Data based about Statistical-Reasoning

• Mean (arithmetic mean)
  • $\bar{x} = \frac{\sum x}{n}$
  • Outliers pull mean toward one end
• Median
  • Midpoint of data set
  • Position: $(n + 1) / 2$
  • Least susceptible to outliers
• Mode
  • Value appearing most often

• Normal distribution
  • 68%, 95%, 99% fall within 1, 2, 3 SDs
  • Standard distribution: mean = 0, SD = 1
• Skewed distributions
  • Tail on one side
  • Negative skew: tail left
  • Positive skew: tail right
  • Left skew → mean < median
• Bimodal distributions
  • Two peaks
  • Don’t always need separate analysis

Range

• $x_{\max} - x_{\min}$
• Strongly influenced by outliers

Interquartile Range (IQR)

• To find Q1: $0.25n$
  • If whole number → average of that position and next
  • If decimal → round up
• To find Q3: $0.75n$
• Outlier rule: more than 1.5 × IQR below Q1 or above Q3

Example

• If $n = 8$:
  • $8 \times 0.25 = 2$→ find 2nd and 3rd values, average them

Standard Deviation

• $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$
• 3 SD from mean = outlier

Outliers may result from:

• True statistical anomaly
• Measurement error
• Data not approximating normal distribution

Independent events

• Don't affect each other (e.g., rolling a die)
• $P(A \text{ and } B) = P(A)P(B)$

At least one event

• $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Dependent events

• Events influence each other (drawing cards)

Mutually exclusive

• Cannot occur at the same time (heads vs tails)

Exhaustive

• No other possible outcomes
• Coin flip: heads or tails

Hypothesis testing

• Null hypothesis (H₀): states equivalence
• Alternative hypothesis (Hₐ): directional or nondirectional
• Z or t tests:
  • Compare test statistic to table → find p-value
  • If $$p < \alpha$$ → reject $H_0$
  • If $p > \alpha$ → fail to reject $H_0$

Errors

• Type I error (α): incorrectly rejecting H₀
• Type II error (β): incorrectly failing to reject H₀

Power

• Probability of correctly rejecting false H₀
• $\text{power} = 1 - \beta$

Confidence

• Probability of correctly failing to reject a true null

Confidence intervals

• Reverse of hypothesis testing
• Start with desired p-value (e.g., 95%) → find t/z score
• Multiply score by SD and add/subtract from mean

Example

• Mean = 30, SD = 3, z = 1.96
• Range: $30 \pm (3 \times 1.96) = 30 \pm 5.88$
• CI = 24.12 to 35.88

Charts

• Circle/pie charts: show raw values or %; become confusing with many variables
• Bar charts: categorical data; avoid scale breaks
• Histograms: numerical data; useful for finding mode

Box plot

• Shows range, median, IQR, outliers

Maps

• Show data geographically
• Can present multiple variables (e.g., location + density)

Graphs and axes

• Linear graphs: linear, parabolic, exponential, logarithmic
• Slope = $y/x$
• Semi-log: one axis logarithmic
• Log–log: both axes logarithmic

Tables

• Just read carefully
• Convert to rough graph/equation if needed

• Correlation: trend between variables
  • + if together, − if opposite
  • Correlation coefficient shows strength
• Correlation ≠ causation
• Need Hill’s criteria for causation
• Must relate findings to broader scientific knowledge