-value & f-value (equal variances assumed)...............................................................36
Eta2 systemic or accidental?..............................................................................................................36
Appendix 1: Table A, Z-values...............................................................................................................37
Appendix 2: Table B, Tail probability....................................................................................................38
Appendix 3: Effect sizes D, Eta2, Δ, r & r2..............................................................................................39
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, By Z. Floren
Statistical techniques
1. Descriptive statistics: describe/summarize the data pertaining to the two groups in:
a. Tables
b. Graphs
c. Metrics
… and draw your conclusions regarding similarities and differences
2. Inductive or inferential statistics: making decisions in the face of uncertainty.
Make the best decision based on incomplete information
a. How to generalize findings based on samples?
b. Is the observed difference more than a coincidence (is the difference statistically significant)?
c. How can past and current data be used to make predictions?
Measuring variables
Numerical Continuous = e.g. distance; you can have all types of values within a range. If
you have a max of 100 meter you can measure all points in between
m/sec, km/h
Scale from 0 to infinity
polygon-chart instead of barchart
Discrete = only inertial numbers, e.g. the year or the number of days
ALWAYS consistent intervals between categories (math)
Ratio = 0 is absolute
E.g. Kelvin or wind speed
Interval = 0 is not absolute
E.g. degrees Celsius, 0 is not absolute
Categoric Ordinal = there are different categories which can be ordered in a rational way
al (data can be organized and ranked)
Beaufort scale (0-12 Bft), intervals not similar
Age
Totally disagree-totally agree
ALWAYS inconsistent interval
Nominal = categories cannot be logically ordered (data can only be organized)
Gender
Sector
Programme
Scale -> interval ranked (ratio)
Weight (kg), distance (m), time (minutes)
Bar chart = graphic representation of the frequency table
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