Applied Data Analysis
Tentamen 16 juni 2021
College 1 : EXPLORING DATA
Literatuur: Field, A. (2018). Discovering statistics using IBM SPSS Statistics (5th edition).
Chapter 2 (§ 2.1 – 2.10)
Chapter 3 (§ 3.1 – 3.7)
Chapter 5 (§ 5.1 – 5.9)
Chapter 6 (p. 243-252, 268-276)
Chapter 19 (§ 19.1 – 19.3.6, 19.7)
In this lecture we will learn how we can explore our data.
Why explore?
Generally, research is (and should be) hypothetical-deductive.
This means we should:
- First formulate a hypothesis (on theoretical grounds) and deduce which pattern of
results should follow from it.
- Then, collect data to test whether these hypotheses apply (hypothesis are always
about the population!).
Usually, this leads to a focused prediction (e.g., females have higher social skills score than
males: µf > µm. In a social skills test, females should score higher than males).
Two reasons to explore our data:
1. We do not want to limit ourselves to only our main prediction! Sometimes, unexpected
results are the most interesting ones (isn’t science about finding out new things)
2. Almost always, we need to check assumptions of hypothesis tests.
Main steps in data analysis
1. Explore. Look what’s in your data.
2. Check assumptions. Significance tests make assumptions about the data, but do they
apply in your case? (and if violated, what has to be done?)
3. Hypothesis testing. Determine if a predicted relationship exists in the sample (e.g. a
correlation between two variables) and if it can be generalized from sample to
population.
4. Interpretation. Analyze the nature of the relationships between variables.
5. Write. Report your results (following APA rules). Preliminary step. Decide which
technique is most suitable for your research question.
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,Preliminary step: Decide which technique is most suitable for your research question
Exploring frequency distributions
Two basic ways of exploration
1. Make pictures (histograms, boxplots)
2. Compute statistics (mean, median, mode, variance, standard deviation, skewness,
kurtosis, Kolgomorov-Smirnov test). We will do both, with emphasis on normality
Remark. Very often the normality assumption is not as important as suggested by Field,
because many tests are robust against violation of this assumption.
Histogram
Histogram. Picture of a frequency distribution (categories on X-axis, numbers of individuals
on Y-axis).
Normality at first sight. From left to right more deviation from normality.
(middle and right histogram are positively skewed. Most clinical scored are positively
skewed, because most people have low scores on for example depression)
Boxplot
Boxplot is exclusively defined in terms of percentiles. Not in means and standard deviations!
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,Boxplot uitleg:
Median: middelste score
75th percentile: 75% van de scores ligt onder dit getal
25th percentile: 25% van de scores ligt onder dit getal
Sticks: either 1,5 times de box height of de laagste/hoogste score. In dit plaatje is de onderste stick de
laagste score en de bovenste stick 1,5 x box height.
Outliers: - Scores die verder dan 1,5 x de box height (sticks) aangegeven met cirkels
- Scores die verder dan 3x de box height (sticks) aangegeven met sterretjes
Warning. Boxplots are based on percentiles (median is 50th percentile). They do not
necessarily give the same results as measures based on means and variances.
No perfect normality or symmetry in previous boxplot, but it can be much worse. Look at
anxiety boxplot.
- Very positively skewed distribution.
- Most people are low on anxiety: more than 25 %
has lowest possible score (→ 25th percentile =
lowest score → no “stick” under box)
- A lot of outliers and extreme scores.
No lower stick means that more than 25% of the
lower scores have exactly the same score
Use boxplots to compare different variables, or to compare different groups on same variable (here:
occupation)
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, Boxplots for different variables are only useful when variables have comparable measuring
scales. DON’T DO THIS! These boxplots are very ugly together because the variables have
different scales. (0-3 and 0-15)
Skewness and kurtosis
Skewness: measure of asymmetry of the distribution.
- perfect symmetry → skewness = 0;
- long tail of distribution to the right → skewness > 0;
- long tail of distribution to the left → skewness < 0.
Normal distribution is always 0 skew. But 0 skew does not mean
normal per se.
Kurtosis: measure of “peakedness” of a distribution (actually whether a distribution is more or
less “peaked” than you would expect on the basis of the standard deviation and the normality
assumption).
- Perfectly normal distribution → kurtosis = 0 (but kurtosis = 0 does not necessarily
imply normal distribution).
- Peak higher than normal → kurtosis > 0 (red: more scores in the middle and in the
tails).
- Peak lower than normal (i.e. distribution to flat) → kurtosis < 0 (green: more
scores between the middle and the tails).
Attention! Positive kurtosis does not only mean a higher peak! It also means more scores
in the tails. Only a higher peak does not have to mean positive kurtosis, but could also
mean a low standard deviation.
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