Week 8 · Chapter

Chapter 8 — Midterm Review & Exam

Introduction to Statistics Free sample edition

The module's primary reading · ~30–40 minutes · Objective 1 (distinguish populations from samples and parameters from statistics, classify variables by type and level of measurement, and identify appropriate sampling methods and study designs, including what each can and cannot conclude), Objective 2 (summarize and display a single quantitative or categorical variable with appropriate tables and graphs, and describe its distribution's shape, center, and spread with the right numerical summaries), Objective 3 (describe the relationship between two variables using scatterplots, the correlation coefficient, and two-way tables, and explain why association alone does not establish causation), Objective 4 (apply the basic rules of probability, including conditional probability and independence, and compute and interpret the expected value and standard deviation of a discrete random variable, including binomial settings) · Read it early this week, before the review activities and well before the midterm.

In plain language

This week you look back at the whole first half of the course. It comes down to four questions about any set of numbers: where they came from, what they look like, whether two of them move together, and how likely an outcome is. You also learn one new tool, the z-score (a count of how many standard deviations a value sits above or below the average of its group). It tells you whether a value is ordinary or unusual for its group, which is called relative standing (where one value sits compared with the rest of its group).

Why you already care about this

A penguin on a field scale weighs 4,000 g. Is that heavy? A friend scores 82 on a chemistry exam and 82 on a history exam. Did both go equally well? A runner finishes a 5 km race in 25 minutes. Fast or slow? You cannot answer any of these from the number alone. You need to know what is typical for the group the number belongs to, and how much that group usually varies.

You have been building those tools for seven weeks. You know how to ask who was measured, how to find a center and a spread, how to read a relationship, and how to put a number on chance. This chapter pulls them together in one place, so the midterm feels like a conversation you have already had. Then it adds the z-score, which uses the mean and the standard deviation you already know to answer "is this value unusual?"

By the end of this chapter you will be able to:

say where a set of data came from, and what that design can and cannot conclude;
choose a center and a spread that fit a distribution's shape, and flag potential outliers with the 1.5 × IQR rule;
read a correlation and a conditional distribution without mistaking association for cause;
apply conditional probability, independence, and the expected value of a random variable, including a binomial count;
compute a z-score and say in words how many standard deviations a value sits above or below its mean;
compare values from different groups by their z-scores.

Carry this sentence with you: subtract the mean, divide by the spread — then name the group.

Where we are

Already covered: Weeks 1 to 3 taught you to describe one variable, Week 4 to relate two variables, and Weeks 5 to 7 to reason about chance with probability rules, random variables, and the binomial model.
This week: a review of Weeks 1 to 7 built around four questions, plus one new idea, the z-score as a measure of relative standing, and the midterm.
How they connect: a z-score combines Week 3's mean and standard deviation into one number, and Week 6's idea of shifting and rescaling explains why it has no units.
Where we're going next: Week 9, The Normal Distribution, starts from the z-score and uses the normal model to turn it into a share of the group.

Each week so far added one piece. Week 1 separated a sample from a population and an observational study from an experiment. Week 2 drew a distribution and named its shape by its tail. Week 3 measured its center and spread. Week 4 read relationships with a scatterplot, the correlation r, and a two-way table. Week 5 put numbers on chance. Week 6 turned chance outcomes into random variables with a long-run average. Week 7 built the binomial model for counts of successes. The midterm covers that ground, and this chapter walks it once more with the course's own pinned numbers. Use the review sections as a checklist: if a step feels shaky, go back to that week's chapter.

1. Where did the numbers come from? Check the source first

Before you trust a number, ask who was measured and how. A restaurant that posts "9 out of 10 diners loved us" might have asked only the diners who chose to leave a review. The number could be correct for those diners and still say little about everyone who ate there.

Three distinctions from Week 1 do this checking for you:

A population is the whole group a question is about; a sample is the part of it you actually observe.
A parameter describes a population; a statistic is computed from a sample and estimates the matching parameter.
In an observational study the researcher records what is there; in an experiment the researcher assigns treatments. Only a randomized experiment supports a cause-and-effect claim.

Worked example — the penguin field log and the die. The penguin field log records 344 penguins on three islands.

Population or sample? The researchers did not weigh every penguin in the region, so the log is a sample, and n (the sample size) is 344 penguins.
Parameter or statistic? The average body mass of the measured birds is computed from the sample, so it is a statistic.
A population you can list: the six faces of a fair die, 1 through 6, are every possible face. So N (the population size) is 6, and their average, 3.5, is a parameter.
Study design: the researchers recorded the penguins they found and assigned nothing. So the field log is an observational study, and its patterns show association, not cause.

In words: the penguin numbers describe a sample of real birds, and any pattern in them is a pattern to explain, not a proven cause.

Misconception check — the bigger-sample fix. Students often think a huge sample cures a biased method: "the review site has 5,000 ratings, so it must be right." It is tempting because more data feels like more truth. But a biased method stays biased at any size. A bigger sample only shrinks the chance wobble from sample to sample. If the method invites only happy diners, 5,000 of them still lean happy.

Memory hook: first ask who was measured, then ask how.

2. What do the data look like? Pick the summary that fits the shape

Picture the 342 penguin body masses as a histogram. Most birds pile up in the middle of the range, and a thinner tail of heavy birds stretches out to the right, toward 6,300 g. That picture tells you which summaries to trust.

Shape is named by the tail, not the pile: a long right tail means right-skewed.
The median is resistant to extreme values; the mean is pulled toward the long tail.
The standard deviation measures a typical distance from the mean in the data's own units; the IQR measures the spread of the middle half.

Worked example — the 342 body masses, start to finish. Every value below is one the course has already pinned.

Center: the sample mean is x̄ (the sample mean) = 4,201.75 g, and the median is 4,050 g.
Shape: the mean sits above the median, because the heavy birds in the right tail pull it up. So the shape is mildly right-skewed, and the median is the more resistant center.
Spread: s (the sample standard deviation) is about 801.95 g. It is in grams, the data's own units. Its square, the variance, would be in squared grams.
Five numbers: the minimum is 2,700 g, Q1 is 3,550 g, the median is 4,050 g, Q3 is 4,750 g, and the maximum is 6,300 g.
IQR: 4,750 − 3,550 = 1,200 g, and 1.5 × 1,200 = 1,800 g.
Fences: 3,550 − 1,800 = 1,750 g and 4,750 + 1,800 = 6,550 g. The minimum and the maximum both sit inside the fences, so no body mass is flagged.

In words: a typical penguin in this sample weighs about 4,050 g. The middle half of the birds spans 1,200 g. No bird is far enough out to flag as a potential outlier.

Misconception check — naming the skew by the pile. Students often say "most birds are on the left, so it is left-skewed." It is tempting because the pile is the first thing your eye lands on. But skew is named by the tail. Here the long tail runs toward the heavy birds on the right, so the shape is right-skewed, and the mean sits above the median.

In practice (your tools). In Google Sheets or Excel, with the masses in column B, =AVERAGE(B2:B343) gives the mean, =MEDIAN(B2:B343) the median, and =STDEV.S(B2:B343) the sample standard deviation. =QUARTILE.INC(B2:B343,1) and =QUARTILE.INC(B2:B343,3) give Q1 and Q3. Different tools round the quartiles a little differently, but here every common method reaches the same verdict: no outliers.

Memory hook: the tail names the skew, and the skew picks the center.

3. Do two variables move together? Read the pattern, then doubt the cause

Think of a growing child. Taller children tend to have bigger feet. That is a pattern between two variables measured on the same individuals, and it is the kind of pattern Week 4 taught you to read.

The correlation coefficient r is a unitless number from −1 to 1 that measures the direction and strength of a straight-line relationship.
A confounding variable is mixed up with the variable you are studying, so you cannot tell which one is behind the pattern.

Worked example — flippers, mass, and species. Every number here is pinned from Week 4.

The pattern: across the 342 measured penguins, r (the correlation) between flipper length and body mass is 0.8712. Close to 1 and positive, so the association is strong, positive, and linear.
A lurking explanation: Gentoo average 5,076.02 g and Adelie average 3,700.66 g. Every Gentoo in the log lives on Biscoe.
What that means for an island comparison: if Biscoe birds weigh more on average, you cannot tell whether the island or the species is behind it. Species is a confounding variable for any comparison of mass by island.

In words: longer flippers go with heavier birds, but the field log is an observational study, so the pattern alone shows association, not cause.

Misconception check — r near 0 means no relationship. Students often read a small r as "no relationship at all." It is tempting because r is the only number on the screen. But r sees only straight-line patterns. A perfect U shape can have r = 0. A small r means no linear relationship, so always look at the scatterplot too.

Memory hook: r measures a line; the design decides the cause.

4. How likely is it, and what should you expect? Use the chance rules

Suppose a friend tags a penguin at random and tells you only that it lives on Biscoe. Your best guess about its species changes. That is conditional probability. Weeks 5 to 7 then added a second question: over many repetitions, what number should you expect on average?

A conditional probability is the chance of one event once you know another has happened. Two events are independent when news about one leaves the chance of the other unchanged.
The expected value of a random variable is its long-run average; it need not be a value the variable can take.
A binomial count has a fixed number of independent trials, two outcomes each, and the same chance of success every time; its mean is np.

Worked example — three chance questions. Each one reuses a pinned value from Weeks 5 to 7.

Conditional probability: P(the penguin is Gentoo | the penguin is on Biscoe) = 124 ⁄ 168 ≈ 73.8%, written P(Gentoo | Biscoe) for short. Across all 344 penguins, P(Gentoo) = 124 ⁄ 344 ≈ 36.0%. Each is rounded to one decimal.
Independence: learning the island changes the chance of Gentoo from about 36.0% to about 73.8%, so species and island are not independent.
Expected value: for one fair die, E(X) (the expected face) is 3.5, and σ (the standard deviation of the face) is about 1.71. Over many rolls, your average face settles near 3.5, yet no single roll can show 3.5.
Binomial count: you guess on 10 four-option questions, so n (the number of questions) is 10 and p (the chance of a lucky guess) is 0.25. The mean is np = 10 × 0.25 = 2.5 correct answers, and the standard deviation is about 1.37 answers.
A tail chance: the chance of 5 or more right by pure guessing is P(X ≥ 5) = 0.0781, about 8 tries in 100.
Rule of thumb: np = 2.5 is below 10, so a normal curve is not an appropriate stand-in for the 10-question guessing count.

In words: knowing the island more than doubles your best guess that the bird is a Gentoo, and a pure guesser averages 2.5 right out of 10 over many quizzes.

Misconception check — the swapped condition. Students often treat P(Gentoo | Biscoe) and P(Biscoe | Gentoo) as the same number. It is tempting because the same two words appear in both. But the condition decides the group you divide by. Every Gentoo lives on Biscoe, so P(Biscoe | Gentoo) = 124 ⁄ 124 = 1, while P(Gentoo | Biscoe) is about 73.8%.

In practice (your tools). In a spreadsheet, =BINOM.DIST(4,10,0.25,TRUE) gives the chance of 4 or fewer right. Subtract it from 1, as in =1-BINOM.DIST(4,10,0.25,TRUE), to get the chance of 5 or more, 0.0781.

Memory hook: divide by the group you were given.

5. Measure relative standing with a z-score

Here is the new idea. A coach looks at one swimmer's lap time and asks, "Is that fast for this team?" The coach does not care about the raw seconds as much as where the time sits compared with teammates. Is it near the team average, or far from it? And "far" compared with what? The natural yardstick is the team's own standard deviation, the typical distance from the mean.

That is exactly what a z-score measures.

Relative standing is where one value sits compared with the rest of its group: above or below the group's mean, and by how much, measured in the group's own standard deviations.
A z-score is the number of standard deviations a value sits above or below the mean of its group, found by subtracting the mean from the value and dividing by the standard deviation; a positive z-score is above the mean, a negative one is below it, and it has no units.

As a formula on one line, for a value in a sample: z = (x − x̄) ⁄ s. For a value against a whole population or a model, the same move uses the population symbols: z = (x − μ) ⁄ σ.

Worked example — a 4,000 g Adelie penguin. The Adelie birds in the field log have a sample mean of 3,700.66 g and a sample standard deviation of 458.57 g.

Find the distance from the mean. x (the bird's mass) minus x̄ (the Adelie mean) is 4,000 − 3,700.66 = 299.34 g.
Measure that distance in standard deviations. Divide by s (the Adelie standard deviation): z = 299.34 ⁄ 458.57 ≈ 0.65, rounded to two decimals.
Read the sign and the size. The sign is positive, so the bird is above the mean. The size is 0.65, a little more than half of one standard deviation.

In words: this bird sits about 0.65 standard deviations above the Adelie mean. It is a little heavier than a typical Adelie, and nothing about it is unusual.

Worked example — a 6 on a fair die. The same idea works against a whole population. The six faces have μ (the mean face) = 3.5 and σ (the population standard deviation) of about 1.71.

Distance: 6 − 3.5 = 2.5 faces.
In standard deviations: z = 2.5 ⁄ 1.71 ≈ 1.46, rounded to two decimals.

In words: a 6 sits about 1.46 standard deviations above the mean face. It is the highest face, but a die has only six values, so even its top face is less than 2 standard deviations out.

Misconception check — the z-score in grams. Students often report "z = 0.65 grams" or read 0.65 as a weight. It is tempting because the bird's mass is in grams. But the subtraction leaves grams on top, and dividing by a standard deviation in grams cancels them. A z-score is a pure count of standard deviations. Say "0.65 standard deviations above the mean" every time.

In practice (your tools). Both Google Sheets and Excel have a function for this. =STANDARDIZE(4000, 3700.66, 458.57) returns about 0.65. Its three inputs are the value, the mean, and the standard deviation, in that order. If the masses sit in a column, =AVERAGE(...) and =STDEV.S(...) can supply the mean and the standard deviation for you.

Honesty note: a z-score measures distance, not rarity. Saying exactly how rare a z of 0.65 is needs a model of the group's shape, and that model arrives next week.

Memory hook: subtract the mean, divide by the spread.

6. Compare across groups: same value, different story

Now take the same 4,000 g and hand it to a different bird. A Gentoo penguin is a larger species. In the field log, the Gentoo birds have a sample mean of 5,076.02 g and a sample standard deviation of 504.12 g. Is a 4,000 g Gentoo heavy too?

A reference group is the group whose mean and standard deviation a z-score measures a value against; the same value can have a very different z-score when it is measured against a different group.

Worked example — a 4,000 g Gentoo penguin. Use the Gentoo mean and standard deviation this time.

Distance from the Gentoo mean: 4,000 − 5,076.02 = −1,076.02 g.
In standard deviations: divide by s (the Gentoo standard deviation): z = −1,076.02 ⁄ 504.12 ≈ −2.13, rounded to two decimals.
Read the sign and the size. The sign is negative, so the bird is below its mean. The size is 2.13, more than two standard deviations.

In words: this bird sits about 2.13 standard deviations below the Gentoo mean. It is unusually light for its species.

Put the two birds side by side. Both weigh exactly 4,000 g. Against Adelie birds, z ≈ 0.65: a little heavy, nothing special. Against Gentoo birds, z ≈ −2.13: far lighter than typical. Same scale reading, two very different stories. The reference group writes the story.

This is why z-scores are useful for comparisons. A z-score has no units, so it lets you compare positions across groups with different centers and spreads. Your friend's two exam scores of 82 can finally be compared, as long as you know each exam's mean and standard deviation.

Misconception check — equal values, equal standing. Students often think two equal raw values must stand in the same place. "Both birds weigh 4,000 g, so both are average-ish." It is tempting because the scale shows one number. But standing depends on the reference group. The same 4,000 g is 0.65 standard deviations above one species' mean and 2.13 standard deviations below the other's.

Caveat: a single z-score never proves a value is a mistake or should be removed. The light Gentoo is a real bird. A large z is a reason to look closer, as February's short month was in Week 3, not a reason to delete.

Pause and think. Choose one and answer it in a sentence or two:

(a) Think of a number from your own life, such as a test score, a paycheck, or a commute time, and name the reference group that would give it the most honest z-score.
(b) Think of a case where two people with the same raw number stand in very different places, and say what reference group explains the difference.

Memory hook: the same number can tell two stories; the group decides which.

Key terms

z-score — The number of standard deviations a value sits above or below the mean of its group, found by subtracting the mean from the value and dividing by the standard deviation; a positive z-score is above the mean, a negative one is below it, and it has no units. (first used: 5. Measure relative standing with a z-score)
relative standing — Where one value sits compared with the rest of its group: above or below the group's mean, and by how much, measured in the group's own standard deviations. (first used: 5. Measure relative standing with a z-score)
reference group — The group whose mean and standard deviation a z-score measures a value against; the same value can have a very different z-score when it is measured against a different group. (first used: 6. Compare across groups: same value, different story)

Summary

Before you trust a number, ask who was measured, how they were chosen, and whether anyone assigned a treatment.
Name the shape by its tail, then report the median under skew and the mean when the shape is roughly symmetric.
A correlation measures a straight-line pattern; an observational study shows association, not cause.
A conditional probability divides by the group you were given, and an expected value is a long-run average.
A z-score says how many standard deviations a value sits above or below its group's mean, and it has no units.
The same value can have very different z-scores in different reference groups, so always name the group.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A campus dining office posts a link on its social page asking, "Do you love the new menu?" It gets 900 replies, and 85% say yes. Who is the sample, what kind of sample is it, and would 9,000 replies fix the problem?
A neighborhood bakery tracks its daily sales for a month. The mean is $412 and the median is $380. What shape do you expect, and which center would you report as a typical day?
Suppose a clinic's visits in one month have a mean wait of 24 minutes with a standard deviation of 6 minutes. One patient waited 33 minutes. Find the z-score and say what it means.
Two runners each finish well. In one club, sprint times over 100 m average 13.0 seconds with a standard deviation of 0.5 seconds, and one member runs 12.0 seconds. In another club, 5 km times average 25.0 minutes with a standard deviation of 2.0 minutes, and one member runs 22.0 minutes. Find both z-scores. Which runner stands out more against their own club?

Answers (worked reasoning — read after attempting)

The sample is the 900 people who chose to reply, which makes it a voluntary-response sample. People with strong feelings are the likeliest to answer, so the 85% may not describe all diners. A bigger sample would not fix it: 9,000 self-selected replies carry the same lean, because size shrinks chance wobble, not bias. A random sample from the list of meal-plan holders would be a better plan.
The mean, $412, sits above the median, $380, so a few very busy days are pulling the mean up. That points to a right-skewed shape. The median is resistant to those big days, so $380 is the better description of a typical day.
The distance from the mean is 33 − 24 = 9 minutes. Divide by the standard deviation: z = 9 ⁄ 6 = 1.5. This patient waited 1.5 standard deviations longer than the clinic's average wait that month. The z-score has no units, so you say "1.5 standard deviations," not "1.5 minutes."
Sprinter: (12.0 − 13.0) ⁄ 0.5 = −1.0 ⁄ 0.5 = −2.0. Distance runner: (22.0 − 25.0) ⁄ 2.0 = −3.0 ⁄ 2.0 = −1.5. Both z-scores are negative, and here negative is good, because a lower time is faster. The sprinter sits 2.0 standard deviations below the club mean and the distance runner 1.5 below, so the sprinter stands out more against their own club. Notice that the raw times, seconds and minutes, could never be compared directly; the z-scores can.

Next up in this module: the lecture, the slide deck, the Readings & Resources page, the practice set, the discussion, and the midterm — and next week the normal model turns a z-score into a share of the group.

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