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Summary Statistics Exam 1 & 2
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All lectures from Statistics Geo2-2217 summarized with pictures and example. Every lecture & all formulas that are used are described in this summary.
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Summary Statistics Lectures GEO2-2217Natuurwetenschap & Innovatiemanagement
March 2021
Contents
Lecture 2 Descriptive Statistics ........................................................................................................... 1
Lecture 3 Explained variation ............................................................................................................. 6
Lecture 4 Theory of estimates and testing ......................................................................................... 12
Lecture 5 Comparing two groups ...................................................................................................... 22
Lecture 6 Comparing more (independent) groups............................................................................. 30
Lecture 7 ANOVA with controls ...................................................................................................... 37
Lecture 9 Association interval and ordinal variables ........................................................................ 43
Lecture 10 Linear regression part I ................................................................................................... 48
Lecture 11 Linear regression part II .................................................................................................. 54
Lecture 12 Association nominal variables ........................................................................................ 61
Lecture 13 Logistic regression .......................................................................................................... 65
Lecture 14 Factor analysis ................................................................................................................. 71
Lecture 2 Descriptive Statistics
- Descriptive statistics: used for processing (a large amount of) data in different situations, e.g.
climate data or experimental data, less commonly used in qualitative research, because this data is
often less structured and therefore less quantitative (open interviews)
Measuring differences in Wind (example)
- Are winds stronger at the coast, compared to the interior?
→ Difficulties; how to measure? How to deal with variability?
- When we do research we limit ourselves in any kind of way with the measuring → need to set
boundaries at your research
- Difference between descriptive and inductive statistics is about the answer you get
Statistical techniques
- Descriptive statistics: describe/summarize the data pertaining to the two groups in tables, graphs
and metrics, and draw your conclusion regarding similarities and differences
→ What do we observe in our sample?
- Inductive statistics: can you generalize the findings for the sample to your population?
1. Is the observed difference more than a coincidence (is the difference statistically
significant?
2. What is the estimated size of the difference between the populations
Different scales
Interval There is a certain order and the intervals between different numbers are
equal, absolute zero is not meaningful (scale in SPSS)
1
, Ratio There is a certain order and the intervals between different numbers are
equal, absolute zero is meaningful (scale in SPSS)
Nominal Categories cannot be ordered, different colors e.g.
Ordinal There is a certain order but the intervals between the different
numbers are not equal
Measuring wind (measurement 1)
- Beaufort scale: from 0 to 12 Bft, based on sailing conditions, looked at wind conditions and saw the
effect of the wind on the sails of the ships → higher score = stronger wind
- Level of measurement: ordinal
Measurement 2
- Wind velocity in m/sec or km/h; scale from 0 to infinity → similar intervals on the scale indicate
similar difference in wind velocity
Level of measurement: interval
- Absolute zero is meaningful in this case
- A score that is p times as high, indicated a wind velocity that is p times as high
Level of measurement: ratio
Measurement 3
- Used for (wind) surfing; 0-4 and every number has their own explanation, there are different groups
but the order is not in line with the strength of the wind, because the zero is stronger than 1-4, we
cannot use order → order of scores not congruent with order in strength of wind
Level of measurement: nominal
Data matrix
- You have to store data in data matrix when you have a lot of data
- Each column is a variable and each row is an observation
- These columns can be presented in different ways → frequency tables, graphs
- Frequency tables: make different classes (of wind velocity), indicate what the number of
observations is in the different months for the certain category
→ From 0.0 to 0.2, 84 observations in march, 86 observations in July & 56 observations in November
- Bar chart: y-ax is number of observations and x-ax is beaufort → in July less wind because it scores
high on 0-1-2 than the higher ones, but there is a problem → there are now separate bars but wind is
not 0-1-2 or 3, because it’s a continuous phenomenon
- Polygons: connected lines, gives a better representation in this case
- What month experiences most wind?: march, because this one is placed more right then the
others, there are higher frequencies
- What month experiences most constant winds?: July, highest observed frequency and
therefore this month has most constant wind
- Any objections against graph?: Beaufort was an ordinal scale, which means that the interval
between different numbers is not equal, so 0-1 is something else than 1-2 etc., should have used a m/s
scale
Cumulative distribution
- Cumulative distribution: to see how large difference between the two groups is, in order to obtain
cumulative distribution we have to look at frequency and add the frequency to the already existing
frequencies
988 + 22443 = 23431 (see picture) → cumulative frequency
2
,CP = cumulative percentages (transformed cumulative frequency)
- Delta is the maximum difference between the lines in the right picture (green line), when this is 0
than the lines are in the same place because there is no difference
Difference between centers relative distribution
- Look at the average of the two groups → Bilt: 2.493 and Held: 3.606 → difference means: 1.113
What does this tell us? What is the meaning of 1.113?
Intermezzo (some important crucial measures in statistics)
Statistical toolbox
- e.g. movies
Mean
- gemiddelde, count everything up and divide by amount of observations
- In this case, movies have the same mean so not very interesting, there are quit some differences
between the observations, this has to do with dispersion
x bar= x → means
Dispersion
- amplitude, look at deviation of the individual observation from the mean, indicated with dev = x –
xbar (x = observation, xbar = mean)
Cannot just sum them up because we will get 0 again
- What metric would say something about deviation for the whole sample?
- Absolute deviation: with two lines on each side, means that you ignore what is in front of the
number (a minus for instance)
- Mean squared deviation: first take the square of the deviation and then take the average of it →
very useful but need some adjustments to use it
3
, Variance
- To calculate the variance for a sample (s2) use:
- Sum of squares: sum of different squared deviations = variation
- Why divide by degrees of freedom?: we have seen that we have 5 observations and that the sum of
the deviations has to be 0 → we can freely choose 4 out 5 of the deviations but the fifth one has to be
fixed than because the sum has to be add up to 0 → n-1 (when we look at variance) so therefore 12.5
divided by 4 instead of 5
- But we squared all the terms to make them positive, so another scale now
4
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