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Q: What are the measures of central tendency for ungrouped data? โ Mean (Arithmetic mean) โ Median โ Mode โ Weighted mean Q: Define ungrouped data. โ Data that gives information on each member of the population/sample individually โ Not summarized in a frequency distribution table Mean Q: How do you calculate the sample mean? โ Q: How do you calculate the population mean? โ Q: Properties of the mean โ Unique and simple to compute โ Sensitive to extreme values (outliers) โ The sample mean varies; the population mean is fixed Q: Example โ Sample mean for July temperatures โ Data: 30.8, 31.7, 30.1, 31.6, 32.1, 27.6, 25. โ Mean = 29.
Q: Example โ Lead concentrations โ Data: 5.40, 1.10, 0.42, 0.73, 0.48, 1. โ Mean = (sum of values) / 6 Median Q: How is the median defined and calculated? โ Median is the middle value when data is sorted โ Odd n: value โ Even n: Mean of the values Q: Median properties โ Not affected by extreme values โ Simple to compute โ Unique Q: Example โ Salaries โ 13 values, median = 7th value = 152 Q: Example โ Employee ages (N=10) โ Sorted data: 27, 32, 39, 40, 44, 49, 53, 57, 60, 61 โ Median = (44 + 49)/2 = 46. Mode Q: What is the mode? โ Most frequent value(s)
Q: Why is variation important? โ Two data sets may have the same mean but different spreads โ Measures of centre alone do not describe data adequately Range Q: How is range calculated? โ R=MaxโMinR = \text{Max} - \text{Min}R=MaxโMin Q: Properties โ Simple to calculate โ Affected by outliers (nonresistant) Q: Example โ Data: -9 to 100 โ Range = 109 Variance and Standard Deviation Q: Formulas โ Sample variance: โ Sample standard deviation.: โ Population variance: ฯ2 = โ(xiโฮผ) โ Population std. dev.: ฯ = Q: Properties โ Always non-negative โ Sensitive to outliers
โ Measures average squared deviation (variance) or root (SD) Q: Example โ DVD prices: Linear Transformation of Mean & SD Q: What happens if we apply y = ax + b? โ โ Coefficient of Variation (CV) Q: What is the CV? โ CV = โ โ Unit-free measure of relative variability Q: Example โ Milk consumption โ USA: CV = 37.5% โ Canada: CV = 33.3% โ Conclusion: USA more variable Q: What is the Empirical Rule (bell-shaped data)? โ 68% within 1 SD โ 95% within 2 SD โ 99.7% within 3 SD
Percentiles Q: How to calculate the k-th percentile (Pk)? โ L = โ If L is whole โ average of L-th and (L+1)-th โ Else, round up L โ Pk = L-th value Q: Percentile rank โ Percentile Rank =
Box-and-Whisker Plot
Q: What is a five-number summary? โ Min, Q1, Median (Q2), Q3, Max Q: What does a box plot show? โ Centre (median), spread (IQR), skewness, outliers Q: How to identify outliers? โ Below: xQ3+1.5รIQRx > Q3 + 1.5 \times IQRx>Q3+1.5รIQR Q: Example โ Household incomes โ Q1 = 77, Q3 = 101, IQR = 24 โ Lower fence = 41, Upper fence = 137 โ Outlier: 144 (above upper fence)
Q1: What is an experiment in probability? โ A process resulting in one and only one outcome โ Can be repeated under similar conditions Q2: What are outcomes and sample space? โ Outcomes: Results of an experiment โ Sample Space (S): Set of all possible outcomes Q3: Define an event. โ A set of one or more outcomes โ Simple event: Exactly one outcome โ Compound event: Two or more outcomes Q4: How many total events can exist in a sample space with n outcomes? โ 2โฟ events Q5: What are examples of experiments and sample spaces? โ Babyโs sex: S = {boy, girl} โ Coin toss: S = {H, T} โ 3 coin tosses: S = {HHH, HHT, ..., TTT} (8 outcomes) โ Child's height: S = all real numbers Q6: What is the tree diagram used for? โ Visualizing possible outcomes โ Example: First two studentsโ genders: branches of {M, F} ร {M, F}
Q12: Classical probability formulas: โ Simple event: P(E) = 1 / total outcomes โ Compound event: P(E) = favorable outcomes / total outcomes Q13: Classical probability examples: โ Coin toss: P(H) = 0. โ Die roll: P(odd) = 3/6 = 0. โ Deck of cards: P(heart) = 13/52 = 0. โ 3 children, exactly 2 boys: Use combinations Q14: Relative frequency probability: โ P(A) = frequency of A / total trials โ Used when outcomes are not equally likely Q15: Relative frequency examples: โ Auto factory: 10 lemons in 500 cars โ P(lemon) = 10/500 = 0. โ Smoking: 52 of 1038 adults โ P(not harmful) = 52/ Q16: Law of Large Numbers โ Repeated trials โ relative frequency โ actual probability โ Example: Coin tosses, car defect rates Q17: Subjective probability examples: โ Steven earning an A โ Jets winning Stanley Cup โ Based on judgment, not rules
Q18: What are key event relationships in set theory? โ Complement (ฤ): Outcomes not in A โ Intersection (A โฉ B): Outcomes in both A and B โ Union (A โช B): Outcomes in A, B, or both โ Mutually exclusive: A โฉ B = โ
Q19: Probability rules for mutually exclusive events: โ P(A โช B) = P(A) + P(B) โ If Aโ โช Aโ โช ... โช A = S and mutually exclusive โ โP(Aแตข) = 1 Q20: Complement rule formulas: โ P(ฤ) = 1 โ P(A) โ P(A) + P(ฤ) = 1 Q21: Complement examples: โ P(not blood type O) = 1 โ 0.461 = 0. โ Deck of cards: P(black) = 1 โ P(red) = 0. Q22: Contingency (Two-way) tables โ Used to display frequency distributions across two variables โ Marginal probability: Row or column total / total observations Q23: Conditional probability formulas:
Q30: Multiplication for independent events: โ P(A โฉ B) = P(A) ร P(B) Q31: Examples of intersection calculations: โ 2 heads in 2 tosses: 0.5 ร 0.5 = 0. โ 2 detectors fail: 0.02 ร 0.02 = 0. โ 2 red cards without replacement: 26/52 ร 25/ Q32: Conditional from joint probability: โ If P(A โฉ B) = 0.2 and P(A) = 0.5 โ P(B|A) = 0.2 / 0.5 = 0. Q33: Titanic example (male given died) โ Conditional probability using contingency data Q34: Happiness vs. Gender survey: โ Are โvery happyโ and โmaleโ independent? โ Check if P(Happy โฉ Male) = P(Happy) ร P(Male) Q35: Independence โ Mutually Exclusive: โ Independent: Events donโt affect each other โ Mutually exclusive: Events canโt happen together โ Example: Coin toss combinations (at least 2 H vs. 3 H or 3 T) Q36: Joint probability for multiple independent events: โ P(Aโ โฉ Aโ โฉ ... โฉ A ) = P(Aโ) ร P(Aโ) ร ... ร P(A ) โ 3 boys: 0.5 ร 0.5 ร 0.5 = 0.
