First-year courses · Mathematics

Introductory statistics

Mean, standard deviation, probability rules, binomial and normal models, sampling distributions, confidence intervals, p-values and regression flashcards.

45 cards7 topicsAbout 20 minutes a roundTypical first-year introductory statistics (descriptive statistics, probability, inference, regression)

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About this deck

Introductory statistics is taken by students in almost every faculty, from science and engineering to business, kinesiology and social sciences. This deck covers the standard first-year path: describing data with centre, spread and shape; probability rules, conditional probability and Bayes' theorem; random variables, expected value, and the binomial and normal distributions; sampling distributions and the Central Limit Theorem; confidence intervals for means and proportions; hypothesis tests, p-values, error types and power; chi-square tests; and correlation and least-squares regression.

Only high school algebra is required. You should be able to evaluate formulas with square roots and summations and read a normal table or use the statistical functions of your calculator or software.

Statistics exams reward correct interpretation as much as correct arithmetic. For each inference card, say aloud both how to compute the quantity and how to explain it in one plain sentence in context. When you flip the card, compare your wording with the definition and look for small but important differences, such as whether you described the population or the sample. Shuffle the Confidence intervals and Hypothesis testing tags together, since they rest on the same sampling distribution ideas.

Mistakes to avoid: saying that a 95% confidence interval has a 95% chance of containing the parameter after it has been computed (the 95% describes the method over many samples); treating a p-value as the probability that the null hypothesis is true; and concluding causation from a correlation in observational data. Another frequent one is using the population formula (dividing by n) when the sample standard deviation (dividing by n - 1) is asked for.

Before you start: High school algebra; no calculus needed.

All 45 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Describing data

Population versus sample: The population is the entire group of interest, described by parameters; a sample is the subset actually measured, described by statistics.Example: The mean height of all Canadian adults is a parameter; the mean of 500 surveyed adults is a statistic.
Mean: The arithmetic average: the sum of the values divided by how many there are. Sensitive to outliers.Example: Data 2, 4, 4, 5, 10: sum 25, divided by 5 gives 5.
Median: The middle value of the ordered data (or the average of the two middle values); resistant to outliers.Example: 2, 4, 4, 5, 10 has median 4; changing 10 to 100 leaves it at 4.
Sample standard deviation: s = sqrt(sum of (x - xbar)^2/(n - 1)): a typical distance of values from the mean. Its square, s^2, is the sample variance.Example: 2, 4, 4, 5, 10: squared deviations sum to 36, 36/4 = 9, so s = 3.
Interquartile range and the 1.5 IQR rule: IQR = Q3 - Q1, the spread of the middle half of the data; values more than 1.5 IQR below Q1 or above Q3 are flagged as possible outliers.Example: Q1 = 6, Q3 = 12: IQR = 6, upper fence 12 + 9 = 21, so a value of 40 is flagged.
Skewness: Lack of symmetry in a distribution. In a right-skewed distribution the long tail is on the right and the mean usually exceeds the median.Example: Household incomes are right-skewed.
z-score: How many standard deviations a value lies above or below the mean: z = (x - mean)/sd.Example: A test score of 85 with mean 70 and sd 10 has z = 1.5.

Probability

Sample space and event: The sample space is the set of all possible outcomes of a random process; an event is any subset of it.Example: Rolling a die: sample space {1, 2, 3, 4, 5, 6}; the event 'even' is {2, 4, 6}.
Complement rule: P(not A) = 1 - P(A).Example: P(at least one head in 3 tosses) = 1 - (1/2)^3 = 7/8.
Addition rule: P(A or B) = P(A) + P(B) - P(A and B); the subtraction removes outcomes counted twice.Example: Drawing a heart or a face card: 13/52 + 12/52 - 3/52 = 22/52, about 0.423.
Conditional probability: The probability of A given that B has occurred: P(A | B) = P(A and B)/P(B).Example: P(king | face card) = (4/52)/(12/52) = 1/3.
Independent events: Events where knowing one occurred does not change the probability of the other: P(A | B) = P(A), equivalently P(A and B) = P(A)P(B).Example: Two fair coin tosses: P(both heads) = 1/2 x 1/2 = 1/4.
Mutually exclusive versus independent: Mutually exclusive events cannot happen together (P(A and B) = 0); events with positive probability that are mutually exclusive are never independent.Example: On one die roll, getting 1 and getting 6 are mutually exclusive and dependent.
Bayes' theorem: P(A | B) = P(B | A) P(A)/P(B); it reverses a conditional probability using prior information.Example: Prevalence 1%, test sensitivity 90%, false positive rate 5%: P(disease | positive) = 0.009/0.0585 = 0.154.

Random variables and distributions

Random variable: A numerical outcome of a random process; discrete if it takes countable values, continuous if it takes any value in an interval.Example: Number of heads in 10 tosses is discrete; waiting time for a bus is continuous.
Expected value: The long-run average of a random variable: sum of each value times its probability (for discrete variables).Example: A fair die: (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5.
Binomial distribution: Counts successes in n independent trials with the same success probability p: P(X = k) = C(n, k) p^k (1 - p)^(n - k).Example: n = 5, p = 0.5: P(X = 2) = 10/32 = 0.3125.
Binomial mean and standard deviation: For X counting successes in n trials with probability p: mean = np, sd = sqrt(np(1 - p)).Example: n = 100, p = 0.3: mean 30, sd = sqrt(21) = 4.58.
Normal distribution: A symmetric, bell-shaped continuous distribution set by its mean and sd; about 68%, 95% and 99.7% of values fall within 1, 2 and 3 sd of the mean.Example: Heights with mean 170 cm and sd 10 cm: about 95% lie between 150 and 190 cm.
Standard normal distribution: The normal distribution with mean 0 and sd 1, used through z-scores and tables to find probabilities.Example: P(Z < 1.96) = 0.975, so 95% of values lie between -1.96 and 1.96.
Poisson distribution: Models counts of events in a fixed interval when events occur independently at a constant average rate lambda: P(X = k) = e^(-lambda) lambda^k/k!.Example: lambda = 2 calls per hour: P(no calls) = e^(-2) = 0.135.

Sampling distributions

Sampling distribution: The distribution of a statistic (such as a sample mean) over all possible random samples of the same size from a population.Example: Repeatedly averaging 30 random students' sleep hours would build one.
Standard error of the mean: The standard deviation of the sample mean's sampling distribution: sigma/sqrt(n). It shrinks as sample size grows.Example: sigma = 12, n = 36: SE = 12/6 = 2.
Central Limit Theorem: For a large enough random sample, the sampling distribution of the sample mean is approximately normal, whatever the population's shape.Example: A common rule of thumb treats n >= 30 as large enough unless the population is very skewed.
Sampling bias: Systematic error caused by a sampling method that favours some outcomes, such as voluntary response or undercoverage; larger samples do not fix it.Example: An online poll answered only by people who feel strongly overstates extreme views.

Confidence intervals

Confidence interval for a mean: xbar plus or minus t* s/sqrt(n): a range of plausible values for the population mean, with t* from the t distribution on n - 1 degrees of freedom.Example: xbar = 50, s = 8, n = 16, 95%: t* = 2.131, margin 4.26, interval (45.74, 54.26).
Meaning of 95% confidence: If we repeated the sampling many times, about 95% of the intervals produced by this method would capture the true parameter.Example: Any one computed interval either contains the parameter or does not.
Confidence interval for a proportion: p-hat plus or minus z* sqrt(p-hat(1 - p-hat)/n), valid when there are enough successes and failures (often at least 10 of each).Example: p-hat = 0.60, n = 400, 95%: SE = 0.0245, margin 0.048, interval (0.552, 0.648).
Sample size for a desired margin of error: For a mean with known sigma: n = (z* sigma/E)^2, always rounded up.Example: sigma = 15, E = 3, 95%: n = (1.96(15)/3)^2 = 96.04, so 97.
t distribution: A symmetric, bell-shaped family with heavier tails than the normal, used when the population sd is estimated by s; it approaches the normal as degrees of freedom grow.Example: 95% critical value: 2.131 for 15 df versus 1.96 for the normal.

Hypothesis testing

Null and alternative hypotheses: H0 states no effect or no difference (a specific parameter value); Ha states the effect the study seeks evidence for.Example: H0: mu = 50 versus Ha: mu > 50.
Test statistic: A standardized measure of how far the sample result is from the null value, in standard errors.Example: xbar = 52, mu0 = 50, sigma = 10, n = 100: z = 2/(10/10) = 2.0.
P-value: Assuming the null hypothesis is true, the probability of getting a result at least as extreme as the one observed. Small values are evidence against H0.Example: z = 2.0, two-sided: p is about 0.046.
Significance level (alpha): A threshold chosen before the test; reject H0 if the p-value is at most alpha. It is also the Type I error rate.Example: With alpha = 0.05, p = 0.046 leads to rejecting H0.
Type I and Type II errors: Type I: rejecting a true H0 (false positive), probability alpha. Type II: failing to reject a false H0 (false negative), probability beta.Example: Convicting an innocent defendant is like a Type I error.
Power of a test: The probability of correctly rejecting a false H0, equal to 1 - beta; it rises with larger samples, larger true effects and larger alpha.Example: Doubling the sample size raises power to detect the same effect.
Two-sample and paired t tests: A two-sample t test compares means of two independent groups; a paired t test analyses differences within matched pairs, such as before and after.Example: Blood pressure before and after a drug in the same patients calls for a paired test.
Chi-square test of independence: Tests whether two categorical variables are associated by comparing observed counts with expected counts (row total x column total/grand total).Example: Row total 60, column total 50, grand total 100: expected count 30.
Statistical versus practical significance: A tiny effect can be statistically significant in a huge sample but too small to matter in practice; report effect sizes and intervals, not only p-values.Example: A diet that lowers weight by 0.1 kg can give p < 0.001 with 100 000 people.

Regression and correlation

Correlation coefficient (r): A number from -1 to 1 measuring the direction and strength of a linear relationship between two quantitative variables.Example: r = 0.9 is a strong positive linear association; r near 0 means little linear association.
Least-squares regression line: The line y-hat = a + bx minimizing the sum of squared residuals, with slope b = r(sy/sx) and intercept a = ybar - b xbar.Example: r = 0.8, sx = 2, sy = 5: b = 2; xbar = 10, ybar = 30: a = 10.
Residual: Observed y minus predicted y for a data point; a plot of these differences against x that shows no pattern supports a linear model.Example: y-hat = 10 + 2x at x = 5 predicts 20; observed 23 gives residual 3.
Coefficient of determination (R^2): The fraction of the variation in y explained by the linear relationship with x; for simple regression it equals r squared.Example: r = 0.8 gives R^2 = 0.64: 64% of the variation in y is explained.
Correlation versus causation: An association between two variables does not prove one causes the other; lurking variables or reverse causation may explain it. Randomized experiments are needed for causal claims.Example: Ice cream sales and drownings both rise in summer because of temperature.
Extrapolation: Using a regression line to predict far outside the range of observed x values; the pattern may not hold there.Example: A child-growth line from ages 2 to 10 would predict absurd adult heights.