3.4 Basic Statistics, Probability, and Data Interpretation

Measures of Central Tendency and Position

Descriptive statistics involves summarizing, organizing, and interpreting quantitative data sets. In educational assessment, teachers regularly analyze student test scores using central tendency and dispersion metrics.

Measures of Central Tendency

1. Mean (Arithmetic Average): The sum of all values divided by the total number of observations. $\bar{x} = \frac{\sum x_i}{N}$
   • Weighted Mean: Used when data values carry different weights (e.g., GPA or weighted grades). $\bar{x}_w = \frac{\sum (w_i \cdot x_i)}{\sum w_i}$
2. Median: The middle value when data is arranged in ascending or descending order.
   • If $N$ is odd, Median = Middle element at position $\frac{N+1}{2}$.
   • If $N$ is even, Median = Average of two middle elements at positions $\frac{N}{2}$ and $\frac{N}{2} + 1$.
   • Key Advantage: The median is resistant to extreme outliers or skewed data.
3. Mode: The value that appears most frequently in a data set. A set can be unimodal, bimodal, multimodal, or have no mode.

Worked Example 1: Calculating Weighted Mean Grade

Problem: A student received the following grades with corresponding course unit weights:

What is the student's final weighted average grade?

Step 1: Compute weighted terms $(w_i \cdot x_i)$.

• Quizzes: $85 \times 0.20 = 17.0$
• Midterm: $88 \times 0.30 = 26.4$
• Final: $92 \times 0.40 = 36.8$
• Project: $90 \times 0.10 = 9.0$

Step 2: Sum the weighted products and divide by total weight ($1.00$).
$\bar{x}_w = 17.0 + 26.4 + 36.8 + 9.0 = 89.2$

Final Answer: $89.2$

Measures of Variability and Standard Scores

• Range: Difference between highest and lowest values: $\text{Range} = X_{\text{max}} - X_{\text{min}}$.
• Variance ($\sigma^2$): The average of squared deviations from the mean.
• Standard Deviation ($\sigma$): The square root of variance, measuring average spread around the mean. $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$
• Z-Score (Standard Score): Indicates how many standard deviations an observation lies above or below the mean. $z = \frac{x - \mu}{\sigma}$

Empirical Rule (68-95-99.7 Rule for Normal Distributions)

• $\approx 68%$ of data falls within $\mu \pm 1\sigma$
• $\approx 95%$ of data falls within $\mu \pm 2\sigma$
• $\approx 99.7%$ of data falls within $\mu \pm 3\sigma$

Combinatorics and Probability Theory

Probability measures the likelihood of an event occurring, bounded on the closed interval $[0, 1]$ (or $0%$ to $100%$).

$\text{Probability of Event } E: P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes in sample space}}$

Fundamental Counting Principle, Permutations, and Combinations

• Fundamental Counting Principle: If task 1 can be done in $m$ ways and task 2 in $n$ ways, both can be performed together in $m \times n$ ways.
• Permutation: An arrangement of items where ORDER MATTERS. $P(n, r) = \frac{n!}{(n - r)!}$
  • Circular Permutations: $(n - 1)!$
  • Distinguishable Permutations: $\frac{n!}{n_1! n_2! \dots n_k!}$
• Combination: A selection of items where ORDER DOES NOT MATTER. $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}$

Worked Example 2: Permutations vs. Combinations

Problem A (Permutation): In how many ways can a President, Vice-President, and Secretary be selected from a committee of 8 teachers?

Solution: Order matters because position titles differ.
$P(8, 3) = \frac{8!}{(8 - 3)!} = \frac{8!}{5!} = 8 \times 7 \times 6 = 336\text{ ways}$

Problem B (Combination): In how many ways can a 3-member committee be chosen from 8 teachers?

Solution: Order does not matter.
$C(8, 3) = \frac{8!}{3!(8 - 3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\text{ ways}$

Compound Probability Rules

1. Addition Rule (Mutually Exclusive vs. Non-Mutually Exclusive):
   • Mutually Exclusive ($A \cap B = \emptyset$): $P(A \cup B) = P(A) + P(B)$
   • Non-Mutually Exclusive: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
2. Multiplication Rule (Independent vs. Dependent Events):
   • Independent Events: $P(A \cap B) = P(A) \times P(B)$
   • Dependent Events: $P(A \cap B) = P(A) \times P(B | A)$

Worked Example 3: Drawing Cards (Probability)

Problem: A standard deck of 52 playing cards is shuffled. What is the probability of drawing either a King or a Spade?

Step 1: Identify individual probabilities.

• Number of Kings $n(K) = 4 \implies P(K) = \frac{4}{52}$
• Number of Spades $n(S) = 13 \implies P(S) = \frac{13}{52}$
• Overlap (King of Spades) $n(K \cap S) = 1 \implies P(K \cap S) = \frac{1}{52}$

Step 2: Apply Non-Mutually Exclusive Addition Rule.
$P(K \cup S) = P(K) + P(S) - P(K \cap S) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$

Final Answer: $\frac{4}{13} \approx 30.77%$