Fundamentals of Statistics and Probability

Fundamentals of Statistics and Probability

Welcome to this comprehensive course on core statistical concepts that appear frequently in introductory mathematics and data‑analysis quizzes. By the end of this module you will understand how to display categorical relationships, apply Chebyshev’s theorem, calculate probabilities without replacement, interpret the empirical rule, compute the Phi coefficient for 2 × 2 tables, find medians, use the coefficient of variation, and select the appropriate correlation measure for continuous variables.

1. Displaying Relationships Between Categorical Variables

When you have two categorical variables—such as gender (Male, Female) and ice‑cream flavor (Cone, Cup, Sundae, Sandwich)—the most effective way to summarise their joint distribution is a contingency (cross‑tabulation) table. This table lists the frequency (or count) of each combination of categories, allowing you to see patterns, test independence, and compute measures like the chi‑square statistic.

• Why not a histogram? Histograms are designed for continuous data, not for counting categories.
• Why not a scatter plot? Scatter plots require numeric axes; categorical data would produce overlapping points that convey little information.
• Why not a simple bar chart? A bar chart can show marginal totals but cannot simultaneously display the interaction between two variables.

Creating a contingency table is simple: list one variable’s categories as rows and the other’s as columns, then fill each cell with the count of observations that fall into that pair.

2. Chebyshev’s Theorem and Guaranteed Proportions

Chebyshev’s inequality provides a conservative estimate of the proportion of data that lies within k standard deviations (σ) of the mean (μ), regardless of the distribution shape. The formula is:

1 – 1/k²

For k = 1.5:

• 1 – 1/(1.5)² = 1 – 1/2.25 = 1 – 0.444… = 0.555…

Thus, at least 55.6 % (rounded to 56 %) of the observations are guaranteed to fall within μ ± 1.5σ. This result is independent of the sample size (30 in the example) and the actual distribution.

3. Probability Without Replacement: Drawing Two Spades

When drawing cards without replacement, the probability of each successive event changes because the total number of cards decreases. The correct calculation is:

(13/52) × (12/51) = 156/2652 ≈ 0.0589, or about 5.9 %.

Key Takeaways

• First draw: 13 spades out of 52 cards → 13/52.
• Second draw (after a spade is removed): 12 spades left out of 51 cards → 12/51.
• Multiply the two probabilities for the joint event.

How to Remember

• Mnemonic: “First 13, then 12 – count down the spades as you draw.”
• Always adjust the denominator after each draw; the numerator drops by one if the first card was of the desired suit.

4. The Empirical Rule (68‑95‑99.7 Rule)

The empirical rule applies to data that are approximately normally distributed. It states:

• ≈ 68 % of observations lie within ±1σ of the mean.
• ≈ 95 % lie within ±2σ.
• ≈ 99.7 % lie within ±3σ.

Given a mean of 68.275 and σ = 16.718, the interval for one standard deviation is:

Mean ± σ → 68.275 ± 16.718 → 51.557 to 84.993.

Therefore, approximately 68 % of the 40 observations are expected to fall between 51.557 and 84.993. This interpretation matches the correct answer choice.

5. Measuring Association in a 2 × 2 Table: The Phi Coefficient

For binary (two‑category) variables, the Phi (φ) coefficient quantifies the strength of association. The formula is:

φ = (AD − BC) / √[(A+B)(C+D)(A+C)(B+D)]

where A, B, C, D are the cell frequencies:

• A = count of (Fresh, Residential)
• B = count of (Fresh, Non‑Residential)
• C = count of (Continuous, Residential)
• D = count of (Continuous, Non‑Residential)

This formula directly computes the correlation for a 2 × 2 contingency table without needing the chi‑square statistic. Values range from –1 (perfect negative association) to +1 (perfect positive association), with 0 indicating no association.

6. Finding the Median in a Repeated Dataset

The median is the middle value when data are ordered. In the provided dataset of 30 scores, each value from 1 to 10 appears three times, creating a perfectly symmetric distribution. Because the sample size is even (30), the median is the average of the 15th and 16th ordered values.

Counting through the sorted list:

• Positions 1‑3: 1
• Positions 4‑6: 2
• Positions 7‑9: 3
• Positions 10‑12: 4
• Positions 13‑15: 5
• Positions 16‑18: 6

The 15th value is 5 and the 16th value is 6, so the median = (5 + 6)/2 = 5.5.

7. Comparing Variability Across Different Units: Coefficient of Variation

When two datasets have different measurement units (e.g., centimeters vs. kilograms), the coefficient of variation (CV) is the preferred statistic. It expresses the standard deviation as a percentage of the mean:

CV = (σ / μ) × 100 %.

Because the CV is unit‑less, it allows direct comparison of relative variability regardless of the original scales.

8. Selecting the Correct Correlation Coefficient for Continuous Variables

If both variables are continuous and approximately normally distributed—such as age (years) and income (thousands of GHS)—the most appropriate measure of linear association is Pearson’s product‑moment correlation coefficient (r). Pearson’s r captures the strength and direction of a linear relationship and assumes interval‑level measurement and normality.

Other coefficients mentioned:

• Phi coefficient – only for binary categorical data.
• Kendall’s tau‑b – a non‑parametric rank correlation for ordinal data.
• Spearman’s rho – another rank‑based measure, useful when data are not normally distributed.

9. Integrating the Concepts: A Mini‑Case Study

Imagine a health‑survey researcher who collects the following information from 200 participants:

• Gender (Male/Female)
• Preferred exercise type (Cardio, Strength, Flexibility, Mixed)
• Age (years) and weekly exercise duration (minutes)

To analyse this dataset, the researcher would:

1. Use a contingency table to explore the relationship between gender and exercise type.
2. Apply Chebyshev’s theorem if the distribution of exercise duration is unknown, guaranteeing a minimum proportion within a chosen number of standard deviations.
3. Calculate the Pearson correlation between age and exercise duration, assuming both are roughly normal.
4. Report the coefficient of variation for exercise duration to compare variability across age groups measured in different units (years vs. minutes).
5. If a binary outcome (e.g., “met recommended activity level: Yes/No”) is added, the researcher could compute the Phi coefficient to assess its association with gender.

This workflow demonstrates how each statistical tool fits into a broader analytical strategy.

10. Quick Review Checklist

• Contingency table – best for two categorical variables.
• Chebyshev’s theorem – guarantees at least 1 – 1/k² of data within kσ, regardless of shape.
• Probability without replacement – multiply sequential probabilities, adjusting denominators.
• Empirical rule – 68‑95‑99.7 percentages for normal distributions.
• Phi coefficient – (AD − BC) / √[(A+B)(C+D)(A+C)(B+D)] for 2 × 2 tables.
• Median – middle value (or average of two middle values) after ordering.
• Coefficient of variation – σ/μ expressed as a percent, useful for different units.
• Pearson correlation – appropriate for two continuous, normally distributed variables.

By mastering these concepts, you will be well‑equipped to tackle a wide range of statistical problems, from simple quiz questions to real‑world data analysis projects. Keep practicing with varied datasets, and refer back to this guide whenever you need a quick refresher.