Free SAT Math Problem-Solving and Data Analysis

SAT Math · Problem-Solving and Data Analysis · free lesson

Bar Charts, Histograms, and Dot Plots

The idea, then worked examples solved step by step, then practice questions whose numbers change every time you try them.

The idea

Pictures of counted data

A bar chart, a dot plot, a histogram and a line graph all turn a table of numbers into heights. In every one of them, the labels along the bottom say what is being counted or measured, and the height says how many or how much. Most mistakes come from mixing the two.

Bar chart

A bar chart has one bar for each category. Read a bar by going from its top straight across to the scale on the left. A class voted on where to go for a trip:

The lines of the scale are 2 apart, so a bar that ends halfway between 8 and 10 stands for 9. The Beach bar ends halfway between 16 and 18. How many more votes did the Beach get than the Museum, and how many students voted?

Dot plot

A dot plot puts one dot above the number line for each value in the data. Eighteen students said how many hours they slept. Watch their answers arrive one at a time:

When all 18 dots are in place, the column above 5 has five dots: five students slept 5 hours. The number under a column is a value. The number of dots in the column is its frequency. No dot stands above 7, so nobody slept exactly 7 hours. The engine counts the same list:

Those are the frequencies of 3, 4, 5 and so on up to 10 hours.

Warning.

"How many students slept 8 hours?" asks for a frequency: count the dots above 8. "What is the greatest number of hours any student slept?" asks for a value: read the label under the last column that has a dot. The tallest column is the most common value, not the greatest one.

Histogram

A histogram looks like a bar chart, but its bottom scale is a number line cut into intervals, and the bars touch. The height of a bar is the number of values that fall inside its interval. Here a value on a boundary belongs to the bar on its right. Twenty people timed themselves on a puzzle. Drag the slider to change the width of the intervals: the data stay the same and the picture changes.

With intervals 10 minutes wide the bars are 2, 5, 7, 4 and 2 high. You can say how many people took at least 40 minutes: add the last two bars. You cannot say what the slowest time was, only that it is between 50 and 60.

For the same reason a histogram does not fix the mean. The mean is smallest if every value sits at the left end of its interval, and it must be less than what you get with every value at the right end:

\begin{aligned} atleast &= 4 + 2 \\ &= 6 \\ leastmean &= \frac{2 \times 10 + 5 \times 20 + 7 \times 30 + 4 \times 40 + 2 \times 50}{20} \\ &= 29.5 \\ topmean &= \frac{2 \times 20 + 5 \times 30 + 7 \times 40 + 4 \times 50 + 2 \times 60}{20} \\ &= 39.5 \end{aligned}

So the mean time is at least $29.5$ minutes and less than $39.5$ minutes, and the picture says no more than that.

Line graph

A line graph joins the points in order, usually over time, so that you can see change. The highest point is the greatest value. The steepest piece is the greatest change. A park counted its visitors, in hundreds, in each of eight months:

The most visitors came in month 4 and the fewest in month 1. The greatest rise from one month to the next is the steep piece from month 3 to month 4.

Note.

On the SAT these are usually the first or second question of a module: the height of one bar, how many items had a given value, where a line graph is highest or lowest, the frequency of one value of a dot plot. The rare hard one gives two histograms and asks what can be said about their means.

Worked examples

Each example is one of the kinds of question the test asks about a chart. The numbers in the working are found by the engine.

Example.

1. The height of a bar, and a difference. The bar chart shows the number of pizzas a shop sold on each of five days. How many did it sell on Wednesday? How many more did it sell on Friday than on Tuesday?

The lines of the scale are 5 apart. Wednesday's bar ends a little above the line for 30, one fifth of the way to 35: it stands for 31. Friday's bar ends at 42 and Tuesday's at 18.

\begin{aligned} more &= 42 - 18 \\ &= 24 \end{aligned}

The shop sold 31 pizzas on Wednesday, and $24$ more on Friday than on Tuesday. When a bar ends between two lines, the test chooses its options far enough apart that a careful reading settles it.

Example.

2. A bar chart of frequencies. A hockey team recorded the number of goals it scored in each game of a season. In how many games did the team score exactly 2 goals? How many games did it play?

The bottom labels are numbers of goals. The heights are numbers of games. Find 2 on the bottom and read the height of its bar: 6 games. Every game is in exactly one bar, so the number of games is the sum of all the heights.

\begin{aligned} games &= 3 + 7 + 6 + 2 + 1 \\ &= 19 \end{aligned}

The team scored exactly 2 goals in 6 games and played $19$ games. Answering "2" to the first question gives back the label, and answering "7" gives the tallest bar. Neither is the bar above 2.

Example.

3. A dot plot. The dot plot shows the scores of 18 students on a 10-point quiz. How many students scored 8? How many scored 8 or higher?

Each dot is one student. The column above 8 has 6 dots, so 6 students scored 8. For "8 or higher", add the columns above 8, 9 and 10.

\begin{aligned} eightormore &= 6 + 3 + 2 \\ &= 11 \end{aligned}

$11$ students scored 8 or higher. The tallest column, above 8, shows the most common score. The greatest score is 10, the label under the last column that has a dot.

Example.

4. A histogram. The histogram shows the heights, in centimeters, of the 30 plants in a greenhouse. A height on a boundary is counted in the bar to its right. How many plants are less than 30 centimeters tall? Can you tell the height of the tallest plant?

The bars are 3, 8, 11, 6 and 2 high. The plants shorter than 30 centimeters are in the two bars to the left of 30.

\begin{aligned} shorter &= 3 + 8 \\ &= 11 \end{aligned}

$11$ plants are less than 30 centimeters tall. The tallest plant is one of the 2 in the last bar, so its height is at least 50 and less than 60 centimeters. The histogram does not say more than that: the single values are gone, and only the count in each interval is left.

Example.

5. A line graph. The line graph shows the number of bikes a shop rented in each of seven weeks. In which week did it rent the most? Between which two weeks next to each other did the number fall the most?

The highest point is above week 4. A fall is a piece of the graph that goes down to the right. There are two: from week 2 to week 3, and from week 4 to week 5. The steeper one is the larger fall.

\begin{aligned} most &= \operatorname{max}(bikes) \\ &= 68 \\ fallone &= 50 - 44 \\ &= 6 \\ falltwo &= 68 - 41 \\ &= 27 \end{aligned}

The most bikes, $68$ , were rented in week 4, and the largest fall, $27$ bikes, is from week 4 to week 5. The answer to "in which week" is a label from the bottom scale, 4, not the height $68$ .

Example.

6. Two histograms and their means. Each histogram shows the ages, in whole years, of the 20 members of a club. A bar from 10 to 20 holds the ages 10 to 19, and so on. What is the smallest possible difference between the mean age of club B and the mean age of club A?

Club A's bars are 8, 7 and 5 high, and club B's are 4, 9 and 7 high, one interval farther to the right. The ages themselves are not shown, so neither mean is fixed. Club B is the older club. The two means are closest when club A is as old as its bars allow (every age at the top of its interval: 19, 29, 39) and club B is as young as its bars allow (every age at the bottom: 20, 30, 40).

\begin{aligned} greatestA &= \frac{8 \times 19 + 7 \times 29 + 5 \times 39}{20} \\ &= 27.5 \\ leastB &= \frac{4 \times 20 + 9 \times 30 + 7 \times 40}{20} \\ &= 31.5 \\ smallest &= leastB - greatestA \\ &= 31.5 - 27.5 \\ &= 4 \end{aligned}

The smallest possible difference is $4$ years. Taking the middle of every interval for both clubs gives a larger number: that is one possible difference, not the smallest.

Practice

Answer SAT-style questions on bar charts, histograms, and dot plots. Each answer is checked, with the full solution, and every try brings new numbers. Practice needs a free account: it keeps your progress.

Practise bar charts, histograms, and dot plots with new numbers