Unit 1 · Topic 1.5 Beta

Graphical Representations for One Quantitative Variable

Dotplots and stemplots show every value of a small quantitative data set; histograms group values into equal bins and show the count in each, and their look depends on the bin width.

Practice 3: Analyze Data

Part 1 · Hook

Why this matters

A teacher asks 25 students how many books they read over the summer. A list of 25 numbers says little. A row of dots over a number line says a lot at once: most students read 2 to 4 books, a few read many more, and one read none. Graphs for quantitative data put the values on a number line so you can see where they fall.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which variable is quantitative?

Number of books read
Favorite author
Library card (yes or no)
Grade level label

Show the answer

Books read is a count, so arithmetic with it makes sense.

Correct: Number of books read:
Favorite author:
Library card (yes or no):
Grade level label:

2. Of 25 students, 12 read four books or more. What proportion is that?

0.48
0.12
2.08
0.25

Show the answer

Part over whole: 12/25 = 0.48.

Correct: 0.48:
0.12:
2.08:
0.25:

Part 4 · See it

See it first

Minutes 30 students spent on homework, from 12 to 75, shown three ways on the same scale from 10 to 80 minutes. Top: a dotplot with one dot per student. Middle: a histogram with bins 5 minutes wide, which shows more detail. Bottom: a histogram with bins 10 minutes wide, which is smoother; the tallest bin, 30 to 40 minutes, holds 9 students. — The same 30 homework times as a dotplot and as histograms with bins 5 and 10 minutes wide. Narrow bins show detail; wide bins give a smoother picture. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Quantitative values have positions on a number linea dotplot puts one dot per value above its place on the line
With many values, dots pile upa histogram groups values into equal bins and draws a bar for each bin’s count
The bins are chosen by whoever draws the graphthe same data can look bumpy with narrow bins or smooth with wide ones
Adding up the relative frequencies from the lefta cumulative relative frequency graph shows the proportion of values at or below each point

Part 6 · Key ideas

Key ideas

A dotplot shows each value as a dot above a number line; repeated values stack. Best for small data sets.
A stemplot splits each value into a stem (leading digits) and a leaf (last digit). It keeps every value; always give a key.
A histogram groups values into equal bins; each bar’s height is the bin’s count or relative frequency, and the bars touch. Its look depends on the bin width.
A cumulative relative frequency graph gives, at each value, the proportion of the data at or below that value. The proportion between two values is the difference of the heights.

Part 7 · Misconception

A common mistake

The wrong idea: A value that lands on a bin edge, like 4 books in bins 2 to 4 and 4 to 6, is counted in both bins.

What actually happens: Each value goes in exactly one bin. With bins that hold their left end, 4 goes in the 4 to 6 bin.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Books read over the summer

A teacher asked 25 students how many books they read over the summer. Each dot is one student (invented data).

1. How many of the students read more than 6 books?

Show the answer

Count the dots to the right of 6: one each at 7, 8, 9, 10 and 12, so 5 students.

7: More than 6 leaves out the two dots at 6.
Correct: 5: Right: one dot each at 7, 8, 9, 10 and 12.
6: 6 is a number of books on the axis; count the dots to the right of it.
12: 12 is the largest value, not a count of students.

2. What proportion of the 25 students read 0, 1, 2 or 3 books?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

Dots at 0, 1, 2 and 3: $1 + 2 + 4 + 5 = 12$ , so the proportion is $\frac{12}{25} = 0.48$ .

Answer: 0.48

3. The teacher makes a histogram with bins 0 to 2, 2 to 4, 4 to 6 and so on, where each bin holds its left end and not its right end. How many students fall in the 2 to 4 bin?

Show the answer

The 2 to 4 bin holds the values 2 and 3: $4 + 5 = 9$ students.

13: A value of 4 is the right end of the 2 to 4 bin, so it belongs in the 4 to 6 bin.
5: The bin starts at 2, so it also holds the four students who read 2 books.
2: The width of a bin is 2 books; the height is the number of students in it.
Correct: 9: Right: the bin holds the 4 students at 2 and the 5 at 3, and the students at 4 go in the next bin.

4. Why is a dotplot a good choice for these data?

There are few values, so every student can be seen
Dotplots work for categorical data such as favorite authors
A dotplot gets rid of the students who read the most books
A dotplot works for any data set, whatever the number of values

Show the answer

For a small quantitative data set, a dotplot shows every value on a number line.

Correct: There are few values, so every student can be seen: Right: with 25 values a dotplot shows each value and how often it occurs.
Dotplots work for categorical data such as favorite authors: A dotplot needs a number line, so it is for quantitative data.
A dotplot gets rid of the students who read the most books: A dotplot shows every value, including the largest ones.
A dotplot works for any data set, whatever the number of values: With hundreds of values the dots pile up; a histogram is then easier to read.

5. A histogram of 500 race finishing times uses bins 10 minutes wide. What happens if the bins are made 2 minutes wide?

The same bars as before, each bar 5 times as tall
More, shorter bars that show finer detail and more bumps
The number of runners changes, since bins change
Fewer, wider bars that give a smoother picture of the data

Show the answer

A histogram’s look depends on its bin width: narrow bins show detail and noise, wide bins smooth it away.

The same bars as before, each bar 5 times as tall: Narrower bins hold fewer values each, so the bars get shorter, not taller.
Correct: More, shorter bars that show finer detail and more bumps: Right: narrow bins split the data more finely, so small ups and downs appear.
The number of runners changes, since bins change: Every runner is still in exactly one bin; only the grouping changes.
Fewer, wider bars that give a smoother picture of the data: That is what wider bins do.

6. Why do the bars of a histogram touch, unlike the bars of a bar chart?

To show that the values were collected one after another in time
Its horizontal axis is a number line split into bins
Because a histogram shows a categorical variable
To make the histogram easier to color and to label

Show the answer

A histogram’s bins are consecutive intervals of a number line, so the bars touch; a bar chart’s categories are separate.

To show that the values were collected one after another in time: The order the data were collected is not shown in a histogram.
Correct: Its horizontal axis is a number line split into bins: Right: the bins cover a number line with no breaks, so the bars meet.
Because a histogram shows a categorical variable: A histogram shows a quantitative variable.
To make the histogram easier to color and to label: Touching bars come from the number line, not from the drawing style.

Part 9 · Summary

Summary

Dotplots and stemplots show every value of a small quantitative data set; histograms group values into equal bins and show the count in each, and their look depends on the bin width. A cumulative relative frequency graph shows the proportion of values at or below each point, so a proportion between two values is a difference of heights.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic