Descriptive Data Analysis

Resources

• Slides
• Exercise Slides

bro just explained data visualization stuff so not renoting it here, slides should be self-explaining

Levels of Measurement

• Nominal: Categories without order
• Ordinal: Categories with order
• Interval: Numeric scales with equal intervals, no true zero
• Ratio: Numeric scales with equal intervals and a true zero

Variable Association

Apparently, No need to learn these formulas by heart

These can also be used to identify outliers or determine if a value is typical for a distribution. Spearman is more robust to outliers than Pearson.

Pearson: Product-Moment Correlation

At least interval data.
[-1 ≤ r ≤ 1] for perfect negative to perfect positive linear relationship

$$r=\frac{{\sum }_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{{\sum }_{i=1}^n(x_i-\bar{x}{)}^2}\sqrt{{\sum }_{i=1}^n(y_i-\bar{y}{)}^2}}$$

where n is the number of cases

This is used to determine the linear slope of correlation lines for e.g. scatter plots

Spearman: Rank Correlation

At least ordinal data.
[-1 ≤ ρ ≤ 1] for perfect negative to perfect positive monotonic relationship

$$\rho =1-\frac{6\sum d_i^2}{n^3-n}$$

where d_i is the difference between ranks of each observation on the two variables, n is the number of cases

Cramér’s V

For when both variables are nominal (categorical).
[0 ≤ V ≤ 1] for no association to perfect association

$$V=\sqrt{\frac{{\chi }^2}{n\ast \min (k-1,r-1)}}$$

Cramér’s V can be used to determine the effect size of a Chi-Square Test.

Interpretation

Interpreting Cramér’s V depends on the degrees of freedom (df), which is calculated as $df=min(k-1,r-1)$, where k is the number of categories in one variable and r is the number of categories in the other variable.

df	Small Effect	Medium Effect	Large Effect
1	0.10	0.30	0.50
2	0.07	0.21	0.35
3	0.06	0.17	0.29
4	0.05	0.15	0.26
5	0.04	0.13	0.23