By Z. Floren
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
2
, 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
3
, 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
4
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
2
, 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
3
, 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
4
