Statistics

2021

René Staritzbichler

Art of statistics

Summarize reality and evaluate reliability

mean / median / mode
range / quantile / standard deviation
significance / confidence
distributions
correlation

Summarize populations

mean: average income
median: income of average person

median = 50th percentile

mode: most common income
normal distribution: mean, median and mode are the same

How to summarize

mean: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i = \frac{1}{n}( x_1 + x_2 + ... + x_n) \]
median:
- sort your population by income
- person in the middle (50%)

Spread or width

range:
unsymmetric data, sensitive to extremes
standard deviation:
symmetric, sensitive
inter quartile range:
unsymmetric, insensitive

How to calculate width

standard deviation (represents 68%): \[ \sigma = \sqrt{ \frac{1}{n} \sum_{i=1}^n (\bar{x} - x_i)^2 } \]
inter quartile range (represents 50%):
- sort your population by income
- take person at 25% and person at 75%

Use of spread

how reliable is the mean/median?
example: error of multiple measurements

Hypotheses

translate your problem into a black/white question
nullhypothesis: no significant relation
alternative hypothesis: significant relation
test whether nullhypothesis can be rejected

p-Value

null hypothesis significance test
most common in scientific publications
highly debated, often misused

Normal distribution

symmetric values
infinite measurements
normal: 68%
total surface = total probability = 1 = 100%

t-distribution

similar to normal distribution
for small populations

Binomial distribution

Probability P that event with probability p is observed
in k out of n observations

\[ P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}, \quad \binom{n}{k}=\frac{n!}{k!(n-k)!} \]

Multiple choice test

10 questions with 4 answers each

Normal to binomial

What is the probability to find 30 out of 100 man having 150-155 cm?

size of men follows normal distribution
range is equivalent to probability
binomial distribution answers it

From observation to significance

throw die 24x
6 x "3"
is the die biased towards "3"?
null-hypothesis: it is not
=> underlying 'true' population $\le 1/6$
determine total probability of cases $\ge 1/6$
binomial test: p = 0.1995
p > 0.05 => no rejection of null hypothesis, no bias

Significance

z-test: normal distribution with known variance
t-test: normal distribution with unknown variance

Significance

Select significance level, generally $ \alpha = 0.05$ or 0.01
Perform e.g. t-test (returns p-value)
$p < \alpha:$ reject Nullhypothesis $\Rightarrow$ significant
$p \geq \alpha:$ Nullhypothesis not rejected $\Rightarrow$ insignificant

Significance level

The maximum percentage of falsely rejected cases (false positives)
For normal distribution the surface that represents the percentage at tail
Non-normal: $\alpha$-quantile

Student t-test

compare two distributions
both with limited number of values

Are Man larger than Women?

Two sample test of independent variables
Nullhypothesis: there are no significant differences in the mean values

Dependencies

Two or more variables

Pearson correlation
Spearman rank correlation

Correlation is no causality!

Correlations

Pearson correlation

\[ r_{xy} = \frac{ \sum_{i=1}^n (x_i - \bar{x}) (y_i - \bar{y}) }{\sigma_x\sigma_y} \]

Distributions

Normal
t-Distribution
Bernoulli
Binomial
Poisson
Exponential
Logarithmic

Crowd wisdom

Bean distribution

Outliers

Example: income of all citizens

many people with low to moderate income
a few exceedingly rich people
resulting issues:

mean value is bad descriptor
not possible to draw

Logaritm

Inverse of exponential function
Can show both small values and very large

Logaritmic scale

In x or y or both
Can show both small values and very large

Beggars and billionaires

Confusion matrix

Measures of trust

Receiver operating characteristics

Biases

framing
priming
rounding
social pressure

Framing

Wording has significant influence:

'giving 16 to 17 yrs old the right to vote': 52%+, 41%-
'reducing the voting age to 16': 37%+, 56%-

Priming

Answers depend on previous questions

10% of young people feel lonely
BBC, after long list of questions: 42%

Rounding

In surveys people tend to use round numbers

Psychology of numbers

98% survival versus
2% death rate

Which sounds better?

Biontech: 20% of german economical growth
Biontech: 5 permille of german GNP

both are equivalent (2.5% growth rate)

Correlation is not causation

99% divorce rate in Maine and per capita consumption of margarine
95% marriage rate in Kentucky and people drowning after falling out of a fisher boat

https://www.tylervigen.com

Reverse causation

"A nearby Waitrose adds £36,000 to house price"
Moderate drinkers live longer than non-drinkers

Lurking factors

Being a pope helps living longer?
Do right handers live longer?

Discrimination or adjustment?

Regression to the mean

Systems aim to return to their mean

Soccer: new trainer, return to normal

Speed cameras after accidents

Placebo

Some may be healed by the belief in something
Some are healed by normal function of the body (return to mean)

Pitfalls

"Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns. There are things we know we know. We also know there are known unknowns. That is to say, we know there are some things we do not know. But there are also unknown unknowns — the ones we don't know we don't know,"

Donald Rumsfeld, 2002