Measures of Central Tendency

1. Mean (Arithmetic Average)

• Data is symmetrically distributed (no significant skew)
• No extreme outliers present
• Continuous or discrete numerical data
• When all data points should contribute to the measure (e.g., regression analysis)

2. Median

1. Sort data in ascending order
2. Find the middle value (or average of two middle values for even n)

• Data is skewed (e.g., income distributions)
• Outliers are present (the median is resistant to them)
• Ordinal data (ranked categories)

3. Mode

• A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes)
• A dataset with no repeating values has no mode

• Categorical/nominal data (e.g., most popular product color)
• Identifying the most common occurrence
• When you need a value that actually exists in the dataset

Comparison Table

Empirical Relationship (Karl Pearson)

Which Measure to Choose?

Other Measures of Central Tendency

• Weighted Mean - assigns different weights to values based on importance: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$
• Geometric Mean - used for rates of change and ratios: $GM = (x_1 \cdot x_2 \cdots x_n)^{1/n}$
• Harmonic Mean - useful for averaging rates/speeds: $HM = \frac{n}{\sum (1/x_i)}$
• Trimmed Mean - removes a percentage of extreme values before calculating the mean, balancing robustness and efficiency

Tests of Significance

Core Concepts (Foundation)

Null Hypothesis (H₀) vs. Alternative Hypothesis (H₁)

Significance Level (α)

• The threshold probability for rejecting H₀
• Commonly set at α = 0.05 (5%) or α = 0.01 (1%)
• Represents the acceptable risk of a Type I error

p-value

• The probability of obtaining the observed result (or more extreme) if H₀ were true
• p < α → Reject H₀ (result is statistically significant)
• p ≥ α → Fail to reject H₀ (insufficient evidence)

Important: A small p-value does NOT prove H₁ is true - it only means the data is unlikely under H₀.

Types of Errors

One-tailed vs. Two-tailed Tests

• Two-tailed: Tests for difference in either direction (H₁: μ₁ ≠ μ₂) - most common
• One-tailed: Tests for difference in a specific direction (H₁: μ₁ > μ₂ or μ₁ < μ₂)

Major Tests of Significance

1. Z-Test

• Population standard deviation (σ) is known
• Sample size n ≥ 30 (large sample)
• Data is normally distributed

• One-sample Z-test: Compare sample mean to population mean
• Two-sample Z-test: Compare means of two independent large samples
• Z-test for proportions: Compare a sample proportion to a known population proportion

2. t-Test (Student's t-Test)

• Population SD is unknown
• Small sample (n < 30), though valid for any n
• Data approximately normally distributed

3. Chi-Square Test (χ²)

• Data must be in frequency/count form
• Expected frequency in each cell ≥ 5
• Observations are independent

4. F-Test / ANOVA (Analysis of Variance)

5. Correlation Tests

6. Non-Parametric Tests

Quick Selection Guide

What type of data?
├── Categorical → Chi-square test (or Fisher's exact)
└── Numerical
    ├── Comparing to a known population mean?
    │   ├── σ known, n ≥ 30 → Z-test
    │   └── σ unknown or n < 30 → One-sample t-test
    ├── Comparing TWO groups?
    │   ├── Independent groups → Independent t-test (or Mann-Whitney)
    │   └── Same group, two time points → Paired t-test (or Wilcoxon)
    └── Comparing THREE or more groups?
        └── ANOVA (or Kruskal-Wallis if non-parametric)

Summary Table

Steps in Performing a Test of Significance

1. State the hypotheses - H₀ and H₁
2. Set the significance level (α = 0.05 or 0.01)
3. Choose the appropriate test (based on data type, sample size, number of groups)
4. Calculate the test statistic (z, t, χ², F)
5. Find the p-value (or compare test statistic to critical value)
6. Make a decision - Reject or fail to reject H₀
7. State the conclusion in context