STATISTICS 2
The Exam will consist of around 40 multiple choice questions and a small number of short essay
questions. These Exam questions cover theory, doing calculations, test selection and execution, and
the interpretation of statistical output.
The mean is the average of a data set. Best for ratio. → sum of all values divided by number of cases.
The mode is the most common number in a data set. Best for nominal. → most frequent value
The median is the middle of the set of numbers. Best for ordinal data. → middle value
Stem and leaf: depict distribution of a variable. Most important info at the bottom. Typically for
smaller datasets than histogram
Error bar plot: mean + confidence interval > compare confidence interval, differences in
groups
Bar chart: show distribution categorical variable, nominal or ordinal data
Boxplot: identify outliers (misleading results for normality)
Q-Q plot: check normality.
Histogram: distribution of interval and ratio variables
Scatterplot: the joint distributions of two (quantitative) variables
Example graph: Large enough sample, 123 cases, more than 30. Histogram correct graph as it
contains ratio variables, and discusses the distribution. Makes mean relevant. Standard deviation
smaller than the mean, not a big spread in the group. 1 outlier. Skewness. Kurtosis.
Exam: discuss the test. Write H0, significant or not and why. Reject or not reject. Variables? Correct
test? Suspicion confirmed yes or no.
Example test: significant in 3 ways; t -3 is lower than -2 + sig. 0.001 lower than 0.05 + 0 excluded in
95% confidence interval.
Week 1: Lecture 2, 08-02-2023
Comparing means between groups
- Comparing more than two groups
- Looking for relationships:
o measures of association
o regression analysis
1
,Statistical tests
› Z-test
› Single sample t-test
› Paired sample t-test
› Two samples t-test
› Or nonparametric alternatives
When to use which test?
Two groups: two samples t-test
More than two groups: analysis of variance
One-way Analysis of Variance (or One-Factor ANOVA)
The factor determines the groups.
Types of Analysis of Variance
› One-way analysis of variance: one independent variable (=factor)
› Two-way analysis of variance: two independent variables (=factors)
Null hypothesis One-Way Analysis of Variance
› About population, never about sample!
› One specific situation - E.g. ‘no difference’, ‘no relation’, etc.
› Population mean is equal between groups
› What is the alternative hypothesis? There is a difference between the groups; but this is very vague
as you have multiple groups. Are all the groups different from one another or just one? So you need
a follow-up test in order to investigate where the difference is.
Null hypothesis (example): In the population, the average travel time home-to-work is equal
between groups based on region of residence.
Requirements One-Way Analysis of Variance
› independent cases
› ratio or interval variable (> comparing means)
› data in more than two groups
- per group: normally distributed or enough cases (if 1 group is weak, the entire analysis is
weak)
- standard deviation groups equal > “homoscedasticity” (compare groups that are similar to
each other)
Central Limit Theorem: Distribution of sample means is normal in large samples > also when the
distribution of the values of the cases is not normal!
Requirements One-Way Analysis of Variance
1. Distribution not normal?
› with enough cases:
- no problem due to the Central Limit Theorem
- rescaling can lead to a better model
› with a small amount of cases:
- does rescaling help?
- if not: nonparametric alternative test
2
, 2. Standard deviation not equal?
› Burt et al (2009)
- Tests biggest vs smallest variance
- Levene’s test
› Rule of thumb
- Biggest standard deviation < 2 * Smallest standard deviation
So 2 ways to check if standard deviation is equal.
If it would be bigger than 56.414, then you wouldn’t be comparing the same thing anymore. So in
example it is good enough, similar enough.
Balanced case is special situation, if all cases are exact same size. So almost always unbalanced
situation.
How the group differs within the group. Tells us how wide the groups are. The noise.
How groups are different from one another.
Test significant when small within and large between. > reject.
Test less likely to be significant when both within and between are large.
3
, 31.44-35.20, square it, multiple by 636. Divide by degrees of freedom; 4-1=3
Group mean minus total mean, square, multiply with number of cases. Divide.
Yi=1 is first group. Yi=2 is second group. Etc.
Number of cases of group minus 1 + standard deviation squared : total number of cases – number of
groups. (in example 4 groups so -4)
4
The Exam will consist of around 40 multiple choice questions and a small number of short essay
questions. These Exam questions cover theory, doing calculations, test selection and execution, and
the interpretation of statistical output.
The mean is the average of a data set. Best for ratio. → sum of all values divided by number of cases.
The mode is the most common number in a data set. Best for nominal. → most frequent value
The median is the middle of the set of numbers. Best for ordinal data. → middle value
Stem and leaf: depict distribution of a variable. Most important info at the bottom. Typically for
smaller datasets than histogram
Error bar plot: mean + confidence interval > compare confidence interval, differences in
groups
Bar chart: show distribution categorical variable, nominal or ordinal data
Boxplot: identify outliers (misleading results for normality)
Q-Q plot: check normality.
Histogram: distribution of interval and ratio variables
Scatterplot: the joint distributions of two (quantitative) variables
Example graph: Large enough sample, 123 cases, more than 30. Histogram correct graph as it
contains ratio variables, and discusses the distribution. Makes mean relevant. Standard deviation
smaller than the mean, not a big spread in the group. 1 outlier. Skewness. Kurtosis.
Exam: discuss the test. Write H0, significant or not and why. Reject or not reject. Variables? Correct
test? Suspicion confirmed yes or no.
Example test: significant in 3 ways; t -3 is lower than -2 + sig. 0.001 lower than 0.05 + 0 excluded in
95% confidence interval.
Week 1: Lecture 2, 08-02-2023
Comparing means between groups
- Comparing more than two groups
- Looking for relationships:
o measures of association
o regression analysis
1
,Statistical tests
› Z-test
› Single sample t-test
› Paired sample t-test
› Two samples t-test
› Or nonparametric alternatives
When to use which test?
Two groups: two samples t-test
More than two groups: analysis of variance
One-way Analysis of Variance (or One-Factor ANOVA)
The factor determines the groups.
Types of Analysis of Variance
› One-way analysis of variance: one independent variable (=factor)
› Two-way analysis of variance: two independent variables (=factors)
Null hypothesis One-Way Analysis of Variance
› About population, never about sample!
› One specific situation - E.g. ‘no difference’, ‘no relation’, etc.
› Population mean is equal between groups
› What is the alternative hypothesis? There is a difference between the groups; but this is very vague
as you have multiple groups. Are all the groups different from one another or just one? So you need
a follow-up test in order to investigate where the difference is.
Null hypothesis (example): In the population, the average travel time home-to-work is equal
between groups based on region of residence.
Requirements One-Way Analysis of Variance
› independent cases
› ratio or interval variable (> comparing means)
› data in more than two groups
- per group: normally distributed or enough cases (if 1 group is weak, the entire analysis is
weak)
- standard deviation groups equal > “homoscedasticity” (compare groups that are similar to
each other)
Central Limit Theorem: Distribution of sample means is normal in large samples > also when the
distribution of the values of the cases is not normal!
Requirements One-Way Analysis of Variance
1. Distribution not normal?
› with enough cases:
- no problem due to the Central Limit Theorem
- rescaling can lead to a better model
› with a small amount of cases:
- does rescaling help?
- if not: nonparametric alternative test
2
, 2. Standard deviation not equal?
› Burt et al (2009)
- Tests biggest vs smallest variance
- Levene’s test
› Rule of thumb
- Biggest standard deviation < 2 * Smallest standard deviation
So 2 ways to check if standard deviation is equal.
If it would be bigger than 56.414, then you wouldn’t be comparing the same thing anymore. So in
example it is good enough, similar enough.
Balanced case is special situation, if all cases are exact same size. So almost always unbalanced
situation.
How the group differs within the group. Tells us how wide the groups are. The noise.
How groups are different from one another.
Test significant when small within and large between. > reject.
Test less likely to be significant when both within and between are large.
3
, 31.44-35.20, square it, multiple by 636. Divide by degrees of freedom; 4-1=3
Group mean minus total mean, square, multiply with number of cases. Divide.
Yi=1 is first group. Yi=2 is second group. Etc.
Number of cases of group minus 1 + standard deviation squared : total number of cases – number of
groups. (in example 4 groups so -4)
4
