Dot Plots and Frequency Tables | Grade 7 Math | LevelKite

Twelve students each answer one question: How many pets live in your home? Here are their answers, in the order they were collected:

1, 0, 2, 1, 3, 1, 0, 2, 1, 5, 2, 1

A list like this is a data set, and each number in it is a data value. Here, each data value is the number of pets for one student.

The list holds all the information, but patterns are hard to see in it. How many students have exactly 1 pet? Which answer is most common? In this lesson you will organize a data set in two ways, so that questions like these are quick to answer.

A dot plot (also called a line plot) shows a data set on a number line. Each data value becomes one mark, usually a dot, above its value on the line. Some plots draw an X instead of a dot, and it means the same thing. When a value appears more than once, its dots stack up in a column.

A dot plot tells you two things at a glance:

The position of a stack on the number line tells you which value it shows.
The height of a stack, which is its number of dots, tells you how many times that value appears.

A plot does not explain everything by itself. You also need to say what the number line measures and what one dot stands for. In this lesson, one dot always stands for exactly one data value.

Worked example

Create a dot plot of the pet data

Twelve students report how many pets live in their homes: 1, 0, 2, 1, 3, 1, 0, 2, 1, 5, 2, 1. Create a dot plot of this data set.

Find the least and greatest data values, because the number line must reach both of them. The least value is 0 pets and the greatest value is 5 pets.
Draw a number line from 0 to 5 in equal steps of 1 pet. Keep 4 on the line even though no student answered 4. Equal steps make distance on the line match the difference between values, so the empty place at 4 shows a real break in the data.
Go through the list in order. For each data value, put one dot above the matching number, and cross the value off the list so that every student gets exactly one dot. The value 0 appears 2 times, 1 appears 5 times, 2 appears 3 times, 3 appears once, 4 never appears and 5 appears once.
Check by counting all the dots. One dot stands for one student, so the number of dots must equal the number of data values, which is 12.
$2 + 5 + 3 + 1 + 0 + 1 = 12$

Answer

The finished dot plot, shown next, has a number line from 0 to 5 pets with stacks of 2, 5, 3, 1, 0 and 1 dots above 0, 1, 2, 3, 4 and 5, for 12 dots in all.

Line plot

The number line shows the number of pets, and each dot is one student. Above 0, 1, 2, 3, 4 and 5 pets, the stacks hold 2, 5, 3, 1, 0 and 1 dots: 12 dots for 12 students.

Now read the dot plot of the pet data instead of the list. Every answer comes from the position of a stack and its height.

How many students have exactly 1 pet? The stack above 1 has 5 dots, so 5 students do.
Which answer is most common? The stack above 1 is taller than every other stack, so 1 pet is the most common value. (If two stacks tied for tallest, both of their values would be most common.)
Is any value missing? No dots sit above 4, but there are dots above 3 and above 5. A gap is an empty stretch of the number line that has dots on both sides of it. This gap shows that the student with 5 pets stands apart from the others.
How many students have 2 or more pets? Add the heights of the stacks above 2, 3, 4 and 5: $3 + 1 + 0 + 1 = 5$ students. Adding works because each student is one dot in exactly one stack, so you count nobody twice.

A dot plot shows counts as heights. A table can show the same counts as numbers.

The frequency of a value is the number of times that value appears in the data set. In the pet data, the value 1 appears 5 times, so the frequency of 1 is 5. Because a frequency is a count, it is always a whole number that is 0 or greater.

A frequency table lists each value in one column and its frequency beside it. Many frequency tables also have a tally column. A tally is a count kept with strokes, one stroke for each item you count. As you read through the list, you draw one stroke in the row for each data value you meet. Tallying lets you read the list just once and still count every data value exactly one time.

Worked example

Create a frequency table of the book data

Twenty students report how many books they read last month: 2, 3, 1, 4, 2, 0, 3, 2, 6, 1, 3, 2, 4, 3, 2, 1, 6, 4, 2, 3. Create a frequency table of this data set.

Find the least and greatest values: 0 books and 6 books. Write each whole number from 0 to 6 in the first column, one row for each value, so that every data value has exactly one row where it belongs. Include a row for 5 too. Its frequency will show that no student read exactly 5 books.
Read the list once, from left to right. For each data value, draw one tally stroke in the row for that value, so you record each student exactly once. For example, the first three data values, 2, 3 and 1, put one stroke each in the rows for 2, 3 and 1.
Count the strokes in each row and write the count in the frequency column. Each stroke is one appearance of the value, so the count is its frequency: the number of students who read that many books. The rows for 0, 1, 2, 3, 4, 5 and 6 books get the frequencies 1, 3, 6, 5, 3, 0 and 2.
Check by adding the frequencies. You tallied each student once, in exactly one row, so the sum must equal the number of data values, which is 20.
$1 + 3 + 6 + 5 + 3 + 0 + 2 = 20$

Answer

The frequency table, shown next, gives the values 0, 1, 2, 3, 4, 5 and 6 books the frequencies 1, 3, 6, 5, 3, 0 and 2, and these frequencies add to 20, one for each student.

Frequency table: books read last month by 20 students

Books read (value)	Tally	Frequency (number of students)
$0$	$\vert$	$1$
$1$	$\vert\,\vert\,\vert$	$3$
$2$	$\vert\,\vert\,\vert\,\vert\,\vert\,\vert$	$6$
$3$	$\vert\,\vert\,\vert\,\vert\,\vert$	$5$
$4$	$\vert\,\vert\,\vert$	$3$
$5$	none	$0$
$6$	$\vert\,\vert$	$2$
Total	$20$ strokes	$1 + 3 + 6 + 5 + 3 + 0 + 2 = 20$

Key concept

One data value, one dot, one count

A dot plot and a frequency table organize a data set in the same way: each one counts every data value exactly once, under its own value.

In a dot plot, each data value is one dot above its value.
In a frequency table, each data value is one tally stroke, which adds 1 to the frequency of its value.

So when both displays show the same data set and one dot stands for one data value, the number of dots above a value equals the frequency of that value. The dot plot that follows shows the book data: its stacks hold 1, 3, 6, 5, 3, 0 and 2 dots, exactly the frequency column of the table. Under these conditions, the sum of the frequencies, the number of dots and the number of data values are all equal.

\text{sum of frequencies} = \text{number of dots} = \text{number of data values}

Line plot

The number line shows the number of books read, and each dot is one student. Above 0, 1, 2, 3, 4, 5 and 6 books, the stacks hold 1, 3, 6, 5, 3, 0 and 2 dots: 20 dots for 20 students.

Worked example

Interpret the book data

Use the frequency table of the book data. What percent of the 20 students read 4 or more books last month?

Decide which rows answer the question. "4 or more books" means the values 4, 5 and 6. The question asks about students, so read the frequencies in those rows: 3, 0 and 2. On the dot plot of the book data, these are the heights of the stacks above 4, 5 and 6.
Add the three frequencies. Each student is counted in exactly one row, so the sum counts each of these students once and nobody twice.
$3 + 0 + 2 = 5$
Compare this part with the whole class. The frequencies add to 20 students in all. Because 20 is not zero, you can write the part as a fraction of the whole: 5 students out of 20.
$\frac{5}{20}$
Write the fraction as a percent, which you know means a number out of 100. Multiply the numerator and the denominator by 5 to make the denominator 100. Multiplying both by the same nonzero number does not change the value of the fraction.
$\frac{5}{20} = \frac{5 \times 5}{20 \times 5} = \frac{25}{100} = 25\%$
Check with a second method. The table after this example writes every frequency as a fraction of 20 and as a percent. The rows for 4, 5 and 6 books share no students and are parts of the same 20 students, so you can add their percents.
$15\% + 0\% + 10\% = 25\%$

Answer

Five of the 20 students read 4 or more books last month, which is $25\%$ of the class.

Each frequency as a fraction and as a percent of the 20 students

Books read (value)	Frequency (number of students)	Fraction of the 20 students	Percent of the class
$0$	$1$	$\frac{1}{20} = \frac{5}{100}$	$5\%$
$1$	$3$	$\frac{3}{20} = \frac{15}{100}$	$15\%$
$2$	$6$	$\frac{6}{20} = \frac{30}{100}$	$30\%$
$3$	$5$	$\frac{5}{20} = \frac{25}{100}$	$25\%$
$4$	$3$	$\frac{3}{20} = \frac{15}{100}$	$15\%$
$5$	$0$	$\frac{0}{20} = \frac{0}{100}$	$0\%$
$6$	$2$	$\frac{2}{20} = \frac{10}{100}$	$10\%$
Total	$20$	$\frac{20}{20} = \frac{100}{100}$	$100\%$

The big idea: a dot plot and a frequency table both sort a data set by value and count each data value exactly once. In a dot plot, you read a value from the number line and its count from the height of the stack. In a frequency table, you read a value from one column and its count, the frequency, from the column beside it. With either display, you can see which values are most common, add counts to answer "how many" questions, and compare a count with the total as a fraction or a percent.

Next, you will create and interpret bar graphs, histograms and box plots.