MAT 141 – Statistics Final Cheat Sheet (Front & Back Summary) ----------------------- 1. INTRODUCTION TO STATISTICS ----------------------- • Statistics:
Science of collecting, organizing, summarizing, analyzing data to draw conclusions. • Population = entire group; Sample = subset of population. •
Parameter = numerical summary of population; Statistic = numerical summary of sample. Types of Variables: - Qualitative (categorical): labels or names. -
Quantitative: numeric values (Discrete: countable; Continuous: measurable). Levels of Measurement: Nominal, Ordinal, Interval, Ratio. -----------------------
2. SAMPLING METHODS ----------------------- • Simple Random Sample (SRS): every sample equally likely. • Stratified: divide into strata, sample from
each. • Cluster: select random clusters, use all individuals in clusters. • Systematic: every kth individual. • Convenience: easily available (biased). •
Voluntary Response: participants self-select (biased). ----------------------- 3. STUDIES ----------------------- • Observational: no manipulation; shows
association only. • Experiment: manipulate variable; can show causation. • Lurking Variable: not accounted for but influences results. • Confounding:
effects of two variables mixed. ----------------------- 4. DESCRIPTIVE STATISTICS ----------------------- Measures of Center: - Mean (µ or x■) = Σx / n -
Median = middle value - Mode = most frequent value Measures of Spread: - Range = max – min - Variance: s² = Σ(x–x■)² / (n–1) - Std. Dev.: s =
√variance - IQR = Q3 – Q1 - Empirical Rule (bell shape): 68%, 95%, 99.7% - Chebyshev: ≥ (1 – 1/k²) within k std. dev. Z-Score: z = (x – µ)/σ → measures
distance from mean. Outlier test: < Q1 – 1.5·IQR or > Q3 + 1.5·IQR ----------------------- 5. GRAPHICAL DISPLAYS ----------------------- • Qualitative: bar
graph, Pareto chart, pie chart. • Quantitative: histogram, stem-leaf, boxplot, dot plot. • Boxplot shows median, quartiles, outliers (5-number summary).
----------------------- 6. PROBABILITY BASICS ----------------------- • Probability (P): between 0 and 1. • Law of Large Numbers: as n ↑, observed proportion →
true probability. • Classical Probability: P(E) = #successes / #outcomes. • Empirical Probability: P(E) ≈ freq(E)/total trials. • Complement: P(E■) = 1 – P(E).
Addition Rules: - Disjoint: P(E or F) = P(E) + P(F). - General: P(E or F) = P(E) + P(F) – P(E and F). Multiplication Rules: - Independent: P(E and F) =
P(E)·P(F). - General: P(E and F) = P(E)·P(F|E). Conditional: P(F|E) = P(E and F)/P(E). Independence test: P(F|E)=P(F). Counting: - n! = n×(n–1)...1 -
Permutation: nPr = n!/(n–r)! - Combination: nCr = n!/[r!(n–r)!] ----------------------- 7. RANDOM VARIABLES ----------------------- • Random Variable: numeric
value of outcomes. • Discrete RV: countable (e.g. # of successes). • Continuous RV: measurable (time, weight). Discrete Mean: µx = Σ[x·P(x)] Discrete
SD: σx = √Σ[(x–µx)²·P(x)] Expected Value = long-run average = Σ[x·P(x)] ----------------------- 8. BINOMIAL DISTRIBUTION ----------------------- Conditions: 1.
Fixed number of trials (n). 2. Independent trials. 3. Two outcomes: success/failure. 4. Constant probability (p). Formulas: P(X=x) = nCx · p^x · (1–p)^(n–x)
Mean = µ = n·p Std Dev = σ = √(n·p·(1–p)) ----------------------- 9. POISSON DISTRIBUTION ----------------------- • For # of occurrences in time/space with
mean rate λ. P(X=x) = (e^(–λ)·λ^x) / x! Mean = λ, SD = √λ ----------------------- 10. NORMAL DISTRIBUTION ----------------------- • Bell-shaped, symmetric
about µ. • z = (x – µ)/σ • Standard Normal: µ = 0, σ = 1 Common areas: P(z < 1) = 0.8413 P(z < –1) = 0.1587 P(–1 < z < 1) = 0.6826 P(–2 < z < 2) =
0.9544 P(–3 < z < 3) = 0.9973 Percentiles: Given z, x = µ + zσ Given x, z = (x–µ)/σ ----------------------- 11. CORRELATION & REGRESSION
----------------------- • Scatterplot shows direction/form/strength. • Correlation (r): –1 ≤ r ≤ 1 - r>0 → positive, r<0 → negative, |r|≈1 → strong. • r =
Σ[(x–x■)(y–■)] / [(n–1)sx·sy] Least Squares Regression Line: ■ = b■x + b■ b■ = r(sy/sx), b■ = ■ – b■x■ Interpret b■: change in y per unit x. Residual =
y – ■ R² = proportion of variation explained by x. ----------------------- 12. EMPIRICAL APPLICATIONS ----------------------- • Use z-tables for probabilities &
percentiles. • Check normality with histogram or normal probability plot. • Outliers affect mean, SD, correlation, regression. ----------------------- TIP
SUMMARY ----------------------- – Use mean & SD for normal data; median & IQR for skewed. – Check independence before using binomial/multiplication
rules. – Always label axes & include units on graphs. – For sampling distributions, larger n → smaller variability. – Never extrapolate far beyond observed
data in regression.