Read distributions and data displays

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Difficulty: Intermediate
Estimated time: 28 minutes
Domains: Problem Solving and Data Analysis
Techniques: Data-displaysFrequency-tablesDot-plotsHistogramsBox-plots

What you’ll learn

Tell what a table, dot plot, histogram, or box plot keeps and what it hides.
Read counts and histogram bars without slipping at the edges of a bar.
Read the five marked values on a box plot and the quarters between them.
Match a display to a possible data set without inventing details it doesn’t show.

Prerequisites

You are ready if you can read a number line and add small whole-number counts.

Why this matters on the SAT

Turn the picture into counts

The SAT can hand you a frequency table, dot plot, histogram, or box plot and ask for one exact fact. Most of the work is reading the picture correctly. Once you have, the math is small.

In this histogram, each bar covers an interval of values, and its height tells you how many values fall in that interval. Start by finding the bars that run from 20 up to, but not including, 40.

Solution to the example

“At least 20 and less than 40” covers exactly two bars: $20\le x<30$ and $30\le x<40$ . Their heights are $5$ and $4$ , so the count is

5+4=9.

Choice C is correct. You never needed the actual values. The bars had already done the counting, so your only jobs were picking the right bars and adding.

SAT example

Read a bar height as the number of values in its interval.

The histogram summarizes a data set. The four bars represent the intervals $10\le x<20$ , $20\le x<30$ , $30\le x<40$ , and $40\le x<50$ , from left to right.

How many values in the data set are at least 20 and less than 40?

A
$4$
B
$5$
C
$9$
D
$15$

Know what each display keeps

A frequency is a count: how many times a value shows up, or how many values land in an interval. All four displays show one variable, one kind of measurement such as quiz scores, but they keep different amounts of detail.

Start by seeing how the display shows frequency:

A frequency table writes each count as a number.
A dot plot draws one dot for each value, stacked when a value repeats.
A histogram draws a bar over each interval, and the bar’s height is the count.
A box plot doesn’t show counts at all. It marks five summary values instead.

Name the display first. That tells you what you can get back from it.

What each one-variable display tells you

Display	How it shows frequency	What you can get back	What it hides
Frequency table	A written count	Each value or group and how often it appears	The order the data was collected in
Dot plot	The number of dots above a value	Every value and how often it appears	The order the data was collected in
Histogram	The height of each interval’s bar	How many values fall in each bar	The exact values inside each bar
Box plot	It doesn’t. It marks the minimum, $Q_1$ , median, $Q_3$ , and maximum	Five summary values and the quarters between them	The individual values, how often each appears, and how many there are

Most display questions come down to one rule: if the display doesn’t show it, you can’t claim it.

Before you solve, run a quick scan:

Read the title and axis labels, so you know what was measured and in what units.
Check the scale. One tick might stand for more than one unit.
Check what the vertical axis shows. It could be a count, a frequency, a percent, or something else.
Put the question’s words into numbers before you add anything. “At least 20 and less than 40” becomes $20\le x<40$ .

Reading the display is often only the first step. When a question goes on to ask for a mean or median, a range or standard deviation, or what happens when values are corrected, added, or removed, see Find and interpret center, Reason with range and standard deviation, or Analyze changed data and outliers.

Example: Connect a dot plot to its data set

A dot plot keeps every single value in view. Where a stack sits tells you the value, and how tall it is tells you how many times that value appears. So you can always write the full list back out.

Worked example

The horizontal position is the score. The stack height is its frequency.

Each dot represents one student's quiz score. Which data set is represented by the dot plot?

A
$2,3,4,5,6,7$
B
$2,3,3,4,4,4,5,6,7$
C
$1,2,2,3,1,1$
D
$2,3,4,4,5,5,6,6,7$

Step 1

Give each dot one value

Each dot is one student’s score. So the one dot above $2$ means one student scored $2$ , and the two dots above $3$ mean two students scored $3$ .

Step 2

Write the list, left to right

Write each score once for every dot above it:

2,\ 3,3,\ 4,4,4,\ 5,\ 6,\ 7.

There are nine dots and nine entries. If those two numbers don’t match, you’ve skipped or doubled a dot.

Step 3

Find the matching choice

Only choice B has every score with the right number of repeats. Choice A lists each score once, so it loses the repeats. Choice D has nine entries, but the wrong repeats, such as two $5$ s where the plot shows one.

Check your understanding:

How many scores in the plot are greater than $3$ and less than $7$ ?

You can write the same plot as a frequency table. Each stack height becomes a count.

The quiz-score dot plot as a frequency table

Score	2	3	4	5	6	7
Frequency	1	2	3	1	1	1

Check your understanding:

Use the table to find how many scores are at least $4$ .

Common mistake:

It’s easy to see a stack of three dots above $4$ and write down $3$ . But that stack means the value $4$ , three times. Read the number under the stack first, then count the dots above it. When you’re done, check that your list has as many entries as there are dots.

Read histogram bars as intervals

A histogram keeps the counts but hides the exact values inside each bar’s interval, called a bin. Go back to the opening histogram. Its first bar has height $4$ , so four values lie somewhere from $10$ up to $20$ . They could be $11$ , $12$ , $12$ , and $19$ , or any other four numbers in that interval. The bar does not say they all equal $10$ , or $15$ , or any other single value.

What about a value that sits exactly on the line between two bars? The picture alone might not tell you, so use the inequalities or interval labels in the question. $20\le x<30$ includes $20$ but not $30$ , and $30\le x<40$ includes $30$ but not $40$ . So a value of $30$ goes in the bar for $30\le x<40$ .

Adding every bar height gives the number of values:

4+5+4+6=19.

But you can’t get the 19 values themselves back from this histogram. Many different data sets have these same four counts.

Check your understanding:

A histogram uses the bins $50\le x<60$ and $60\le x<70$ . Which bin contains $x=60$ ?

Common mistake:

A bar stands for a whole interval, not the value at its left edge, middle, or right edge. If your count comes out wrong, write the question’s condition as an inequality, find the bars that fit completely inside it, and add their heights. Your answer can never be more than the total of all the bars.

Read the five marked values on a box plot

A box plot starts from the data sorted from least to greatest, and it marks five values:

the minimum, at the left end of the left whisker;
the first quartile, $Q_1$ , at the left edge of the box;
the median, also called $Q_2$ , at the line inside the box;
the third quartile, $Q_3$ , at the right edge of the box; and
the maximum, at the right end of the right whisker.

These marks cut the sorted data into four parts, and each part holds about a quarter of the values, however long or short it looks. The box, from $Q_1$ to $Q_3$ , holds the middle half.

Both plots share one scale, so read each mark straight down to it.

The five marked values for each group

Group	Minimum	$Q_1$	Median	$Q_3$	Maximum
A	2	4	6	8	10
B	3	5	6	7	9

Check your understanding:

Which group’s box covers the wider stretch of hours?

Now for what a box plot leaves out. It doesn’t show the mean, the individual values, how often any value appears, or how many values there are. So don’t claim any of those unless the question gives you more information.

Common mistake:

Group A’s box is wider than Group B’s, so it can look like it holds more people. The width doesn’t tell you that. Each box holds the middle half of its own group. A longer box or whisker only means the marked values sit farther apart on the number line. Before you compare two lengths, name each one’s endpoints, like $4$ to $8$ and $5$ to $7$ .

Read it, tally it, or use Desmos

Pick the form that keeps what the question needs.

How to start an SAT data-display question

What you’re given	Best first move	Why
A finished table, dot plot, histogram, or box plot	Read it by eye.	Rebuilding it in Desmos adds steps and gives you a chance to mistype a value.
A short list of raw values	Tally it by hand, or write it in order.	You can see every value and check the count quickly.
A longer list you need to turn into a display	Use Desmos to build or check the display.	You type the list once, and dotplot, boxplot, or histogram draws it, so you don’t plot each mark by hand.

Desmos is also a good way to see how a display depends on the data: change one value, and the picture changes right away.

Here are the quiz scores from the dot plot example:

L=[2,3,3,4,4,4,5,6,7] stores the scores as a list named $L$ .
dotplot(L) draws one dot for each score.
Change the last $7$ to a $4$ . The dot at $7$ disappears, and the stack at $4$ grows.
Change it back to $7$ , then replace dotplot with boxplot. The dots are gone, because a box plot keeps only its five marked values.

You can also try histogram(L,1), where the $1$ sets the width of each bar. All three commands work in the current College Board graphing calculator.

Check your understanding:

Which display lets you read all nine scores back: the dot plot or the box plot?

Calculator loads as you approach

Edit the list, then try dotplot, boxplot, and histogram. Reset brings back the original dot plot.

Practice

The first problem is a straight count. The next two ask what a display can prove, which takes more care.

Count from a frequency table

Practice problem

The frequency table summarizes a data set.

Value	3	4	5	6	7
Frequency	2	5	4	3	1

How many values in the data set are greater than $4$ ?

Calculator loads as you approach

Add only the frequencies of the values that fit.

What a histogram guarantees

Practice problem

A bar tells you how many values fall in its interval, not what those values are.

The histogram summarizes a data set. The four bars represent the intervals $10\le x<20$ , $20\le x<30$ , $30\le x<40$ , and $40\le x<50$ , from left to right.

Which statement must be true?

Calculator loads as you approach

Read the bar heights as counts.

Compare two box plots

Practice problem

Use only the five marked positions and their shared scale.

The box plots summarize the weekly screen time for Group A and Group B.

Which statement must be true based only on the box plots?

Calculator loads as you approach

Compare the five marked values.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

In a frequency table, add the counts in the frequency row.
In a dot plot, where a stack sits is the value, and its height is how many times that value appears.
In a histogram, each bar counts the values in an interval. The exact values inside stay hidden.
In a box plot, read the minimum, $Q_1$ , median, $Q_3$ , and maximum. The box holds the middle half of the sorted values.
If the display doesn’t show it, you can’t claim it. That includes a mean, the number of values, exact values, and counts it doesn’t draw.
Read a finished display by hand. Use Desmos when raw data has to become a display, or to watch how a display changes when one value changes.

Next lesson

Find and interpret center

Use the exact data or frequencies in a display to calculate and interpret mean and median.

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Practice

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135 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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