Quiz 7 — Statistics and Probability

The Guide

Which number stands for the group?

Any pile of numbers can be squeezed down to one number that supposedly represents the pile. There are three ways to do that squeezing, and they can give wildly different answers about the same pile. Knowing which one somebody used, and which one they should have used, is the whole skill.

Four words

Mean: Add everything up and divide by how many there are. This is what most people mean by "average." It uses every number, which is usually a strength and is sometimes exactly the problem.
Median: Line the numbers up in order and take the one in the middle. The middle is not found by looking — it is found by counting inward from both ends at the same time, one step at a time, until you run out of room. With an odd count you land on one number. With an even count you land on two, and the median is the mean of those two. The median does not care how extreme the extremes are.
Mode: The value that appears most often. A set can have no mode, or more than one. It is the only one of the three that works for things that are not numbers — the most common blood type, the most frequent complaint.
Range: Largest minus smallest. Not an average at all — it describes how spread out the numbers are, which is a different question from where their center sits.

One extreme value moves the mean and not the median. This is the most useful idea on this page. Add one very large number to a set and the mean jumps while the median barely shifts. So when a set contains one huge value — one enormous salary, one disastrous delivery, one very old patient — the median usually describes the group better, and the mean describes a person who may not exist.

Two groups can share a mean and share almost nothing else. The mean tells you where the middle is. The range tells you how far apart people are. A class averaging 80 where everyone scored between 78 and 82 is a completely different room from a class averaging 80 where scores ran from 60 to 100 — and no average of any kind will tell you that. You have to look at the spread.

Probability

The probability of something is the number of ways it can happen divided by the number of things that could happen at all. Four red seeds in a packet of ten gives a probability of 4 ÷ 10, which can be written as 0.4, or as 40 percent, or as two fifths. All three say the same thing.

Probability runs from 0 to 1. Zero means it cannot happen; one means it must. Nothing is ever less than zero or more than one, so an answer of 120 percent is a signal that something went wrong.

Independent events multiply. If one event does not affect the next — a coin flip, a die roll, a seed drawn and put back — the chance of both happening is the two probabilities multiplied. Half and half gives a quarter. Notice that multiplying two numbers below one always gives something smaller: two things both happening is always less likely than either one alone.

A probability is not a promise. A 60 percent chance does not mean six out of every ten. It means that over a great many tries, the proportion settles near six in ten. In ten tries you might get four, or eight, and neither is surprising. Small runs wander. This idea comes back hard in Part IV, where a genetics ratio of 3 to 1 predicts nothing whatever about any particular four children.

Worked Examples

Two questions, worked through

Data

A courier recorded how long five deliveries took, in minutes: 12, 15, 15, 18, 60.

Example 1. Find the mean, median, mode, and range.

Mean: 12 + 15 + 15 + 18 + 60 = 120, and 120 ÷ 5 = 24 minutes. Median: the numbers are already in order, and the middle one is 15 minutes. Mode: 15 appears twice, so 15 minutes. Range: 60 − 12 = 48 minutes.

Put them in order first, every time. It costs ten seconds and it is the only way the median is findable.

Example 2. Which number best describes a typical delivery?

The mean is 24 minutes. But four of the five deliveries took 18 minutes or less. The mean is longer than almost every actual delivery, because one 60-minute run is dragging it upward.

Answer: the median, 15 minutes. When one value sits far away from the rest, the mean stops describing the group and starts describing the outlier's influence on the group. The 60-minute delivery is still worth knowing about — it just should be reported as its own fact, not folded into an average.

The Quiz

Ten questions

Answer all ten, then press the button at the bottom. A calculator is fine.

Data A — questions 1 to 4

A small print shop employs seven people. Their annual pay, in dollars:

Position	Annual pay
Press assistant	32,000
Bindery worker	34,000
Press operator	35,000
Press operator	35,000
Designer	38,000
Shop supervisor	42,000
Owner	260,000

1.What is the median pay at the shop?

A$32,000
B$35,000
C$38,000
D$68,000

2.What is the mean pay at the shop?

A$35,000
B$38,000
C$68,000
D$228,000

3.Which figure better describes what a typical worker at this shop earns?

AThe median, because the owner's pay pulls the mean above what six of the seven people actually earn.
BThe mean, because it uses every number in the set.
CThe range, because it covers everybody.
DNeither one; pay cannot be summarized.

4.A job advertisement for the shop says "average pay $68,000." How should this be judged?

AIt is accurate and gives a fair picture of the job.
BIt is false, because the arithmetic is wrong.
CThere is not enough information to judge it.
DThe arithmetic is correct, but the claim misleads, since six of the seven employees earn far less than $68,000.

Data B — questions 5 and 6

Each dot is one student. Class A scored 78, 79, 80, 81, 82. Class B scored 60, 70, 80, 90, 100.

5.What is the range of the Class B scores?

6.What do the two rows together show?

ATwo groups can have the same mean and very different spreads, so a mean alone does not describe a group.
BClass B is the stronger class.
CClass A is the stronger class.
DThe two classes performed identically.

Passage C — questions 7 to 9

A seed packet holds 40 seeds. The grower states that 24 of them will produce red flowers and 16 will produce white. The seeds are identical to look at and are thoroughly mixed.

A gardener reaches in without looking and takes one seed.

7.What is the probability that the seed will produce a red flower?

A24%
B40%
C60%
D62.5%

8.The gardener puts the seed back, mixes the packet again, and draws a second time. What is the probability that both draws were red-flowering seeds?

A36%
B30%
C60%
D120%

9.The gardener plants 10 seeds from the packet and 7 come up red. Does this show the grower's stated proportion was wrong?

AYes. The packet must have been mislabeled.
BNo. With only 10 seeds, results wander around the expected proportion by chance, and 7 is an ordinary result.
CYes, because 7 out of 10 is more than 6 out of 10.
DNo, because probability does not apply to living things.

Question 10

A plant breeder crosses two plants and expects the offspring to be tall and short in a ratio of 3 to 1. She plants four seeds from the cross. All four grow tall.

10.What should she conclude?

AThe expected ratio of 3 to 1 must be wrong.
BShe made an error in performing the cross.
CExactly three should have been tall and one short.
DNothing yet. A 3-to-1 ratio predicts a long-run proportion, not the outcome of any particular four plants.

Send this line to your teacher

The line records which questions you missed and which answer you chose. That is more useful to your teacher than the score, because it shows where a question went wrong. If a question felt unclear even though you got it right, add its number with a question mark — for example 5? — before you send it.

Score ______ / 10 Missed — write the question number and the letter you chose:
______________________________________________________________

The Key

Answers, and the trap in each one

1. B — $35,000. Seven people, so the middle one is the fourth from either end. The table is already in order, and the fourth line is the second press operator at $35,000. D is the mean, sitting in the choices to be grabbed by anyone who read "median" as "average." The two words look alike and mean different things, and this is the most common single error on the topic.

2. C — $68,000. The seven figures total $476,000, and 476,000 ÷ 7 = 68,000. D is the range, $260,000 − $32,000. Notice how far the mean sits from the median: $68,000 against $35,000, nearly double. One number did that.

3. A — the median. Six of the seven people earn between $32,000 and $42,000. The mean of $68,000 describes nobody in the shop except, loosely, the owner. The median lands in the middle of where people actually are. B gives a true reason for the wrong answer: the mean does use every number, and that is normally its advantage. Here it is the reason it fails. A method being thorough does not make it appropriate.

4. D — correct arithmetic, misleading claim. Nobody added wrong. The advertisement takes a real figure and lets it stand for something it does not stand for, which is what a person applying for a bindery job would earn. B assumes that a misleading claim must contain a false number. Most of them do not. The number is usually correct and the framing does the work — which is the same shape as the bike lane in Quiz 1 and the water filter in Quiz 5.

5. C — 40. 100 − 60 = 40. The bracket under the row shows the same thing. A is Class A's range, on the row above. When two data sets share a picture, check which row the question named.

6. A — same mean, different spread. Both rows average 80, and the dashed line proves it by passing through the middle of each. But in Class A everyone is within two points of the average, and in Class B the lowest and highest students are forty points apart. Those are two different rooms and two different teaching problems. B and C both try to rank the classes, which the data will not support. Equal means, and nothing here says which spread is preferable. That is a judgment about what you want, not a fact the numbers contain.

7. C — 60%. 24 red out of 40 total. 24 ÷ 40 = 0.6, which is 60 percent, or three fifths. A takes the 24 and adds a percent sign. D is 25 ÷ 40, which is what you get from miscounting by one. The denominator is the total of everything, not the total of the other group.

8. A — 36%. The seed went back in, so the second draw faces the same packet as the first. Independent events multiply: 0.6 × 0.6 = 0.36. D adds instead of multiplying and produces 120 percent, which is impossible and should stop you before you finish reading it. Nothing is more certain than certain. Any probability above 1, or above 100 percent, is a signal to go back.

9. B — no, small runs wander. Six red out of ten was the expected number, and seven is one more than that. Getting seven when you expect six is roughly as unremarkable as flipping a coin ten times and seeing six heads. C treats the expected proportion as a quota that the seeds are obliged to meet. They are not. If the gardener planted all 40 the count would land much closer to 24, and if she planted 4,000 it would land closer still. The prediction gets more reliable as the numbers get bigger, which is the same reason a study needs a large enough sample.

10. D — nothing yet. Four tall plants from a 3-to-1 cross happens roughly a third of the time by chance alone. It is not evidence of anything having gone wrong. She needs a great many more plants before the count means much. C is the belief that a ratio must reproduce itself exactly in every small batch, and it is the single most common misunderstanding in genetics. Hold onto this item. Part IV is built on it, and students who arrive there expecting three tall and one short in every family of four spend a long time confused.

Your Score

What the number means

8 to 10	Solid. Check topic 7 on your map. That completes Part I.
6 or 7	Close. Read the whole key, then take this again in a few days before you check the box.
5 or fewer	Worth real time, and it pays twice: every one of these ideas appears on the mathematics test as well.

Misses on 1, 2, 5, or 7 are arithmetic and vocabulary, and both are learnable in an afternoon. Misses on 3, 4, or 6 mean a summary number was allowed to stand for a group it does not describe. And a miss on 9 or 10 is the one to sit with, because the belief that a probability owes you a result is the misunderstanding that will cost the most points later.

That is Part I

Seven topics, and none of them was a science fact. You have not yet been asked to know what a mitochondrion does or why the seasons change. What you have been building is the equipment: how to read a passage for what it says rather than what you assume, how an investigation is put together and how it fails, how to read four kinds of chart and four kinds of diagram, how to tell a finding from an explanation, how to keep a unit attached to a number, and how to spot an average that is describing nobody.

From here the content starts. Part II opens on the cell. Every quiz after this one leans on the seven behind it, which is why they came first.

The plate for this quiz

The gallery image is Florence Nightingale's diagram of army mortality in the Crimea, drawn in 1858. It is remembered as a nursing story and it is really a statistics story: she was elected the first woman member of the Royal Statistical Society that same year.

The diagram is a circle divided into months, with a wedge for each. The area of each wedge shows how many soldiers died, and the shading separates deaths from wounds received in battle from deaths caused by preventable disease inside the hospitals. The disease wedges dwarf the others. She could have published a table with the same numbers in it and been ignored. She published a shape that could be understood across a room by men who did not want to hear it, and the sanitary reforms followed. It is public domain and available at high resolution.