Lecture4-Note

Logic and Probability For Statistics

Descriptive Statistics

What is a Mean (M) ?
- Measurement of central tendency
- Mathematical midpoint (average) of a data set

Standard Deviation (SD)

SD = \sqrt\frac{\sum(X -\overline{X})^2}{N}

Measurement of variability
How variable is the data

How close to the mean is a given value

   Target of descriptive statistics:
   Descriptive statistics can be run on the entire target and take a sample is the usually method.
   Use the sample M and SD of an enough sample size to represent the entire target's statistics.

NHST (Null hypothesis significance testing)
A systematic procedure for deciding whether the outcome of a study (results from a sample) support a particular theory (which is thought to apply to a population)

If you want to draw a conclusion about a population, your sample better be from that population.
Probability
- Definition: The expected relative frequency of a particular outcome
  - Relative frequency - # of times something happens relative to # of times it could have, like 4/10
  - Expected relative frequency (probability)
- Probability = $p$ = $\frac{X\,ways\,to\,get\,an outcome}{Y\,possible\,outcomes}$ = expected relative frequency
- proportion: 0 - 1
NHST uses deductive reasoning
- Deductive Reasoning: From general to specific - Theory to hypotheses
- Inductive Reasoning: From specific to general - Observations to theories
- P<0.05
```
   We have to find one false case to disprove
```
Normal Curve
- Unimodal, symmetrical, bell-shaped curve
- Mathematical (or theoretical) distribution
- 34-14-2
- p<0.05 (below -1.96 & above 1.96)
Z-Scores (标准化)
A Z-score is the number of standard deviations the score is above or below the mean.
- Standardization: Process of converting raw scores into Z-scores
- Formula: $\frac{X-\overline{X}}{SD}$
Sampling Distribution
```
One individual -> distribution of individual scores
Mean -> distribution of means
```
- Standard Error (SE)
  Measurement of variability of samples (Mean of means)
  Z-score computing formula:
$Z = \frac{(M - \mu_m)}{\sigma_m}$

$\mu_m$ = mean of the distribution of means
$\sigma_m$ = standard deviation of the distribution of means aka “standard error (SE)”

Inferential Statistics

Practical Significance
- Statistical significance does not guarantee the result is meaningful
- Strong significance (p<0.001) means a reliable effect, not necessarily a strong effect
Effect Size
An effect size is a measure of the strength of the relationship between two variables. Most commonly reported effect size is Cohen’s $d$

Small effect Medium effect Large effect

$d$ 0.2 0.5 >0.8

	Small effect	Medium effect	Large effect
$d$	0.2	0.5	>0.8

Types of statistical tests

	Continuous predictor (e.g, how much)	Categorical predictor (e.g, exp vs control)
Continuous outcome(e.g, how much)	Correlation or regression	T-test or ANOVA
Categorical outcome (e.g, yes/no)	logistical regression	chi-square test or loglinear

t-tests
```
Simple test for comparing groups
independent samples t-test vs. paired samples t-test
```
e.g, t(18) = 19.06, p = 0.0001 “18” is degrees of freedom, “19.05” is t-value, “0.0001” is significance
- Independent Samples t-test
  - Compares two independent groups
  - One example:
- Paired-Samples t-test
  - Compares one (the same) group that has been tested twice
  - Before and after; under two different conditions
  - One example:
Analysis of Variance (ANOVA)
```
Used when experiment has a more complex design.
(variables with more than two levels, multiple predictor variables)
```
- One-way ANOVA
  - Used when the predictor variable has more than two levels
  - One example:
- Repeated measures ANOVA
  - Used when group has been tested more than twice
  - One example:
- Factorial ANOVA
  - Used for multiple predictor variables (one of more can be repeated measures)
  - Each variable has a main effect
  - Variables can have interactions effects
Chi-Square
- Used to analyze categorical (count or proportion) data
Correlation coefficient
- Used to represent the relationship between two continuous variables.