Univariate Data (One Variable)

Analysing

Overall Pattern
- Shape
  - Skewness
    - Left Skewed: Mean < Median < Mode
    - Symmetrical: Mean = Median = Mode
    - Right Skewed; Mead > Median > Mode
  - Peaks
- Centre
  - Mean, Median, Mode
- Spread
  - IQR
  - Standard Deviation
  - Range (Biggest value - smallest value)
Deviations
- Outliers
  - Defined as more than $1.5 \times$ IQR outside the IQR (e.g. $Q_{3} + 1.5 \times$ IQR)

A cross in the middle of the box is the mean

The five numbers are: Smallest, Q1, Q2, Q3, Largest

Variables that have a deterministic relationship
- Given the value of one variable, the exact, unique value of the other variable can be calculated
- e.g. Relationship between Fahrenheit and Celsius
Statistical or non-deterministically related variables
- Given the value of one variable, the average value of the other variable can be calculated

Overall Pattern
- Direction
  - Positive/negative relationship or no relationship (Correlation Coefficient r)
- Form
  - Linear, non-linear, quadratic, exponential
- Strength
  - Strong, weak, moderate (Correlation Coefficient r)
Deviations from the pattern
- Outliers
  - No common way to identify such data points

To predict A using B, we need to use the equation of the regression line of A against B
- i.e. can only predict the y axis value from the x axis value
The input value to predict must be within range of the dataset for the predicted value to be valid

The measure of the linear association between two variables ( $- 1 < r < 1$ )
The sign of $r$ is the direction of association
- $r > 0$ means positive association, so when one variable increases, the other will tend to increase
The magnitude of $r$ is the strength of the association
- $r = - 1$ or $r = 1$ : perfect [+ve/-ve] association
- $- 1 < r < - 0.7$ or $0.7 < r < 1$ : Strong [+ve/-ve] association
- $- 0.7 < r < - 0.3$ or $0.3 < r < 0.7$ : Moderate [+ve/-ve] association
- $- 0.3 < r < - 0$ or $0 < r < 0.3$ : Weak [+ve/-ve] association
- $r = 0$ : No association
To compute
- $r = \frac{1}{n - 1} \sum_{i = 1}^{n} \frac{( x _{i} - x ˉ )}{S _{x}} \frac{( y _{i} - y ˉ )}{S _{y}}$
- Where $S_{x}$ is the standard deviation of $x$ and $\overset{x}{ˉ}$ is the mean of $x$
- Note: $\frac{( x _{i} - x ˉ )}{S _{x}}$ is the standard unit for the $i$ th $x$
- $r = \frac{S _{x}}{S _{y}} m$ , where $m$ is the gradient of the regression line
Properties
- Not affected by:
  - Interchange of axes
  - Addition of numbers
  - Multiplication of +ve numbers

Computed based on aggregates (groups of individuals) rather than on individuals

Ecological Fallacy
- Trends observed at aggregate level → Inferences at individual level
Atomistic Fallacy
- Trends observed at individual level → Inferences at aggregate level