Before you begin
Family 8 has two halves. Quiz 28 did the first: the mean, the median, the mode and the range — four numbers that stand in for a pile of numbers. This quiz does the second half, which is the pictures.
The pictures are worth more of your time than the formulas, for a plain reason: the GED asks about them constantly, and almost every question is answered by reading carefully rather than by calculating. There is exactly one piece of arithmetic in this whole quiz, and it comes at the end.
So the skill being built here is not a mathematical one in the usual sense. It is the skill of looking at a picture someone else made and working out what it actually shows — which, as the last section of this page argues, is not a small thing to be able to do.
Read the furniture first
Before you look at a single bar, read the parts of the graph that are not the data. There are four of them and it takes ten seconds.
The title. What is this a picture of?
Both axis labels. What is along the bottom, and what is up the side?
The units, and the scale. Dollars, or thousands of dollars? People, or percent? And — the important one — where does the number axis start? If it does not start at zero, every comparison your eye makes is wrong.
The key, when there is one. Which color or pattern is which.
Nearly every wrong answer in this family comes from skipping that ten seconds. The graph is designed to be readable; the question is designed to catch someone who did not read it.
Bar graphs: comparing separate things
A bar graph puts separate categories along one axis and a number along the other. The bars can stand up or lie sideways; it makes no difference.
To read a value, find the top of the bar and run your eye straight across to the number axis. If the top falls between two marked numbers, work out what the gap between marks is worth and estimate. On the test that estimate is usually enough, because the answer choices are far enough apart.
The things you will be asked: which is largest or smallest, what is the difference between two of them, how many altogether, how many more than, and sometimes the ratio of one to another. All of them are reading followed by one subtraction or one addition.
A double bar graph puts two bars in each category so you can compare two sets at once. The key tells you which is which, and it is the first thing to read.
With two sets there is a second kind of question: not just which day was busiest, but where the two sets disagree. Here they run together except on Wednesday, where last week beat this week — and that one reversal is exactly the sort of thing a question asks about.
Line graphs: change over time
A line graph nearly always has time along the bottom. It is the picture for one thing measured again and again, and the question it answers is what happened, and when.
You can read a single value the same way as on a bar graph. But the question the test prefers is about the steepness, not the height.
“Between which two years did the rent rise the most?” is not asking where the line is highest. It is asking where the line climbs most steeply. Here the line is highest at 2026, but the steepest climb is from 2022 to 2024, where the rent went up $250 in two years — more than it rose in the six years before that put together.
Histograms: how a pile of numbers is spread out
A histogram looks like a bar graph and is not one, and telling them apart matters.
The difference is what is along the bottom. A bar graph has separate categories — Monday, Tuesday, Wednesday — which could be put in any order without breaking anything. A histogram has a number line, cut into intervals. The intervals are in order because numbers are in order, and they cannot be rearranged.
That is why the bars of a histogram touch and the bars of a bar graph do not. The touching is not decoration. It is saying that there is no gap between 25 and 26 — the intervals run into one another and cover everything.
Each bar's height is how many values landed in that interval. So this histogram says seven students in the class are between 16 and 25, nine are between 26 and 35, and so on down to two who are over 55.
Dot plots: the same job, when the numbers are few
A dot plot puts one dot above the number line for each value, stacked when values repeat.
Because every value is a separate dot, you can do things here that a histogram will not let you. You can count the dots to find how many there are altogether — twelve. You can see the mode by looking for the tallest stack — three. You can find the median by counting in from both ends until you meet in the middle. And you can read the range straight off: from 1 to 7, so 6.
Dot plots are for small sets. Twelve values make a readable picture; two hundred would make a mess, and that is when a histogram earns its place.
Scatter plots: two things at once
Every graph so far has shown one measurement. A scatter plot shows two, and each dot is one thing measured twice.
Here each dot is one student: how many hours they studied, and what they scored. The dot at 5 hours and 88 is a person; so is the dot at 5 hours and 80. Two people studied the same amount and did not get the same result, which is worth noticing.
What you look for in a scatter plot is the direction, and there are only three answers.
If the dots tend to rise from left to right, the two things go up together — a positive relationship, which is what this one shows. If they tend to fall from left to right, one goes up as the other goes down — negative. The price of a used car against its age looks like that. And if the dots are a shapeless cloud with no lean at all, there is no relationship, which is also a finding.
The straight line drawn through the dots is the line of best fit. It does not pass through them all; it is the single line that comes closest to the lot of them, and it stands for the general trend with the individual variation smoothed away. Its use is prediction: run up from a value on the bottom axis to the line, then across to the side axis.
Four hours of study lands the line at about 74. That is not a promise about any particular person — the two students who studied 5 hours scored 80 and 88, neither of them exactly on the line. It is a statement about what the pattern suggests.
A pattern is not a cause
This gets its own heading because the test asks about it and because it is the single most useful thing in this family outside the exam room.
A scatter plot can show two things moving together. It cannot show that one makes the other happen.
The standard example: towns with more firefighters have more fire damage. The dots rise clearly. Nobody thinks firefighters start fires. The explanation is a third thing neither axis mentions — town size. Big towns have more of everything, including fires and the people paid to put them out.
When a question shows you a relationship and offers you a cause, it is usually checking whether you will reach for the third thing. Ask yourself what else might be going on that would produce this picture without one side causing the other.
The weighted mean
Now the one calculation, and it is a small change to something you already do.
An ordinary mean treats every value as counting once: add them up, divide by how many. A weighted mean lets some values count for more than others, which is what happens whenever a course syllabus says tests are 60% of the grade.
The method is: multiply each value by its weight, add all of those up, and divide by the total of the weights.
A course counts tests as 60, homework as 30, and attendance as 10. A student has 80 on tests, 90 on homework, 100 on attendance.
(80 × 60) + (90 × 30) + (100 × 10) = 4800 + 2700 + 1000 = 8500.
The weights add to 100, so divide by 100: the grade is 85.
The ordinary mean of 80, 90 and 100 would have been 90. The weighted mean is lower because the student's weakest grade is the one the course cares about most. That gap is the whole reason weighting exists.
Do not average the averages
Here is the same idea wearing a disguise, and it catches almost everybody the first time.
One class of 20 students averaged 70 on a test. Another class of 10 students averaged 85. What did all 30 students average?
It is tempting to say (70 + 85) ÷ 2 = 77.5. That is wrong, and it is wrong because the two classes are not the same size. Twenty people scoring 70 should pull the combined average down harder than ten people scoring 85 can pull it up.
Do it as a weighted mean, with the class sizes as the weights:
(70 × 20) + (85 × 10) = 1400 + 850 = 2250, and 2250 ÷ 30 = 75.
The real answer is 75, not 77.5. Averaging averages is only safe when the groups are the same size, and questions in this family go out of their way to make them different sizes.
The guard
Five things, and the first is worth the other four.
Read the axes, the units and the key before the data. Ten seconds.
Check where the number axis starts. If a bar chart's axis begins at 90 instead of 0, the bars are showing you only the top slice of each value, and their relative heights mean nothing. Here are the same four numbers — 92, 93, 95, 96 — drawn twice.
Nothing has been falsified. Both graphs carry the correct figures, and the second one labels its axis honestly. But in the first the four months are almost indistinguishable, which is the truth of it, and in the second April towers over January. A four-percent rise has been made to look like a tripling, using nothing but the choice of where to begin the axis. This is the most common way an honest number is made to tell a lie, and question 8 is about it.
Touching bars means histogram. Gaps mean separate categories. Knowing which one you are looking at tells you what questions it can answer.
Steepness, not height, for any question about increase or decrease.
Two things moving together is not one causing the other. Look for the third thing.
Three to study before you start
From the double bar graph above: on how many days did this week beat last week, and what was the largest gap in this week's favor?
Read the key first: gold is this week, gray-green is last week. Now go day by day and subtract.
Monday 40 against 30, this week ahead by 10. Tuesday 55 against 35, ahead by 20. Wednesday 35 against 50, behind by 15. Thursday 60 against 45, ahead by 15. Friday 75 against 65, ahead by 10.
So this week beat last week on four days, and the largest gap in its favor was 20 packages, on Tuesday. Going day by day and writing each difference down is slower than eyeing it and is the reason you get it right; the eye is poor at comparing two bars that are not side by side.
From the rent graph above: between which two listed years did the rent rise most, and by how much?
Take the differences in order. 2016 to 2018: 980 − 900 = 80. 2018 to 2020: 70. 2020 to 2022: 150. 2022 to 2024: 250. 2024 to 2026: 150.
The largest is 2022 to 2024, a rise of $250.
Notice you did not need the picture to answer it once you had the numbers — but the picture is how you would spot it in one glance, because that stretch is visibly the steepest. The graph is for seeing; the subtraction is for being sure.
A job pays differently for different shifts. Last week a worker did 20 hours at $18 an hour and 10 hours at $27 an hour. What was the average hourly pay for the week?
Not $22.50. The hours are the weights, and there are twice as many hours at the lower rate.
(18 × 20) + (27 × 10) = 360 + 270 = 630, and the total hours are 30, so 630 ÷ 30 = $21 an hour.
The check that this is right: the answer must sit between 18 and 27, and it must sit nearer to 18, because more of the week was spent there. It does.
Now you
Work on paper. A calculator is fine. Read the title, the axes and the key of every graph before you answer anything about it. Questions 6 and 10 have two parts.
- Use the bar graph below for questions 1 and 2. How many more packages were delivered on Friday than on Wednesday?
- How many packages were delivered over the five days altogether?
- Use the line graph below. Between which two listed years did the rent rise the least, and by how much?
- Use the histogram below. How many students in the class are under 36?
- A) 9
- B) 16
- C) 21
- D) 26
- Use the dot plot below. What is the mode of these household sizes, and what is the range?
- A course counts tests as 50% of the grade, projects as 40%, and attendance as 10%. A student scores 72 on tests, 95 on projects, and 100 on attendance. (a) What is the student's course grade? (b) What would the plain average of those three numbers have been?
Reading A reading rest stop, for the stubborn ones.
Ten minutes with the last section of this page — The Company, below the answer key — before you finish the quiz. It is about sixty hand-drawn charts carried to Paris in 1900 to argue with the world about what Black Americans had done in thirty-five years of freedom. It will not help you with question 7. It is the best case I know for why a graph is worth learning to read.
No photograph at this rest stop. The graphs in this quiz are all above, and the ones the reading describes are worth looking up afterward.
- Use the scatter plot below. Describe in a sentence what it shows, and use the line of best fit to estimate the score of a student who studies for 6 hours.
- A shop's newsletter prints a bar graph of monthly sales. The bars are labeled 92, 93, 95 and 96, and the number axis starts at 90. The headline says sales have almost doubled since January. Explain what is wrong, and say what the actual increase is.
- A scatter plot shows that towns with more firefighters have more fire damage each year. Which conclusion does the graph support?
- A) Firefighters cause fire damage.
- B) Hiring fewer firefighters would reduce fire damage.
- C) Larger towns tend to have both more fires and more firefighters.
- D) The data must have been collected wrongly.
- One class of 20 students averaged 70 on a test. A second class of 10 students averaged 85. (a) What is the average of the two class averages? (b) What is the actual average score of all 30 students? Say why the two answers differ.
Check your work
Sixty charts, carried to Paris
In 1900 the Exposition Universelle opened in Paris, a world's fair on a scale that is hard to picture now: fifty million visitors over seven months, the Eiffel Tower eleven years old and still the tallest thing anyone had built. Inside it, in a small room, hung an exhibit about Black Americans, and a large part of it was graphs.
They had been made in Atlanta by W. E. B. Du Bois, then a young professor at Atlanta University, working with his students. There were around sixty of them, drawn by hand in ink and paint on boards about the size of a poster. They are among the most striking data graphics anyone has ever made, and they were made to win an argument.
The argument was about what had happened in the thirty-five years since emancipation. The prevailing account, in the American press and in a good deal of what passed for science at the time, was that not much had, and that not much could. Du Bois's answer was not an essay. It was counting.
The charts counted how many Black Americans could read and write, and how that had changed decade by decade. They counted what people did for a living, and what they owned, and the value of the land they owned. They counted city and country populations, births and deaths, how households spent their money, how many were in school. One set was about Georgia alone; another covered the country.
And they were designed, in a way nothing else on this page is. Bars that spiral. Radiating wedges. Stacked blocks in red, gold and blue-green. On one famous board, a bar representing a quantity too large for the space does not shrink to fit — it runs to the edge and then folds back on itself, turning a corner and continuing, and then does it again, coiling around the board. The number was too big for the page, so the page gave way. Du Bois did not adjust the data to fit the drawing.
That decision is worth holding onto, because it is the opposite of the trick in question 8 of this quiz. A truncated axis makes a small difference look large. Du Bois's folded bar is a large quantity refusing to be made to look small.
He won a gold medal at the Exposition for his part in the exhibit. The boards came home and eventually reached the Library of Congress, where they sat largely unlooked-at for most of a century before being rediscovered, reproduced and studied. You can find them online now, and they are worth ten minutes of anyone's time.
Two things to take from it.
The first is what a graph actually is. Every one of these decisions — what to count, which years, whether to show a total or a rate, where to start the axis — is a choice made by a person, and the choices add up to a claim. A graph is an argument in a form that can be checked. That is its virtue and it is also why the ten seconds of reading the furniture matters: you are reading somebody's case, and you are entitled to see how it was built.
The second is about who made them. Du Bois is named on the work and deservedly. The students at Atlanta University who sat with the rulers and the ink and drew the lines are, for the most part, not named anywhere. The counting was theirs too. A great deal of what we know about the world arrived that way — assembled by people doing careful work for a wage or for a grade, whose names did not survive the filing.