[07-A] Descriptive Statistics and data exploration

‹#› Het begint met een idee
Data Analysis
Descriptive Statistics and data exploration
Ivano Malavolta

Vrije Universiteit Amsterdam
2
Quick Recap
Experiment
scoping
Experiment
planning
Idea
Experiment
operation
Analysis &
interpretation
Presentation &
package

3
Analysis and Interpretation
● Understanding the data
▪ descriptive statistics
▪ exploratory data analysis (EDA, e.g. boxplots, scatter plots)
● (Optional) data reduction
● Hypothesis testing
● Results interpretation

4
Descriptive Statistics
● Goal: get a ‘feeling’ about how data is distributed
● Properties:
▪ Central tendency (e.g. mean, median)
▪ Dispersion (e.g. frequency, standard deviation)
▪ Dependency (e.g., correlation)

5
Parameter vs. statistic
● Parameter: feature of the population
▪ μ: mean
▪ σ: standard deviation
● Statistic: feature of the sample
▪ : mean
▪ s: standard deviation
● Statistics are an estimation of parameters

6
Central Tendency
● Arithmetic mean:
● Geometric Mean:
• It is like the arithmetic mean, but with multiplication
à used when collected data is not ”additive”, but “multiplicative”
• Less sensible to outliers
• Try it when the range of the considered values is very large

7
Central Tendency: example
● Average of scores:
6 - 7 - 8 - 9 - 10
● Arithmetic mean: 8
● Geometric mean: ~7.87

8
Central tendency: example
● Average of returns of investments:
90% ; 10% ; 20% ; 30% ; -90%
● Arithmetic mean:
(90+10+20+30-90)/5= 12%
● Geometric mean:
[(1.9 x 1.1 x 1.2 x 1.3 x 0.1) ^ 1/5] - 1 =0.2008= -20.08%

9
Central tendency
● Median (or 50% percentile): middle value separating the
greater and lesser halves of a data set
X = [13, 18, 13, 14, 13, 16, 14, 21, 13]
Xsort = [13, 13, 13, 13, 14, 14, 16, 18, 21]

10
Central tendency
● Mode: most frequent value in data set
X = [13, 18, 13, 14, 13, 16, 14, 21, 13]
Mox = 13

11
Central tendency - Skewness

12
Dispersion
● Sample variance:
● Standard Deviation:
● Standard Deviation is dimensionally equivalent to the data
Informally: everything which is within 1 SD from
the mean is “normal”
Informally: it gives an idea about how ”sparse” is
data

13
Dispersion - three-sigma-rule
"Empirical Rule" by Dan Kernler - Own work. Licensed under CC BY-SA 4.0 via Wikimedia Commons -
http://commons.wikimedia.org/wiki/File:Empirical_Rule.PNG#/media/File:Empirical_Rule.PNG

14
Dispersion - three-sigma-rule
● Range:
● Coefficient of variation:
(in percentage of mean)
● Coefficient of variation only has meaning if all values are
positive (ratio scale, not interval scale e.g. temperatures)
It is useful if you want to compare the dispersion
of variables with different units of measure

15
Dispersion - example
● Dataset: [100, 100, 100]
Mean: 100
● Variance: 0
● Standard Deviation: 0
● Coeff. Variation: 0
● Range: 0

16
● Dataset: [90, 100, 110]
Mean: 100
● Sample Variance: 100
● Standard Deviation: 10
● Coeff. Variation: 10%
● Range: 20

17
● Dataset: [1, 5, 6, 8, 10, 40, 65, 88]
Mean: 27.875
● Sample Variance: 1082.69
● Standard Deviation: 32.9
● Coeff. Variation: 1.18%
● Range: 87

18
Basic visualizations
Box Plot
Median
3rd quartile
1st quartile

19
Box Plot
Minimum/maximum values THAT ARE NOT OUTLIERS

20
Box Plot
By Gbdivers (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0
(http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
outliers positive
skewness

21
Dependency: correlation
● Sample correlation coefficient (Pearson):
● Meaningful when comparing paired values/datasets

22
Dependency: correlation
● Spearman’s rank correlation coefficient:
● also good for ordinal data
● Kendall’s rank correlation coefficient:
▪ smaller values
▪ more accurate on small samples
● Pearson correlation coefficient assumes normally distributed
data

23
Dependency: example
Age vs. body fat %
● Pearson: r = 0.7921
● Spearman: 𝜌 = 0.7539
● Kendall: 𝜏 = 0.5762

24
Basic Visualizations
Scatter Plot

25
Positive VS negative correlation
https://statistics.laerd.com/statistical-guides/pearson-correlation-
coefficient-statistical-guide.php

26
Scatter plots per different values of r
r = Pearson
rs = Spearman
https://www.researchgate.net/publication/224915794_Improving_standa
rds_in_brain-behavior_correlation_analyses

27
Correlation does NOT imply causation!
● Spurious Correlations: http://tylervigen.com/

● Now you know how to explore trends within your data
● but you cannot reject null hypotheses yet
● You can have a “feeling” about
● how disperse-correlated is your data
● what is “standard” in your data
● You can quickly visualize interesting trends
● box plots
● scatterplots28
What this lecture means to you?

29 Ivano Malavolta / S2 group / Experiment design
Readings
Chapter 10

[07-A] Descriptive Statistics and data exploration

More Related Content

What's hot (20)

Similar to [07-A] Descriptive Statistics and data exploration (20)

More from Ivano Malavolta (20)

Recently uploaded (20)

[07-A] Descriptive Statistics and data exploration