Key Formulas & Equations
Core Concepts
Descriptive vs Inferential Statistics
Descriptive statistics summarize data through measures of center (mean, median, mode) and spread (range, variance, standard deviation). Inferential statistics use sample data to make generalizations about populations through hypothesis testing and confidence intervals.
Normal Distribution
The bell-shaped curve where data is symmetrically distributed around the mean. ~68% of data falls within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ. The Central Limit Theorem guarantees normality of sample means for large n.
Hypothesis Testing
A formal procedure for evaluating claims. Set up H₀ (null) and H₁ (alternative). Choose α (significance level). Compute test statistic. If p-value < α, reject H₀. Type I error: rejecting true H₀; Type II error: failing to reject false H₀.
Correlation vs Causation
Correlation measures the strength of a linear relationship (-1 to +1). It does NOT imply causation. A third confounding variable may explain the relationship. Only controlled experiments can establish causation.
Sampling Methods
Random sampling gives every member an equal chance, reducing bias. Stratified sampling divides the population into subgroups. Convenience sampling is easy but biased. The larger and more representative the sample, the more reliable the inference.
Probability Distributions
Mathematical functions describing the likelihood of different outcomes. Discrete: Binomial, Poisson. Continuous: Normal, t-distribution, Chi-square, F-distribution. Each has specific use cases based on data type and assumptions.
Worked Examples
Example 1: Z-Score
Problem: Test scores: μ = 75, σ = 10. What is the z-score for a score of 92?
Step 1: z = (x - μ) / σ
Step 2: z = (92 - 75) / 10 = 17/10
Step 3: z = 1.70
Answer: A score of 92 is 1.70 standard deviations above the mean.
Example 2: Confidence Interval
Problem: Sample of n=36, x̄=50, s=8. Find 95% CI for the population mean.
Step 1: Standard error: SE = s/√n = 8/√36 = 8/6 = 1.33
Step 2: For 95% CI, z* = 1.96
Step 3: Margin of error: ME = 1.96 × 1.33 = 2.61
Step 4: CI = 50 ± 2.61 = (47.39, 52.61)
Answer: We are 95% confident the true mean is between 47.39 and 52.61.
Example 3: Regression
Problem: Given data points (1,2), (2,4), (3,5), find the regression line ŷ = a + bx.
Step 1: x̄ = 2, ȳ = 11/3 ≈ 3.67
Step 2: b = Σ(xᵢ-x̄)(yᵢ-ȳ) / Σ(xᵢ-x̄)² = [(1-2)(2-3.67)+(2-2)(4-3.67)+(3-2)(5-3.67)] / [1+0+1] = [1.67+0+1.33]/2 = 3/2 = 1.5
Step 3: a = ȳ - bx̄ = 3.67 - 1.5(2) = 0.67
Answer: ŷ = 0.67 + 1.5x
Example 4: Binomial Probability
Problem: A fair coin is flipped 10 times. What is P(exactly 6 heads)?
Step 1: n=10, k=6, p=0.5
Step 2: P(X=6) = C(10,6)·(0.5)⁶·(0.5)⁴
Step 3: C(10,6) = 210
Step 4: P = 210 × (0.5)¹⁰ = 210 × 0.0009766 = 0.205
Answer: The probability is approximately 0.205 (20.5%).
Practice Problems
Problem 1
Find the mean and standard deviation of: 2, 4, 6, 8, 10.
Answer: Mean = 6. Variance = 40/4 = 10. SD = √10 ≈ 3.16
Problem 2
A poll of 400 voters shows 220 support Candidate A. Find the 95% CI for support proportion.
Answer: p̂ = 0.55, SE = √(0.55×0.45/400) = 0.0249. CI = 0.55 ± 1.96(0.0249) = (0.501, 0.599)
Problem 3
Test H₀: μ=100 vs H₁: μ≠100 with n=16, x̄=98, s=5. Use α=0.05.
Answer: t = (98-100)/(5/4) = -1.6. Critical t (df=15, α=0.05 two-tailed) = ±2.131. Since |−1.6| < 2.131, fail to reject H₀.
Problem 4
If P(A) = 0.3, P(B) = 0.4, and P(A∩B) = 0.12, are A and B independent?
Answer: Check: P(A)·P(B) = 0.3×0.4 = 0.12 = P(A∩B). Yes, independent.
Problem 5
Calculate the correlation coefficient for: (1,1), (2,3), (3,4), (4,6).
Answer: r = 0.99 (strong positive linear relationship)
Problem 6
In a normal distribution (μ=200, σ=25), what percentage of values lie between 175 and 225?
Answer: 175 and 225 are each 1σ from the mean → ~68% of values