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Week 8 · Midterm study guide

Midterm Exam — Study Guide

Introduction to Statistics Free sample edition

Coverage — what the midterm covers

The midterm is cumulative over Weeks 1–7. It opens in the Week 8 module and closes at the due time your instructor sets. There is no time limit, and you get one attempt. The questions appear one at a time, and you cannot go back once you move on.

Format: 25 questions, and every question counts the same. Eighteen ask you to pick one option, four ask you to type a short result such as a number, two ask you to judge a statement as true or false, and one asks you to pick every option that applies. Bring a basic calculator. Each question gives you the data it needs.

What is in scope, with the number of midterm questions on each course objective:

Midterm coverage by course objective: the weeks that teach it, its main skills, and its number of questions out of 25
Course objective Weeks Main skills Questions
Where the data came from 1, used again in 2 Population and sample; parameter and statistic; variable types and levels of measurement; sampling methods and bias; observational studies and experiments 7
Describing one variable 2–3, used again in 4 Frequency tables and graphs; shape named by the tail; mean, median, and mode; range, variance, standard deviation, and the IQR 6
How two variables move together 4, used again in 5 Scatterplots and r; marginal and conditional distributions; association versus causation 4
Probability and random variables 5–7 Probability rules; conditional probability and independence; expected value and spread of a random variable; the binomial model 8

Probability and random variables carry the most questions, and where the data came from is close behind, so put the most review time there. The z-score from Week 8 is not on the midterm.

Topic-by-topic review

Each topic gives the key idea in plain words, the course's own definitions, the mistake students make most often, and its cure. The deeper review for every topic is that week's chapter, lecture page, and slide deck.

Week 1 — Where the data came from

  • Key idea: before you trust a number, ask who was measured and how they were chosen. A statistic only speaks for a population if the sample was chosen well.
  • Definitions: a population is the entire group of individuals that a question is about, and a sample is the part of a population that is actually observed or measured. A parameter is a number that describes a whole population. A statistic is a number computed from a sample, and it estimates the matching parameter.
  • Course example: the penguin field log is a sample of n = 344 penguins, so the 4,201.75 g average body mass of its 342 measured birds is a statistic. The six faces of a fair die are a whole population with N = 6, so their average of 3.5 is a parameter.
  • Levels of measurement: nominal (labels with no order), ordinal (an order but unequal gaps), interval (equal gaps but no true zero), and ratio (equal gaps and a true zero). The level decides which summaries make sense.
  • Sampling methods: a simple random sample, a stratified sample (a simple random sample from every group), a cluster sample (whole groups chosen at random), and a systematic sample (a random start, then every k-th individual). A convenience sample and a voluntary-response sample both build in bias.
  • Study design: in an observational study, researchers record what they find without assigning any treatment. Only a randomized experiment, where chance decides who gets which treatment, supports a cause-and-effect conclusion.
  • Predictable mistake → cure: thinking a bigger sample fixes a biased method. Bias is a built-in tendency to miss the truth in the same direction every time, and a bigger sample does not remove it. Cure: judge how the sample was chosen before you look at how big it is.

Week 2 — Tables and graphs

  • Key idea: a table or graph shows a variable's distribution: which values occur and how often each one occurs.
  • Categorical variables: a frequency table counts each category, and a relative frequency divides each count by the total. For the penguins, Adelie make up 152 ⁄ 344 ≈ 44.2% of the field log. Show categories with a bar chart, or with a pie chart when the categories together make one whole.
  • Quantitative variables: show them with a dot plot, a stem plot, or a histogram. Then describe the shape: symmetric, right-skewed, or left-skewed, and unimodal or bimodal.
  • Two-way tables: read each cell, each row total, and each column total, and check that the totals agree.
  • Misleading graphs: a truncated axis starts above zero and makes small differences look large. An area distortion scales an icon in both height and width, so doubling an amount draws 2 × 2 = 4 times the area.
  • Predictable mistake → cure: naming the skew by where the pile is instead of where the tail is. A right-skewed distribution has its long tail stretching to the right, toward the larger values. Cure: find the thin, stretched-out end first, and name the skew after that side.

Week 3 — Center and spread

  • Key idea: a measure of center says where the middle sits, and a measure of spread says how far values typically sit from it. You need both.
  • Center: the mean is the sum of all the values divided by how many values there are, and it needs interval or ratio data. The median is the middle value once the data are put in order. The mode is the most common value, and it is the only center that works for nominal data.
  • Resistance: the median is a resistant measure and the mean is not. If a die's 6 is mistyped as 60, the mean jumps from 3.5 to 75 ⁄ 6 = 12.5, while the median stays 3.5.
  • Skew moves the mean: skew pulls the mean toward the long tail. The month lengths have mean 30.4167 days below their median of 31, so they are left-skewed. The penguin body masses have mean 4,201.75 g above their median of 4,050 g, so they are right-skewed.
  • Spread: the range is the largest value minus the smallest. The standard deviation describes the typical distance of a value from the mean, in the data's own units. The variance is in squared units.
  • Sample or population: for a sample, divide the squared deviations by n − 1 (Bessel's correction); for a whole population, divide by N. On the same six die faces, s² = 17.5 ⁄ 5 = 3.5 while σ² = 17.5 ⁄ 6 ≈ 2.9167.
  • Five-number summary and the 1.5 × IQR rule: the minimum, Q1, the median, Q3, and the maximum. The IQR is Q3 − Q1. A value more than 1.5 × IQR below Q1 or above Q3 is flagged as a potential outlier. For the penguin body masses, the IQR is 1,200 g and the fences are 1,750 g and 6,550 g, so no value is flagged.
  • Predictable mistakes → cures: (1) dividing by N for a sample. Cure: ask "is this every individual, or only some?" before you divide. (2) mixing up variance and standard deviation units. Cure: the standard deviation is in the data's units; the variance is in squared units. (3) computing a mean of category labels. Cure: check the level of measurement first.

Week 4 — Two variables

  • Key idea: do two variables move together? Name the explanatory variable x (the one that may help explain the other) and the response variable y (the one you want to explain).
  • Scatterplots: describe direction, form, strength, and unusual points. For the penguins, flipper length and body mass show a strong, positive, linear pattern.
  • The correlation coefficient r: a unitless number between −1 and 1 that measures the direction and strength of a straight-line relationship. For the penguins, r = 0.8712. Changing units does not change r. One unusual point can move it a lot, and it does not detect curved patterns.
  • Two-way tables: a marginal distribution comes from the totals. A conditional distribution comes from one row or one column, divided by that row's or column's total. Gentoo are 36.0% of all penguins but 73.8% of the Biscoe penguins, so species and island are associated.
  • Predictable mistakes → cures: (1) "r near 0 means no relationship." Cure: r near 0 means no linear relationship; look at the scatterplot for a curve. (2) "correlation means causation." Cure: name a lurking variable that could drive both. Temperature drives both ice-cream sales and drownings; age drives both shoe size and reading level in children.

Week 5 — Probability

  • Key idea: a probability is a number from 0 to 1 that gives the long-run share of repetitions in which an event happens. Short runs swing widely; long runs settle down. That settling is the law of large numbers.
  • Counting equally likely outcomes: two dice land in 6 × 6 = 36 ordered outcomes, and six of them sum to 7, so P(sum = 7) = 6 ⁄ 36 = 1 ⁄ 6.
  • The rules: the complement rule, P(not A) = 1 − P(A). The addition rule for mutually exclusive events, P(A or B) = P(A) + P(B). The general addition rule subtracts the overlap: P(A or B) = P(A) + P(B) − P(A and B). The multiplication rule for independent events, P(A and B) = P(A) × P(B).
  • Conditional probability: P(A | B), read "A given B", keeps only the outcomes in B and finds the share of them that are also in A. For the penguins, P(Gentoo | Biscoe) = 124 ⁄ 168 ≈ 0.7381, while P(Gentoo) = 124 ⁄ 344 ≈ 0.3605. Learning the island changes the chance of Gentoo, so the two events are dependent.
  • Predictable mistakes → cures: (1) the gambler's fallacy, believing an outcome is "due" after a streak without it. Cure: a die has no memory. (2) treating P(A | B) as P(B | A). Cure: say the condition out loud; P(Biscoe | Gentoo) = 1, but P(Gentoo | Biscoe) is only 124 ⁄ 168. (3) thinking mutually exclusive means independent. Cure: if two events cannot happen together, learning one happened drops the other's chance to 0, so they are dependent.

Week 6 — Random variables

  • Key idea: a random variable is a number whose value is set by the outcome of a chance process. A discrete random variable has values you can list with gaps; a continuous random variable can take any value in a range.
  • Legitimacy check: a probability distribution is legitimate when every probability is between 0 and 1 and the probabilities add to exactly 1. Miscount eight 31-day months and you get 1 ⁄ 12 + 4 ⁄ 12 + 8 ⁄ 12 = 13 ⁄ 12, which is more than 1, so that table fails.
  • Expected value: multiply each value by its probability and add. For one die, E(X) = 3.5; for the sum of two dice, E(S) = 7. The expected value is a long-run average and need not be a possible value.
  • Spread: Var(X) weights each squared deviation by its probability; SD(X) is its square root, in the variable's own units. For the sum of two dice, SD(S) ≈ 2.4152.
  • Shifting and rescaling: adding a constant moves the mean and leaves the SD alone. Multiplying by a positive constant multiplies both the mean and the SD.
  • Density curves: for a continuous random variable, probability is the area under the curve, and any single exact value has probability 0.
  • Predictable mistake → cure: "the expected value is what you should expect on one play." Cure: E(X) is what you would average over many plays. No die ever shows 3.5.

Week 7 — The binomial model, and a first look at the normal curve

  • Key idea: a binomial setting has a fixed number of trials n, trials that are independent of one another, exactly two outcomes per trial, and the same probability of success p on every trial. The binomial random variable X counts the successes.
  • Probabilities: multiply the number of ways the k successes can land by p once for each success and by 1 − p once for each failure. For "at least", use the complement: P(X ≥ k) = 1 − P(X ≤ k − 1).
  • Course example: guessing on a 10-question, four-option quiz gives n = 10 and p = 0.25. P(X ≤ 2) = 0.5256 and P(X ≥ 5) = 0.0781. The mean is μ = np = 2.5 correct answers, and the standard deviation is σ = √(np(1 − p)) ≈ 1.37.
  • Shape and the normal curve: the course's working rule of thumb says a normal curve may stand in for a binomial when np and n(1 − p) are both at least 10. For the 10-question quiz, np = 2.5 falls short of 10, which rules it out. The normal curve is a model you assume, never a fact about the data.
  • Predictable mistake → cure: using the binomial when the trials are not independent or p changes, as when drawing without replacement from a small group. Cure: check all four conditions out loud before you compute anything.

Representative practice

These are fresh variants, not the midterm's questions. Work each worked example yourself first, then compare your steps. Every number below is computed and checked in advance.

Worked example — choosing a sample (Week 1)

A fruit orchard wants the average weight of every apple picked this season. Its 1,200 crates are listed in the order they were packed. A worker picks a random crate among the first 40, then every 40th crate after it, and weighs the apples in each chosen crate.

  • Population and sample: the population is every apple picked this season. The sample is the apples in the chosen crates.
  • Method: a random start, then every 40th crate, is a systematic sample with step k = 40. It chooses 1,200 ⁄ 40 = 30 crates.
  • Statistic or parameter: the average weight of the sampled apples is a statistic. It estimates the parameter, the average weight of every apple picked this season.
  • Variable types: apple weight is a continuous quantitative variable at the ratio level, because 0 grams means none. The orchard row a crate came from is a categorical variable at the nominal level.

Why it matters: the sample was chosen by a rule with a random start, not by whoever was easiest to reach. So the orchard has no built-in reason to expect a biased average.

Worked example — center, spread, and a potential outlier (Weeks 2–3)

A ferry's nine crossings one day took 21, 23, 23, 24, 25, 26, 28, 28, and 45 minutes, already in order. Treat them as a sample.

Mean. Add and divide by n = 9:

x̄ = 243 ⁄ 9 = 27 minutes

Median. With n = 9, the median is the 5th value: 25 minutes.

Shape. The mean, 27, sits above the median, 25, because the 45-minute crossing stretches a tail to the right. The crossing times are right-skewed.

Five-number summary. Q1 is the median of the lower four values, (23 + 23) ⁄ 2 = 23. Q3 is the median of the upper four, (28 + 28) ⁄ 2 = 28. So the summary is 21, 23, 25, 28, 45 minutes.

The 1.5 × IQR rule.

IQR = 28 − 23 = 5 minutes, and 1.5 × 5 = 7.5 minutes

fences: 23 − 7.5 = 15.5 and 28 + 7.5 = 35.5 minutes

The 45-minute crossing sits above 35.5, so it is flagged as a potential outlier. It stays in the data unless you learn it was recorded wrong.

Standard deviation. Subtract the mean from each value:

Deviations and squared deviations of the nine ferry crossing times from their mean of 27 minutes
x (minutes) x − x̄ (x − x̄)²
21 −6 36
23 −4 16
23 −4 16
24 −3 9
25 −2 4
26 −1 1
28 1 1
28 1 1
45 18 324

The squared deviations add to 408. This is a sample, so divide by n − 1 = 8:

s² = 408 ⁄ 8 = 51 square minutes, and s = √51 ≈ 7.14 minutes

A typical crossing lands about 7.14 minutes from the mean of 27 minutes.

Resistance check. Drop the 45 and the mean of the other eight falls to 198 ⁄ 8 = 24.75 minutes. The median moves only from 25 to 24.5 minutes. Why it matters: for a typical crossing time, report the median, because one slow crossing drags the mean.

Worked example — a two-way table, read both ways (Weeks 4–5)

A hiking club asks its 80 members whether they prefer short or long trails.

Hiking club members counted by membership status (rows) and preferred trail length (columns)
Member status Short trail Long trail Total
New 22 8 30
Returning 15 35 50
Total 37 43 80
  • Marginal: P(long) = 43 ⁄ 80 = 0.5375, so about 53.8% of all members prefer long trails.
  • Conditional: P(long | returning) = 35 ⁄ 50 = 0.70, but P(long | new) = 8 ⁄ 30 ≈ 0.2667. The conditional distributions differ, so membership status and trail preference are associated.
  • Independence check: P(long | returning) = 0.70 is not P(long) = 0.5375, so the two events are not independent.
  • Order matters: P(returning | long) = 35 ⁄ 43 ≈ 0.8140, a different number from P(long | returning) = 0.70.
  • General addition rule: P(long or returning) = 43 ⁄ 80 + 50 ⁄ 80 − 35 ⁄ 80 = 58 ⁄ 80 = 0.725.

Why it matters: the club's survey is observational, so the association does not show that being a returning member causes a taste for long trails. Fitness built over time could be a lurking variable.

Worked example — expected value and spread (Week 6)

A bowling alley runs a promotion. On each visit, X is the number of free games a player wins: P(X = 0) = 0.6, P(X = 1) = 0.3, and P(X = 2) = 0.1.

Legitimate? Each probability is between 0 and 1, and 0.6 + 0.3 + 0.1 = 1. Yes.

Expected value.

E(X) = 0 × 0.6 + 1 × 0.3 + 2 × 0.1 = 0.5 free games

Over many visits, a player averages half a free game per visit, even though no single visit gives half a game.

Variance and standard deviation.

Var(X) = (0 − 0.5)² × 0.6 + (1 − 0.5)² × 0.3 + (2 − 0.5)² × 0.1 = 0.15 + 0.075 + 0.225 = 0.45

SD(X) = √0.45 ≈ 0.67 free games

Rescaling. Each free game is worth $4, so the prize value is Y = 4 × X dollars. Its mean is 4 × 0.5 = $2.00 per visit, and its SD is 4 × 0.67 ≈ $2.68. Why it matters: rescaling multiplies both the mean and the SD by the same constant.

Worked example — a binomial count (Week 7)

A help desk resolves each call on the first try with probability p = 0.8, independently of other calls. X counts the first-try fixes among the next n = 4 calls.

Conditions. n = 4 is fixed, the calls are independent, each call is fixed or not, and p = 0.8 stays the same. So X is binomial.

Exactly 3 fixes. The one miss can be any of the 4 calls, so there are 4 ways:

P(X = 3) = 4 × 0.8 × 0.8 × 0.8 × 0.2 = 0.4096

All 4 fixes. There is 1 way:

P(X = 4) = 0.8 × 0.8 × 0.8 × 0.8 = 0.4096

At least 3. Add the two: P(X ≥ 3) = 0.4096 + 0.4096 = 0.8192. In about 82 of every 100 such runs of 4 calls, at least 3 are fixed on the first try.

Center and spread. μ = np = 4 × 0.8 = 3.2 fixes, and σ = √(4 × 0.8 × 0.2) = √0.64 = 0.8 fixes.

Rule of thumb. np = 3.2 falls short of 10, which rules out a normal stand-in here.

Self-check questions

Try each one, then read the check and the one-line why.

  1. A wildlife app invites its users to report every bird they see, and 9,000 reports come in. Can these reports estimate how common each bird is in the region? Check: not reliably. This is a voluntary-response sample. Why: keen birders who choose to report are not like everyone else, and 9,000 reports do not remove that bias.
  2. A music shop sold 50 instruments last month: 20 guitars, 15 keyboards, 10 drum kits, and 5 violins. What is the relative frequency of keyboards? Check: 15 ⁄ 50 = 30.0%. Why: a relative frequency divides one category's count by the total.
  3. A laptop study finds r = −0.82 between hours of daily use and battery health. Describe the relationship. Check: strong, negative, and linear. Why: r near −1 means points falling from left to right along a line. It still does not prove that use causes the decline.
  4. A playlist has 40 songs, and 12 come from one decade. What is the chance that a song picked at random is not from that decade? Check: 1 − 12 ⁄ 40 = 28 ⁄ 40 = 0.7. Why: the complement rule.
  5. A table lists P(X = 1) = 0.2, P(X = 2) = 0.5, and P(X = 3) = 0.4. Is it a legitimate probability distribution? Check: no. Why: 0.2 + 0.5 + 0.4 = 1.1, and a legitimate distribution adds to exactly 1.
  6. A study group of 8 people, 5 of them nursing majors, draws 3 names without replacement to present. Is the number of nursing majors drawn binomial? Check: no. Why: each draw changes who is left, so p changes from draw to draw.

Check yourself — choose one. Write two or three sentences to a friend.

  • (a) Explain why the ferry's median is a better summary of a typical crossing than its mean.
  • (b) Explain why the hiking club's association does not show cause and effect.

Either one rehearses an explain-why skill the midterm checks.

Watch before the exam

These two short videos come from this week's Readings & Resources page, and both have captions. Use them to review and to connect weeks, not to learn anything new.

Study plan

A short countdown sized to a course that meets twice a week. Week 8's chapter and its z-score are new, so leave room for them too.

  • At the start of Week 8: take the practice exam once, cold and closed-note. Note which objectives cost you the most.
  • Before the first class meeting of Week 8: reread the topic reviews above for your weakest two objectives. Redo the matching worked examples from scratch, without looking.
  • Between the two class meetings: run the paired exam-prep tutorial (the copy-and-paste chatbot tutor) for active recall. Tell it which areas the practice exam exposed.
  • The day before you sit the midterm: retake the practice exam, redo every miss by hand, and work the self-check questions above out loud.
  • Right before: skim the "predictable mistake → cure" lines for all seven weeks. Then stop studying.

This guide is for review. The exam-prep tutorial and the practice exam are where you rehearse. Retrieval practice, not rereading, is what moves scores.

Exam format

Every one of the 25 questions counts the same, and the midterm checks your own understanding without AI. The questions come one at a time, and there is no going back, so settle each one before you move on.

Test-taking strategies for this material:

  • Name the design before you judge the claim. For a cause-and-effect claim, ask first whether the study assigned treatments at random.
  • Sort before you take a median, and find the tail before you name a skew.
  • Ask "every individual, or only some?" That one question settles parameter versus statistic, N versus n, and dividing by N versus n − 1.
  • Read the condition in every probability. Underline the part after "given", and keep only those individuals.
  • Check all four binomial conditions before you use n and p.
  • Keep extra decimals until the end, then round probabilities to four decimals and percentages to one, as the course does.
  • Use scratch paper for every computation, even a short one. Writing each step catches most slips.

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