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Transcript
Tools in Cognitive Science II:
Basic Statistics for Cognitive Scientists
Some basic concepts in statistics
Philipp Mitteröcker
Basic terms
Statistics
from Latin »statisticum« (about the state) and
Italian »statista« (statesman, politician).
Historical roots
17th century
Handling of demographic and economic data (“political arithmetic”)
John Graunt (1662) “Observations on the Bills of Mortality”
Development of Probability Theory by Pascal, Fermat, and Bernoulli
1794
The method of least squares was described by Carl Friedrich Gauss
19th and early 20th century
Francis Galton, Florence Nightingale, Karl Pearson, Ronald A. Fischer
Historical roots
Basic terms
Applied statistics
Descriptive statistics
Inferential statistics (hypothesis tests, confirmatory a.)
Exploratory analysis, modeling, data mining
Mathematical statistics
Basic terms
Biometrics, psychometrics, econometrics,
morphometrics...
metron = measurement
Basic terms
Measurement
The process of assigning a number to an attribute
(or phenomenon) according to a rule or set of rules.
Sample
A collection of individual observations selected by a
specific procedure.
Population
Totality of individual observations about which inferences
are to be made
Data (sing. Datum), Information, Knowledge
Theory, Hypothesis
Basic terms
Variable
A symbol that stands for a value that may vary.
Univariate statistics
Multivariate statistics
Bivariate statistics
Measurements
Precision (Präzision)
Degree to which repeated
measurements show the same results
(reproducibility, repeatability)
accuracy (Genauigkeit)
Closeness of measurements of a
quantity to the quantity‘s actual (true)
value.
bias (Verzerrung)
Difference between the average of the
measurements and the reference value
Measurements
3FGFSFODFWBMVF
1SPCBCJMJUZ
EFOTJUZ
"DDVSBDZ
1SFDJTJPO
7BMVF
Measurements
Estimating measurement error by repeated measures
Random error
Systematic error
Measurements
Outliers
Mistake or important measurement?
Measurements
Longitudinal versus cross-sectional data
Measurement scales
nominal scale (categorial data)
e.g., gender, nationality, habitat
ordinal scale
e.g., school grades, rank order, Likert scale
interval scale
no natural zero point, i.e., we can compute differences but no ratios
e.g., degree Celsius, coordinates
ratio scale
e.g., body height, counts, frequencies, degree Kelvin
Measurement scales
Discrete data (meristic data)
e.g., natural numbers, rank order, number of fish in a pond,
scale from 1 to 7
Continuous data
e.g., real numbers, cm, kg, degree Celsius
Descriptive statistics
Central tendency
mean, weighted mean
arithmetic, geometric, harmonic mean
mode, median
Dispersion, spread
range, variance, standard deviation, quantiles
coefficient of Variation
Descriptive statistics
The problem of multimodal distributions and outliers
Descriptive statistics & measurement scales
nominal scale
mode, frequencies (contingency tables)
ordinal scale
median, percentile
interval scale
mean, standard deviation, correlation, regression,
analysis of variance
ratio scale
geometric mean, coefficient of variation, logarithms
Descriptive statistics
How to describe a bivariate distribu6on?
Bivariate statistics
Covariance, Correlation
Correlation
-1 < r < 1
r = 0 ... no linear relationship
r = 1 or -1 ... perfect linear relationship
1 ... positive relationship
-1 ... negative relationship
Bivariate distribution
s12 = 0.647
Equal frequency ellipses
Data matrix
Var. 1 Var. 2 Var. 3
Var. 4
...
Case 1 Case 2 Case 3
Case 4
Case 5
...
Multivariate spaces
A
B
0.7
0.6
2
2
1
B
0.7
0.6
3
0.5
0.5
0.2
0.4
0.4
0.3
0.3
0.1
0.2
3
0.1
0.0
0.2
0.1
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.2
0.3
A
0.7
Q-space
R-space
Multivariate distribution
variance-covariance matrix
s12
s12
s21
s22
sn1
s1n
sn2
0.4
0.5
0.6
1
Multivariate distribution
correlation matrix
1
r12
r21
1
rn1
r1n
1
Multivariate distribution
0.950
0.647
0.647
1.535
0
0.820
0
0.235
Principal Component Analysis (diagonalization of a covariance matrix)
Multivariate distribution
Discriminant function analysis