Before you begin
A week of hours, a month of rent, fifteen skulls in a museum drawer: sometimes you have many numbers and need one. This quiz is about four ways to get that one number, and about what each of them tells you and hides. Three of them, the mean, the median and the mode, each answer the question what is typical? in a different way. The fourth, the range, answers a different question: how spread out are these? The GED asks for all four by name, and it also asks the harder thing, which is to choose the right one and to read them off a table or a chart. Take this Guide slowly. Every one of these is a small idea, and the trouble comes from mixing them up.
The mean: the fair share
Rosa's hours for one week were 8, 6, 8, 9 and 4. If those thirty-five hours had been spread evenly over the five days, how many would each day have had? Add them up and divide by how many there are: 8 + 6 + 8 + 9 + 4 = 35, and 35 ÷ 5 = 7. The mean is 7 hours a day. It is Quiz 5's fair share: the total, dealt out equally. That is why the mean is the one you want whenever the total matters, and why, if you know the mean and the count, you can get the total back: 7 × 5 = 35.
The median: the middle, after sorting
Put the same five days in order: 4, 6, 8, 8, 9. The one in the middle is 8. That is the median: half the days were at or below it, half at or above. The first move is always the sort, and the sort is Quiz 2's ordering, nothing more. Skip it and you will pick the middle of the list as it happened to be written, which means nothing.
With an even count there is no single middle, so the median is halfway between the two middle values. Six numbers sorted, 9, 11, 12, 15, 15, 20, have 12 and 15 in the middle, and halfway between them is (12 + 15) ÷ 2 = 13.5. A median can be a number that is not in the list.
Why have a second kind of typical? Because the mean can be dragged. A landlord's block has eleven apartments: ten rent for about $1,800 and one penthouse rents for $14,000. The mean rent is over $2,900, and no one on the block pays anything like it. The median is $1,800, which is what living there costs. One wild value pulls the mean toward itself, and it cannot move the median past the next value in line. When you hear an average that seems too good or too bad to be true, ask which one it was.
The mode: the most common
The mode is the value that appears most often. In Rosa's week, 8 appears twice and the others once, so the mode is 8. A list can have no mode, if nothing repeats, or two modes, if two values tie. The mode is the only one of the four that works on things that are not numbers: the most common shoe size a store sells, the answer most people in a survey gave. It is also the flimsiest. In a small list, one repeat can make a mode out of a coincidence, and you will see that happen to fifteen skulls below.
The range: how spread out
The range is the largest value minus the smallest: Rosa's 9 − 4 = 5 hours. It says nothing about what is typical. It says how far the list stretches. Two workers can share a mean of 7 hours a day, one steady at 7, 7, 7, 7, 7 and one swinging 2, 12, 4, 10, 7, and the range is what tells them apart: 0 against 10.
Fifteen skulls, one question
In 1910 Ray Lankester had a question he could not settle by looking. Some okapi skulls seemed broad and some seemed narrow, and he wanted to know whether there were two kinds of okapi or one. So he measured fifteen skulls, lettered A to P, from museums in London, Edinburgh, Paris, Rome, Madrid, Genoa and Tervuren and two private collections, and for each one he divided the width across the eyes by the length of the face and printed the answer per hundred, the ratio of Quizzes 23 to 25. Here are his fifteen numbers, sorted.
| Skull | Ratio | Skull | Ratio | Skull | Ratio |
|---|---|---|---|---|---|
| M | 150.0 | F | 172.7 | A | 189.0 |
| H | 162.8 | L | 173.4 | E | 189.0 |
| N | 166.8 | P | 174.2 | O | 190.4 |
| G | 170.4 | D | 178.3 | J | 197.7 |
| K | 171.7 | B | 179.3 | C | 198.9 |
Read down each column in turn: the five smallest, then the middle five, then the five largest. The same three groups are the ones Example 3 adds.
Now the four numbers. The median is the eighth of fifteen: 174.2, skull P. The mode is 189.0, because two skulls share it and nothing else repeats. The range is 198.9 − 150.0 = 48.9. The mean takes the most work, and Example 3 does the addition in full: the fifteen add to 2,664.6, and 2,664.6 ÷ 15 = 177.64. Notice that the mean sits above the median. Skull M, in Rome, at 150.0 is the only value far from the rest, and it should pull the mean down, but the five broad skulls between 189 and 199 pull harder. The guard holds: 177.64 lies between 150.0 and 198.9.
And Lankester's question? Look at the dots, not the numbers. Two kinds of skull would make two clusters with a gap between them. These dots spread from 150 to 199 without any gap that splits them into two groups. The widest empty stretch is at the bottom, from 150.0 to 162.8, about thirteen units, and it sets one skull apart, not a kind; the next, from 179.3 to 189.0, is about ten; the next, from 190.4 to 197.7, about seven. Gaps that shrink step by step like that are the look of one spread. Broad and narrow turn out to be the two ends of one spread, not two kinds. The range is real; the two kinds are not. That is what a table and a chart are for on the GED: not to hold numbers, but to show a shape that a single number cannot.
Reading a table
The last skill is the plainest: finding the number a question asks for in a table someone else made. The table below is copied from the San Diego Zoo Wildlife Alliance's okapi fact sheet, which Quiz 27's rest stop pointed at. Each cell holds a range, smallest to largest, and every question about it is answered by reading the right row and the right column before doing any arithmetic.
| Measurement | Males | Females |
|---|---|---|
| Shoulder height | 1.40 to 1.55 m | 1.42 to 1.59 m |
| Weight | 180 to 260 kg | 240 to 356 kg |
The female shoulder heights run from 1.42 to 1.59 meters, so their range is 1.59 − 1.42 = 0.17 meters, seventeen centimeters, and that subtraction is Quiz 14's. Read the row, read the column, then compute. Most table errors on the test are not arithmetic; they are the right arithmetic on the wrong cell.
Three to study before you start
Diego counted the minutes he waited for the bus on seven mornings: 9, 14, 6, 11, 14, 8, 15. Sort first: 6, 8, 9, 11, 14, 14, 15. Mean: the seven add to 77, and 77 ÷ 7 = 11. Median: the fourth of seven is 11. Mode: 14 appears twice, so 14. Range: 15 − 6 = 9. Mean and median agree here because nothing in the list is wild, and the guard is satisfied: 11 sits between 6 and 15.
One brand of rice at six stores: $2.49, $2.79, $2.49, $3.19, $2.99, $2.49. Sorted: 2.49, 2.49, 2.49, 2.79, 2.99, 3.19. Mean: the six add to $16.44, and 16.44 ÷ 6 = $2.74. Median: two middles, 2.49 and 2.79, and halfway between them is (2.49 + 2.79) ÷ 2 = $2.64, a price no store charges. Mode: $2.49, three times. Range: 3.19 − 2.49 = $0.70. Three different answers to what does this rice cost?, and each is right about something: the mean is what you would pay per bag if you bought one at every store, the median is the middle of the market, and the mode is the price you are most likely to meet.
Adding fifteen decimals in one go invites mistakes, so add in threes of five. The five smallest: 150.0 + 162.8 + 166.8 + 170.4 + 171.7 = 821.7. The middle five: 172.7 + 173.4 + 174.2 + 178.3 + 179.3 = 877.9. The five largest: 189.0 + 189.0 + 190.4 + 197.7 + 198.9 = 965.0. Then 821.7 + 877.9 + 965.0 = 2,664.6, and 2,664.6 ÷ 15 = 177.64. Check it two ways: the mean lies between 150.0 and 198.9, and 177.64 × 15 = 2,664.6 gives the total back, as a mean always does.
Now you
Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. Before every median, sort. After every mean, check it lies inside the list.
- Four subway waits this week, in minutes: 4, 8, 6, 2. What was the mean wait?
- Seven quiz scores: 72, 88, 65, 90, 88, 75, 81. What is the median score?
- A landlord says the average rent on his block is $2,900. Ten of the eleven apartments rent for about $1,800; the penthouse rents for $14,000. Which measure would describe what an apartment there typically costs, and why?
- A) The median, because one very large value pulls the mean up but cannot pull the median past the next value in line.
- B) The mean, because it uses every one of the eleven rents.
- C) The range, because it shows how far apart the cheapest and dearest are.
- D) The landlord's figure, because an average is an average.
- A clinic logged the wait, in minutes, for six patients: 12, 15, 9, 20, 15, 11. What is the median wait?
- For the same six waits, what are the mode and the range?
- Rosa's tips on four days were $40, $55, $30 and $60. Her mean for the five-day week was $50. How much did she make on the fifth day?
ReadingA reading rest stop, for the stubborn ones.
The fifteen skulls are in the atlas itself, which you can page through on the Internet Archive. Turn to the plates numbered 13 to 17: three skulls to a plate, each with its ratio printed beneath it, the numbers this quiz has been using. The two pages before Plate 13 say where every skull came from.
The drawing above is from the same book, Plate 14, in a different library's scan; the link opens the American Museum of Natural History's copy.
- Using the dot plot in the Guide, how many of the fifteen skulls have a ratio above 180?
- Lankester wanted to know whether okapi skulls come in two kinds, broad and narrow. Which statement does the dot plot support?
- A) There are two kinds: the dots form two separate clusters with a clear gap between them.
- B) The mode, 189.0, shows that most skulls are alike.
- C) The mean, 177.64, is the size of a typical skull, so every skull is close to it.
- D) The dots spread from 150 to 199 without a clear gap, so the data do not show two separate kinds.
- Skull M, at 150.0, sits far below the others. Leave it out. What is the median of the remaining fourteen ratios?
- What is the range of all fifteen ratios?
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 6 | The mean, forward and backward | Add, then divide by the count. Mean times count gives the total back, which is how question 6 is solved. |
| 2, 4, 9 | The median | Sort first, every time. Odd count: the middle one. Even count: halfway between the two middles. |
| 5, 10 | Mode and range | The mode is the repeat; the range is largest minus smallest. Neither says what is typical. |
| 3, 8 | Choosing a measure; reading a shape | One wild value drags the mean, not the median. Two kinds show as two clusters with a gap. |
| 7 | Reading a chart | Count the dots, and count a stack as what it is: one dot per skull. |
Eight or more right: the four measures are in hand, and Quiz 29 goes back to Quiz 4's toolkit for the gold diagonal, the perfect squares. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. One habit for a week: whenever a number is called an average, ask which of the four it is and whether one wild value could be pulling it. Then come back to these same ten.
This is where the okapi leaves us. Next, opening a new leg called Creatures in Flight: Quiz 29 — Exponents. The gold diagonal from Quiz 4's multiplication table, the perfect squares, comes back, and with it the other half of what the giraffe owed Quiz 27: squares and cubes, written the way the test writes them.
One kind, or two
Lankester had reasons to want two kinds. In 1902 he had looked at Johnston's skull and skin, decided they were different enough from the others, and given them a name of their own, Okapia erikssoni, after the lieutenant in the Congo Free State's service who had obtained them. If there were two species, he had found one. So he did the honest thing, which was to measure. Fifteen skulls, one ratio each, all of it printed so anyone could check.
The fifteen numbers did not oblige him. They spread from 150 to 199 with no gap in them anywhere, the median sitting at 174, the mean at 178, and the only repeat a coincidence between two skulls in London. A range is not two kinds. Broad and narrow were the two ends of one animal's variation, the way tall and short are the ends of one species of person. The second species was quietly dropped, and every okapi alive is Okapia johnstoni, one kind.
That is worth carrying out of this quiz. A single number, a mean or a median, tells you where the middle of something is. Only the whole spread, laid out where you can see it, tells you what shape the thing has, and shape is what settles questions like Lankester's. The dot plot did in one glance what the names could not do in eight years. It is the same lesson the series keeps finding: put the record where it can be counted, and count it.
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