Data Visualization and Statistical Thinking

Lesson: Categorical vs. Numerical Data

When we collect information, we call that information data.

Data helps us answer questions. For example, we might ask, “What is the class’s favorite fruit?” or “How tall are the plants?”

There are two important kinds of data we will learn about today:

• Categorical data
• Numerical data

Knowing the difference helps us sort, count, compare, and make charts.

1. What is categorical data?

Categorical data is data that is put into groups or categories.

These categories tell what kind of thing something is, not how much.

Examples of categorical data are:

• favorite color
• type of pet
• kind of weather
• favorite lunch choice

If students say their favorite color is red, blue, or green, those are categories. We can count how many students chose each one.

2. What is numerical data?

Numerical data is data that uses numbers.

These numbers tell how many, how much, or how big.

Examples of numerical data are:

• height of a plant in inches
• number of books read
• how many apples are in a basket
• how long a jump is

If four plants are 3 inches, 5 inches, 5 inches, and 7 inches tall, that is numerical data because the information is given with numbers.

3. A simple way to tell the difference

Ask yourself this question:

• Is the data telling which group something belongs to? Then it is categorical.
• Is the data telling a number that can be counted or measured? Then it is numerical.

Another way to think about it:

• Categorical = names of groups
• Numerical = numbers

4. Why does this matter?

Different kinds of data help us answer different questions.

Categorical data helps us compare groups. We might ask, “Which lunch choice is the most popular?”

Numerical data helps us compare amounts or sizes. We might ask, “Which plant is tallest?” or “How many books did students read?”

Both kinds of data can be shown in charts, tables, and pictures. But first, we need to know what kind of data we have.

Worked Example 1: Sorting data by type

Look at each example and decide if it is categorical or numerical.

1. Favorite ice cream flavor
2. Number of crayons in a box
3. Type of shoe
4. Length of a pencil

Step-by-step thinking:

• Favorite ice cream flavor: This is a group name like chocolate or vanilla. Categorical.
• Number of crayons in a box: This tells how many. Numerical.
• Type of shoe: This is a group such as sneakers, boots, or sandals. Categorical.
• Length of a pencil: This is measured with a number. Numerical.

Answer:

• 1: Categorical
• 2: Numerical
• 3: Categorical
• 4: Numerical

Worked Example 2: Looking at class data

A teacher asks students for their favorite fruit.

The answers are:

apple, banana, apple, orange, banana, apple

Is this categorical or numerical data?

Step-by-step thinking:

• The answers are names of fruits.
• Fruit names are categories.
• So this is categorical data.

We can count each category:

• apple: 3
• banana: 2
• orange: 1

This helps us see that apple is the most chosen fruit.

Worked Example 3: Measuring plant heights

Four plants have heights of 2 inches, 4 inches, 3 inches, and 5 inches.

Is this categorical or numerical data?

Step-by-step thinking:

• The data uses numbers: 2, 4, 3, and 5.
• These numbers tell how tall the plants are.
• Height is something we measure.
• So this is numerical data.

We can compare the numbers:

• Shortest plant: 2 inches
• Tallest plant: 5 inches

We can also find the difference between the tallest and shortest plant:

$$5 - 2 = 3$$

The tallest plant is 3 inches taller than the shortest plant.

Worked Example 4: One question, two kinds of data

Suppose we study classroom pets.

Question A: What kind of pet is each one?

Answers: fish, hamster, fish, turtle

This is categorical data because the answers are group names.

Question B: How old is each pet?

Answers: 1 year, 2 years, 1 year, 3 years

This is numerical data because the answers use numbers.

This shows that even when we are talking about the same thing, we can collect different kinds of data.

5. Let’s compare categorical and numerical data

• Categorical data: tells the kind, type, or group
• Numerical data: tells the number, amount, or size

Here are some more examples:

• Favorite season → Categorical
• Number of students in class → Numerical
• Color of a car → Categorical
• Weight of a pumpkin → Numerical
• Kind of bird seen outside → Categorical
• How many steps you walked → Numerical

6. Watch out for this tricky idea

Sometimes we count categories, but the original data is still categorical.

For example, if students choose favorite colors, the original answers are:

red, blue, red, green, blue

Those answers are categorical because they are color groups.

After we count them, we may write:

• red: 2
• blue: 2
• green: 1

The survey question was about categories, so the data is still categorical data.

7. How this helps with graphs and charts

When data is categorical, we often make a chart to compare how many are in each group.

When data is numerical, we often line up the numbers to compare which are greater, smaller, longer, taller, or heavier.

For example:

• Categorical data can help us see which choice is most popular.
• Numerical data can help us see which measurement is greatest.

8. Try these on your own

Decide whether each one is categorical or numerical:

1. Favorite sport
2. Number of pets
3. Size of a T-shirt
4. How many minutes you read

Answers:

• 1: Categorical
• 2: Numerical
• 3: Categorical
• 4: Numerical

Summary

Data is information we collect.

Categorical data puts information into groups, like favorite color or type of pet.

Numerical data uses numbers to tell how many or how much, like height, age, or number of books.

If the answer is a group name, it is categorical. If the answer is a number, it is numerical.

Now you can look at data and decide what kind it is!

Constructing Frequency Tables

Sometimes we collect information, like favorite fruits, pets, or the number of books students read. At first, the information may look messy because it is just a list of answers. A frequency table helps us organize that information so it is easy to read.

A frequency table shows what the answers are and how many times each answer appears. The word frequency means how often something happens.

When we make a frequency table, we usually use three parts:

• Category — the answer or item
• Tally — quick marks to count
• Frequency — the total number

Here is what tally marks look like:

•  1 is |
• 2 is ||
• 3 is |||
• 4 is ||||
• 5 is ||||/ because the fifth mark goes across the first four

Grouping tally marks in fives makes counting faster and easier.

How to make a frequency table

1. Look at the list of data.
2. Find the different categories.
3. Make a tally mark each time you see a category.
4. Count the tally marks.
5. Write the total in the frequency column.

Let us try it step by step.

Example 1: Favorite Pets

A class was asked, “What is your favorite pet?” The answers were:

dog, cat, dog, fish, dog, cat, bird, dog, fish, cat

First, find the categories:

• dog
• cat
• fish
• bird

Now add tally marks.

Dogs: 4 times

Cats: 3 times

Fish: 2 times

Bird: 1 time

Here is the frequency table:

This table helps us see that dog was the most popular pet.

Example 2: Number of Books Read

Here is a list showing how many books some students read last month:

2, 1, 3, 2, 4, 2, 1, 3, 2, 1, 4, 2

The categories are the numbers of books:

• 1 book
• 2 books
• 3 books
• 4 books

Now count each one:

• 1 appears 3 times
• 2 appears 5 times
• 3 appears 2 times
• 4 appears 2 times

Here is the frequency table:

We can tell that 2 books happened most often.

Check your work

After making a frequency table, it is smart to check it. Add all the frequencies together. The total should match the number of data items in the list.

In Example 2, the frequencies are:

$$3+5+2+2=12$$

There were 12 data items in the list, so the table is correct.

Example 3: Favorite School Lunch

A survey asked 15 students to pick their favorite lunch:

pizza, taco, pizza, sandwich, taco, pizza, pizza, sandwich, taco, salad, pizza, taco, sandwich, taco, pizza

Step 1: Find the categories.

• pizza
• taco
• sandwich
• salad

Step 2: Count with tally marks.

• Pizza: 6
• Taco: 5
• Sandwich: 3
• Salad: 1

Here is the frequency table:

Check the total:

$$6+5+3+1=15$$

That matches the 15 students, so the table is correct.

Tips for making a good frequency table

• Write each category only once.
• Be careful to tally every answer.
• Count tally marks slowly so you do not skip any.
• Use groups of 5 tally marks when needed.
• Check that the total frequency matches the total number of answers.

Example 4: Measuring Pencil Lengths by Group

A teacher sorted pencil lengths into groups. The data showed:

short, medium, short, long, medium, short, medium, medium, long, short, medium, short

The categories are:

• short
• medium
• long

Count each category:

• Short: 5
• Medium: 5
• Long: 2

Frequency table:

Check the total:

$$5+5+2=12$$

This tells us there were 12 pencils altogether.

Why frequency tables are helpful

Frequency tables make data easier to understand. They help us answer questions like:

• Which category has the most?
• Which category has the least?
• How many answers are there in all?

They also help us get ready to make picture graphs or bar graphs later.

Summary

A frequency table is a chart that shows categories and how many times each one appears. We can make one by listing the categories, using tally marks to count, and writing the total frequency. Always check your work by adding the frequencies to make sure they match the number of data items.

Scaled Picture Graphs help us show data using pictures. A picture graph uses small icons or drawings to stand for numbers.

Sometimes, one picture stands for just 1 thing. But in a scaled picture graph, one picture stands for more than 1. This is called the scale.

For example, if 1 star means 2 books, then 3 stars mean 6 books. We find the total by multiplying or skip-counting.

$$3 \times 2 = 6$$

Learning to read scaled picture graphs helps us answer questions like:

• How many are there in all?
• Which group has more?
• How many more does one group have than another?

Parts of a Scaled Picture Graph

• Title tells what the graph is about.
• Labels name each group.
• Pictures or icons show the data.
• Key tells what each picture means.

Always look at the key first. The key is very important because it tells the value of each picture.

Here is a small example of a key:

Key: 1 apple picture = 4 apples

If a row has 2 apple pictures, that does not mean 2 apples. It means:

$$2 \times 4 = 8$$

How to Read a Scaled Picture Graph

1. Read the title.
2. Look at the labels.
3. Check the key to see what each picture means.
4. Count the pictures in one row.
5. Multiply the number of pictures by the scale, or skip-count by the scale.

Worked Example 1: Reading a Simple Scaled Picture Graph

A class keeps track of pets.

Key: 1 picture = 2 pets

• Dogs: 4 pictures
• Cats: 3 pictures
• Fish: 2 pictures

Let’s find how many pets are in each group.

Dogs: 4 pictures, and each picture means 2 pets.

$$4 \times 2 = 8$$

So there are 8 dogs.

Cats: 3 pictures, and each picture means 2 pets.

$$3 \times 2 = 6$$

So there are 6 cats.

Fish: 2 pictures, and each picture means 2 pets.

$$2 \times 2 = 4$$

So there are 4 fish.

Now let’s compare.

• Which group has the most? Dogs
• Which group has the fewest? Fish
• How many more dogs than fish? $$8 - 4 = 4$$ So, 4 more dogs.

Worked Example 2: Finding the Total

A snack shop sold fruit cups.

Key: 1 picture = 5 fruit cups

• Monday: 2 pictures
• Tuesday: 4 pictures
• Wednesday: 3 pictures

First, find each day’s total.

Monday:

$$2 \times 5 = 10$$

Monday has 10 fruit cups.

Tuesday:

$$4 \times 5 = 20$$

Tuesday has 20 fruit cups.

Wednesday:

$$3 \times 5 = 15$$

Wednesday has 15 fruit cups.

Now find the total sold on all 3 days.

$$10 + 20 + 15 = 45$$

The snack shop sold 45 fruit cups in all.

Worked Example 3: Comparing Two Groups

A library tracked books read by 3 groups.

Key: 1 book picture = 3 books

• Group A: 5 pictures
• Group B: 3 pictures
• Group C: 4 pictures

Find the total for each group.

Group A:

$$5 \times 3 = 15$$

Group B:

$$3 \times 3 = 9$$

Group C:

$$4 \times 3 = 12$$

Now answer some questions.

How many more books did Group A read than Group B?

$$15 - 9 = 6$$

Group A read 6 more books than Group B.

How many books did Group B and Group C read together?

$$9 + 12 = 21$$

They read 21 books together.

Worked Example 4: Making a Scaled Picture Graph

Now let’s make one.

Suppose these are the numbers of balloons sold:

• Red: 8
• Blue: 12
• Green: 4

We want to draw a scaled picture graph. Choose a scale that matches all the numbers well.

A good choice is:

Key: 1 balloon picture = 4 balloons

Now divide each total by 4 to find how many pictures to draw.

Red:

$$8 \div 4 = 2$$

Draw 2 pictures.

Blue:

$$12 \div 4 = 3$$

Draw 3 pictures.

Green:

$$4 \div 4 = 1$$

Draw 1 picture.

Your graph would have:

• Red: 2 pictures
• Blue: 3 pictures
• Green: 1 picture

This is a good scaled picture graph because each picture stands for 4 balloons, and the graph is easy to read.

Tips for Success

• Always check the key first.
• Do not just count pictures. Each picture may stand for more than 1.
• Use multiplication or skip-counting to find totals.
• Use subtraction to find how many more or fewer.
• Use addition to find how many in all.

Common Mistakes to Avoid

• Mistake: Saying 3 pictures means 3 items.
  Fix: Look at the key. If 1 picture = 2 items, then 3 pictures = 6 items.
• Mistake: Forgetting the scale when comparing groups.
  Fix: First find the real total for each group, then compare.
• Mistake: Picking a scale that does not fit the data when making a graph.
  Fix: Choose a number that works well with the totals.

Let’s Practice Thinking

If a graph shows 6 sun pictures, and the key says 1 sun = 2 sunny days, then the total is:

$$6 \times 2 = 12$$

So the graph shows 12 sunny days.

If another row shows 4 sun pictures, that means:

$$4 \times 2 = 8$$

To find how many more sunny days the first row has:

$$12 - 8 = 4$$

The first row has 4 more sunny days.

Summary

A scaled picture graph uses pictures to show data, and each picture stands for more than 1 item.

The key tells the scale. To read the graph, count the pictures and multiply by the number in the key.

Then you can add to find totals, subtract to compare groups, and even make your own scaled picture graphs.

Creating and Interpreting Scaled Bar Graphs

A bar graph is a picture that uses bars to show data. Data means information we collect, count, and compare.

In a bar graph, each bar stands for a category, such as favorite fruit, pets, books read, or types of weather. The height of each bar shows how many are in that category.

Sometimes the numbers in the data are small, so we can count by ones on the side of the graph. But sometimes the numbers are larger. Then we use a scaled bar graph.

A scaled bar graph is a bar graph where the scale does not count by ones. Instead, it might count by 2s, 5s, 10s, or another equal amount.

For example, a graph scale might go:

0, 2, 4, 6, 8, 10

or

0, 5, 10, 15, 20

Using a scale helps us fit larger numbers on a graph and read the data more easily.

Parts of a Scaled Bar Graph

• Title tells what the graph is about.
• Categories tell what each bar stands for.
• Scale shows the counting pattern on the side of the graph.
• Bars show the amount for each category.

When you look at a scaled bar graph, always ask:

1. What is the graph about?
2. What are the categories?
3. What number does each step in the scale stand for?
4. How high is each bar?

Why the Scale Matters

If the scale counts by 1s, then each line means 1. If the scale counts by 5s, then each line means 5. This changes how we read the height of the bars.

For example, if a bar reaches the line marked 15 on a graph that counts by 5s, then the value is 15, not 3.

How to Read a Scaled Bar Graph

1. Read the title.
2. Look at the labels on the bottom to find the categories.
3. Check the scale on the side.
4. Match the top of each bar to the correct number on the scale.
5. Compare the bars to answer questions.

You can answer many kinds of questions from a graph:

• Which category has the greatest value?
• Which category has the least value?
• How many more is one category than another?
• How many in all?

How to Make a Scaled Bar Graph

1. Collect the data.
2. List the categories.
3. Choose a scale that fits the largest number.
4. Draw the graph with a title and labels.
5. Draw one bar for each category up to the correct value.

When choosing a scale, pick equal steps that make the graph easy to read. If your greatest number is 24, a scale counting by 2s or 5s could work well.

Worked Example 1: Reading a Simple Scaled Bar Graph

A class asked students which snack they liked best. The bar graph uses a scale counting by 2s.

• Apples: 8
• Pretzels: 4
• Popcorn: 10
• Crackers: 6

Let’s answer some questions.

1. Which snack was chosen the most?

Popcorn has 10. That is the greatest number, so Popcorn was chosen the most.

2. Which snack was chosen the least?

Pretzels has 4. That is the smallest number, so Pretzels was chosen the least.

3. How many more students chose Popcorn than Crackers?

Find the difference:

$$10 - 6 = 4$$

So, 4 more students chose Popcorn than Crackers.

4. How many students were asked in all?

Add all the choices:

$$8 + 4 + 10 + 6 = 28$$

So, 28 students were asked.

Worked Example 2: Paying Attention to the Scale

A bar graph shows how many books students read in one month. The scale counts by 5s:

• Lena: 10
• Marco: 15
• Sara: 20
• Jay: 5

Because the graph counts by 5s, each line stands for 5 books.

1. How many books did Sara read?

Sara’s bar reaches 20, so she read 20 books.

2. How many more books did Marco read than Lena?

$$15 - 10 = 5$$

Marco read 5 more books than Lena.

3. How many books did Lena and Jay read together?

$$10 + 5 = 15$$

Together, they read 15 books.

Important reminder: If Jay’s bar goes to the first line on a graph counting by 5s, that means 5, not 1.

Worked Example 3: Making a Scaled Bar Graph

Suppose we have this data about pets owned by students:

• Dogs: 12
• Cats: 8
• Fish: 4
• Birds: 16

Let’s make a scaled bar graph.

Step 1: Choose a title.

A good title is Pets Owned by Students.

Step 2: Write the categories.

The categories are Dogs, Cats, Fish, and Birds.

Step 3: Choose a scale.

The greatest number is 16. A scale counting by 2s works well:

0, 2, 4, 6, 8, 10, 12, 14, 16

Step 4: Draw the bars to the correct heights.

• Dogs goes to 12
• Cats goes to 8
• Fish goes to 4
• Birds goes to 16

Now the graph is easy to read and compare.

Which category has the tallest bar?

Birds has 16, so Birds has the tallest bar.

How many more dogs than fish are there?

$$12 - 4 = 8$$

There are 8 more dogs than fish.

Worked Example 4: Using a Graph to Compare and Find Totals

A graph shows the number of cans collected for a food drive. The scale counts by 10s.

• Class A: 30
• Class B: 50
• Class C: 40
• Class D: 20

1. Which class collected the most cans?

Class B collected 50 cans, which is the greatest number.

2. Which class collected the fewest cans?

Class D collected 20 cans, which is the least number.

3. How many more cans did Class B collect than Class D?

$$50 - 20 = 30$$

Class B collected 30 more cans than Class D.

4. How many cans were collected in all?

$$30 + 50 + 40 + 20 = 140$$

So, 140 cans were collected altogether.

Tips for Success

• Always check the scale first.
• Make sure the scale goes up by equal amounts.
• Read the title to know what the graph shows.
• Look carefully at where the top of each bar ends.
• Use addition to find totals.
• Use subtraction to find how many more or how many fewer.

Common Mistakes to Avoid

• Forgetting the scale: On a graph counting by 5s, the second line is 10, not 2.
• Unequal scale steps: A scale must be even, like 0, 5, 10, 15.
• Reading the wrong category: Make sure each bar matches the correct label.
• Skipping labels: A graph should have a title, categories, and a scale.

Let’s Review

A scaled bar graph uses bars to show data, but the scale may count by numbers other than 1. We can use scaled bar graphs to show larger numbers clearly.

To read a scaled bar graph, look at the title, categories, and scale. Then match each bar to its value.

To make a scaled bar graph, choose a good scale, label the graph carefully, and draw bars to the correct heights.

When you understand the scale, you can compare categories, find differences, and add to find totals.

Scaled Bar Graphs help us show data when the numbers are too big to count by 1s easily.

In a regular bar graph, each line on the side might stand for 1. But in a scaled bar graph, each line can stand for 2, 5, 10, or another equal amount.

This helps us fit larger numbers on the graph and read the data more quickly.

What is a bar graph?

A bar graph uses bars to show how many are in each group. The length or height of each bar tells the number for that group.

A bar graph usually has:

• A title that tells what the graph is about
• Labels for the groups
• A scale that tells what each mark means
• Bars that show the data

What does scaled mean?

Scaled means the numbers on the graph go up by the same amount each time.

For example, a scale might count:

• , 4, 6, 8, 10
• 0, 5, 10, 15, 20
• 0, 10, 20, 30, 40

Each time, the graph goes up by an equal step.

If the scale is counting by 5s, then each line stands for 5.

If the scale is counting by 10s, then each line stands for 10.

Why do we use scaled bar graphs?

We use scaled bar graphs when the numbers are large.

Imagine making a graph for 50 books. Counting every line by 1 would take too much space. Counting by 5s or 10s makes the graph neat and easy to read.

How to read a scaled bar graph

1. Look at the title.
2. Read the labels for each group.
3. Check the scale on the side.
4. Find where the top or end of each bar reaches.
5. Use the scale to tell the number.

Important: Do not just count the lines. You must read what each line stands for.

For example, if the scale goes 0, 5, 10, 15, 20, then a bar reaching 15 means 15, not 3.

How to make a scaled bar graph

1. Collect the data.
2. Choose a scale that fits the numbers well.
3. Draw the side numbers using equal steps.
4. Write the group names.
5. Draw one bar for each group.
6. Make sure each bar matches the correct number.

Choosing a good scale

A good scale should:

• Use equal jumps, like 2, 5, or 10
• Be large enough to fit the greatest number
• Make the graph easy to read

Suppose your data numbers are 4, 8, 12, and 14.

You could count by 2s:

$$0, 2, 4, 6, 8, 10, 12, 14$$

Suppose your data numbers are 10, 25, 30, and 40.

You could count by 5s or 10s.

Worked Example 1: Reading a simple scaled bar graph

A class counts how many apples, bananas, and oranges students chose for snack.

The graph scale counts by 2s:

$$0, 2, 4, 6, 8, 10$$

The bars show:

• Apples: 6
• Bananas: 8
• Oranges: 4

Questions:

• How many students chose bananas?
• Which fruit was chosen the most?

Answer:

The bananas bar reaches 8, so 8 students chose bananas.

The tallest bar is bananas, so bananas were chosen the most.

Worked Example 2: Comparing amounts

A scaled bar graph shows the number of books read by four students. The scale counts by 5s:

$$0, 5, 10, 15, 20, 25$$

The bars show:

• Ava: 10
• Ben: 15
• Cara: 20
• Diego: 5

Question 1: Who read the most books?

Answer: Cara read the most because 20 is the greatest number.

Question 2: How many more books did Ben read than Diego?

We subtract:

$$15 - 5 = 10$$

Ben read 10 more books than Diego.

Question 3: How many books did Ava and Diego read altogether?

Add the numbers:

$$10 + 5 = 15$$

Together, Ava and Diego read 15 books.

Worked Example 3: Making a scaled bar graph

Here is some data about pets owned by students:

• Dogs: 12
• Cats: 8
• Fish: 4
• Birds: 10

Step 1: Choose a scale.

The largest number is 12. A scale counting by 2s works well:

$$0, 2, 4, 6, 8, 10, 12$$

Step 2: Write the labels.

The groups are Dogs, Cats, Fish, and Birds.

Step 3: Draw the bars.

• Dogs goes to 12
• Cats goes to 8
• Fish goes to 4
• Birds goes to 10

Step 4: Check your work.

Make sure the scale uses equal steps and each bar ends at the correct number.

Worked Example 4: Be careful with the scale

A graph shows favorite games. The scale counts by 10s:

$$0, 10, 20, 30, 40$$

The soccer bar reaches 30.

Question: Does that mean 3 students or 30 students?

Answer: It means 30 students, because each line stands for 10.

This is a common mistake. Always read the scale first.

Tips for success

• Always check what the scale counts by.
• Each step must be equal.
• Read the bar all the way to the number it reaches.
• Use addition or subtraction to compare groups.
• If you make a graph, choose a scale that fits the largest number.

Things to watch out for

• Thinking each line means 1 when it really means 2, 5, or 10
• Using a scale that does not go high enough
• Skipping numbers in a way that is not equal
• Reading the wrong bar label

Lets practice thinking

If a graph counts by 5s and a bar reaches 25, then the value is 25.

If one bar is 20 and another is 15, then the difference is:

$$20 - 15 = 5$$

If two bars are 10 and 30, then the total is:

$$10 + 30 = 40$$

Summary

A scaled bar graph is a bar graph where the scale counts by equal amounts greater than 1, such as 2, 5, or 10.

Scaled bar graphs help us show larger numbers in less space.

To read one correctly, first check the scale, then read where each bar ends.

To make one, choose a good scale, label the graph, and draw each bar to match the data.

Line Plots for Fractional Data

Sometimes we measure things, like pencils, leaves, or pieces of string. The measurements are not always whole numbers. A pencil might be \(3\frac{1}{2}\) inches long, or a leaf might be \(2\frac{1}{4}\) inches long.

A line plot helps us show this data on a number line. We place an X above each measurement. When more than one item has the same measurement, we stack the Xs.

In this lesson, you will learn how to read and make line plots with halves and quarters.

1. What is fractional data?

Fractional data means measurements that include fractions. In 3rd grade, we often use:

• halves: \(\frac{1}{2}\)
• quarters: \(\frac{1}{4}\) and \(\frac{3}{4}\)

These are the small parts between whole numbers.

Between 1 and 2, the fractional marks are:

\(1,\ 1\frac{1}{4},\ 1\frac{1}{2},\ 1\frac{3}{4},\ 2\)

Each step of \(\frac{1}{4}\) is one quarter more.

2. Reading the number line

Before making a line plot, we must read the number line carefully. Look at the spaces between whole numbers.

If there are 4 equal parts between 1 and 2, then each part is worth \(\frac{1}{4}\).

That means:

• the first mark after 1 is \(1\frac{1}{4}\)
• the second mark after 1 is \(1\frac{1}{2}\)
• the third mark after 1 is \(1\frac{3}{4}\)

If the number line is marked by halves, then the middle mark between 1 and 2 is \(1\frac{1}{2}\).

3. How to make a line plot

Follow these steps:

1. Read all the measurements.
2. Look at the number line and make sure you know what each mark means.
3. Put one X above the correct measurement for each item.
4. If the same measurement happens more than once, stack the Xs above each other.

Each X stands for one piece of data.

4. How to read a line plot

To read a line plot:

• look at where the Xs are placed
• count how many Xs are above each number
• find which measurement happens the most
• find which measurement happens the least
• find the total number of items

A line plot helps us answer questions about data quickly.

Worked Example 1: Plotting data with halves

Here are the lengths of 5 pencils in inches:

\(2\frac{1}{2},\ 3,\ 2\frac{1}{2},\ 3\frac{1}{2},\ 3\)

We use a number line marked by halves:

\(2,\ 2\frac{1}{2},\ 3,\ 3\frac{1}{2}\)

Now place the Xs:

• \(2\frac{1}{2}\): 2 pencils
• \(3\): 2 pencils
• \(3\frac{1}{2}\): 1 pencil

The line plot would look like this:

\(2\)    \(2\frac{1}{2}\)    \(3\)    \(3\frac{1}{2}\)
       X       X
       X       X       X

What do we learn?

• The most common lengths are \(2\frac{1}{2}\) inches and \(3\) inches.
• There are 5 pencils total.

Worked Example 2: Plotting data with quarters

Here are the lengths of 6 leaves in inches:

\(1\frac{1}{4},\ 1\frac{3}{4},\ 1\frac{1}{2},\ 1\frac{1}{4},\ 2,\ 1\frac{1}{2}\)

The number line is marked in quarters:

\(1,\ 1\frac{1}{4},\ 1\frac{1}{2},\ 1\frac{3}{4},\ 2\)

Count each measurement:

• \(1\frac{1}{4}\): 2 leaves
• \(1\frac{1}{2}\): 2 leaves
• \(1\frac{3}{4}\): 1 leaf
• \(2\): 1 leaf

The line plot would look like this:

\(1\)   \(1\frac{1}{4}\)   \(1\frac{1}{2}\)   \(1\frac{3}{4}\)   \(2\)
    X         X
    X         X            X      X

What do we learn?

• \(1\frac{1}{4}\) inch and \(1\frac{1}{2}\) inch are the most common lengths.
• There are 6 leaves total.

5. Be careful with the spaces

A very important idea is that the marks on the number line must be equal spaces. If the line is divided into quarters, each jump is \(\frac{1}{4}\).

From \(2\) to \(3\), the quarter marks are:

\(2,\ 2\frac{1}{4},\ 2\frac{1}{2},\ 2\frac{3}{4},\ 3\)

From \(4\) to \(5\), the quarter marks are:

\(4,\ 4\frac{1}{4},\ 4\frac{1}{2},\ 4\frac{3}{4},\ 5\)

Do not guess. Count the spaces carefully.

Worked Example 3: Answering questions from a line plot

A class measures ribbons. The line plot shows:

• \(2\frac{1}{4}\): 1 ribbon
• \(2\frac{1}{2}\): 3 ribbons
• \(2\frac{3}{4}\): 2 ribbons
• \(3\): 1 ribbon

Question 1: How many ribbons are there in all?

Add the Xs:

$$1+3+2+1=7$$

There are 7 ribbons.

Question 2: Which length is most common?

The tallest stack is above \(2\frac{1}{2}\).

So the most common length is \(2\frac{1}{2}\) inches.

Question 3: How many ribbons are longer than \(2\frac{1}{2}\) inches?

Lengths longer than \(2\frac{1}{2}\) are \(2\frac{3}{4}\) and \(3\).

Add those counts:

$$2+1=3$$

So 3 ribbons are longer than \(2\frac{1}{2}\) inches.

Worked Example 4: Making a line plot from measurement data

Here are the lengths of 8 crayons in inches:

\(4\frac{1}{4},\ 4\frac{1}{2},\ 4\frac{1}{4},\ 4\frac{3}{4},\ 4\frac{1}{2},\ 5,\ 4\frac{1}{2},\ 4\frac{3}{4}\)

Step 1: Write the number line marks.

\(4,\ 4\frac{1}{4},\ 4\frac{1}{2},\ 4\frac{3}{4},\ 5\)

Step 2: Count each length.

• \(4\): 0 crayons
• \(4\frac{1}{4}\): 2 crayons
• \(4\frac{1}{2}\): 3 crayons
• \(4\frac{3}{4}\): 2 crayons
• \(5\): 1 crayon

Step 3: Plot the Xs.

\(4\)   \(4\frac{1}{4}\)   \(4\frac{1}{2}\)   \(4\frac{3}{4}\)   \(5\)
     X         X         X
     X         X         X
               X             X

What do we learn?

• The most common length is \(4\frac{1}{2}\) inches.
• There are 8 crayons total.
• No crayon is exactly \(4\) inches long.

6. Tips for success

• Read the scale first. Make sure you know if the line is marked by halves or quarters.
• Plot one X for each item. Do not skip any data.
• Stack Xs neatly. This makes the plot easy to read.
• Count carefully. Check your total number of Xs.
• Look for patterns. Which measurements happen the most? Which happen the least?

7. What line plots help us do

Line plots help us organize measurement data. They make it easier to compare lengths and answer questions.

With a line plot, we can:

• find the total number of items
• find the most common measurement
• compare two measurements
• see which measurements are greater or less

Summary

A line plot shows data on a number line using Xs. When the data has fractions, we must read the number line carefully to see if it is marked in halves or quarters.

To make a line plot, place one X above each measurement and stack Xs when measurements are the same. To read a line plot, count the Xs and use them to answer questions about the data.

Extracting Statistical Measures means looking at a graph and finding important facts about the data. We can look for the mode, the range, outliers, and clusters.

These words may sound big, but the ideas are simple. They help us answer questions like: Which number happens the most? How far apart are the smallest and biggest numbers? Is there a number far away from the others? Are many numbers gathered together in one place?

When we study a graph, we are not just reading it. We are also thinking about what the data is telling us.

Let’s learn the 4 main statistical measures.

• Mode: the number or category that appears the most.
• Range: the difference between the greatest number and the smallest number.
• Outlier: a value that is far away from the other values.
• Cluster: a group of values close together.

1. Finding the Mode

The mode tells us what shows up the most. On a bar graph, it is often the tallest bar. On a picture graph or line plot, it is the value with the most marks.

If we have the numbers \(2, 3, 3, 4, 5\), then 3 is the mode because it appears more than the other numbers.

2. Finding the Range

The range tells us how spread out the data is. To find the range:

1. Find the smallest value.
2. Find the greatest value.
3. Subtract the smallest from the greatest.

For example, if the smallest number is \(2\) and the greatest number is \(8\), then:

$$8 - 2 = 6$$

So the range is 6.

3. Finding an Outlier

An outlier is a value that does not seem to fit with the others because it is much smaller or much greater than most of the data.

For example, in \(4, 5, 5, 6, 20\), the number 20 is an outlier. It is far away from the rest of the numbers.

4. Finding a Cluster

A cluster is a group of data values that are close together. A cluster shows where many of the data points are gathered.

For example, in \(2, 3, 3, 4, 8\), the numbers \(2, 3, 3, 4\) make a cluster because they are close together.

How to read a graph for these measures

When you look at a completed graph, try these steps:

1. Look for the tallest bar or the most marks to find the mode.
2. Look for the smallest and greatest values to find the range.
3. See if one value is far away from the others to find an outlier.
4. Look for values bunched together to find a cluster.

Worked Example 1: Finding the Mode

A line plot shows how many books students read last month:

\(1, 2, 2, 3, 4\)

Let’s find the mode.

• The number \(1\) appears 1 time.
• The number \(2\) appears 2 times.
• The number \(3\) appears 1 time.
• The number \(4\) appears 1 time.

The number that appears the most is 2.

Mode = 2

This means 2 books was the most common number of books read.

Worked Example 2: Finding the Range

A graph shows the heights of plants in inches:

\(3, 5, 5, 6, 7\)

Let’s find the range.

• Smallest value: \(3\)
• Greatest value: \(7\)

Now subtract:

$$7 - 3 = 4$$

Range = 4

This means the plant heights are spread across 4 inches.

Worked Example 3: Finding an Outlier

A graph shows how many minutes students spent reading:

\(10, 11, 12, 12, 13, 25\)

Most of the values are between \(10\) and \(13\). But \(25\) is much bigger than the rest.

So 25 is the outlier.

This tells us one student read much longer than most of the others.

Worked Example 4: Finding a Cluster and More Than One Measure

A completed graph shows the number of marbles 6 students have:

\(4, 5, 5, 6, 6, 12\)

Let’s study the data.

• Mode: \(5\) and \(6\) each appear 2 times, so both are most common.
• Range: greatest is \(12\), smallest is \(4\).

$$12 - 4 = 8$$

• Outlier: \(12\), because it is far from the others.
• Cluster: \(4, 5, 5, 6, 6\), because these values are close together.

This data has a group of values together and one value far away.

Helpful Tips

• Mode means most. Both words start with m.
• For range, remember: biggest minus smallest.
• An outlier looks lonely because it is far from the group.
• A cluster looks like a bunch of values packed together.

Things to watch out for

• Do not confuse the greatest value with the mode. The mode is the most common, not always the biggest.
• Do not find the range by counting all the numbers. Find it by subtracting.
• Not every set of data has an outlier.
• A data set can have one cluster, more than one cluster, or no clear cluster.

Let’s Review

When you look at a graph, ask yourself these questions:

• Which value happens the most? That is the mode.
• What are the smallest and greatest values? Subtract to get the range.
• Is one value far away from the others? That may be an outlier.
• Are many values close together? That is a cluster.

These ideas help us understand data better. They help us describe what a graph is showing in a smart and careful way.

Solving Graph-Based Word Problems means using information from a graph to answer questions.

Graphs help us see data quickly. A graph can show how many books students read, how many apples were sold, or how many votes each pet got.

When we solve graph-based word problems, we often need to:

• read the graph carefully
• find the numbers
• add to find a total
• subtract to find how many more or how many fewer
• compare two or more groups

Let’s learn how to do this step by step.

Step 1: Read the title.

The title tells what the graph is about.

Step 2: Look at the labels.

The labels tell what each bar, picture, or line stands for.

Step 3: Look at the scale.

The scale tells what each mark means. Sometimes each mark means 1. Sometimes each mark means 2, 5, or 10.

For example, if the scale counts by 2s, then the numbers may be:

$$0, 2, 4, 6, 8, 10$$

Step 4: Find the number for each category.

Read the graph carefully so you know the correct amount.

Step 5: Solve the question.

Ask yourself:

• Do I need to add?
• Do I need to subtract?
• Do I need to compare more than two groups?

Main Idea 1: Find the total by adding.

If a problem asks how many in all or how many altogether, you add.

Example: If one bar shows 4 and another bar shows 6, then the total is

$$4 + 6 = 10$$

Main Idea 2: Find how many more by subtracting.

If a problem asks how many more, subtract the smaller number from the larger number.

Example: If one category has 9 and another has 5, then

$$9 - 5 = 4$$

So one category has 4 more.

Main Idea 3: Solve multi-step problems.

Sometimes you must do more than one step.

You may need to add two categories first, and then compare that total to another category.

For example:

Category A = 3, Category B = 4, Category C = 10

First add A and B:

$$3 + 4 = 7$$

Then compare with C:

$$10 - 7 = 3$$

So Category C is 3 more than Categories A and B together.

Worked Example 1: Reading a simple bar graph

A class made a graph of favorite fruits.

• Apples = 5
• Bananas = 3
• Grapes = 4

Question: How many students chose apples and grapes altogether?

Step 1: Find the two numbers.

Apples = 5 and Grapes = 4

Step 2: Add.

$$5 + 4 = 9$$

Answer: 9 students chose apples and grapes altogether.

Worked Example 2: How many more

A graph shows how many pets students have.

• Dogs = 8
• Cats = 6

Question: How many more students have dogs than cats?

Step 1: Find both numbers.

Dogs = 8, Cats = 6

Step 2: Subtract.

$$8 - 6 = 2$$

Answer: 2 more students have dogs than cats.

Worked Example 3: A scale that counts by 2s

A bar graph shows books read during one month. The scale goes up by 2s.

• Liam = 6 books
• Maya = 8 books
• Noah = 4 books

Question: How many books did Liam and Noah read in all?

Step 1: Read the scale carefully.

The graph counts by 2s, so a bar at 6 means 6 books, not 3.

Step 2: Find the numbers.

Liam = 6, Noah = 4

Step 3: Add.

$$6 + 4 = 10$$

Answer: Liam and Noah read 10 books in all.

Question: How many more books did Maya read than Noah?

Step 1: Find the numbers.

Maya = 8, Noah = 4

Step 2: Subtract.

$$8 - 4 = 4$$

Answer: Maya read 4 more books than Noah.

Worked Example 4: A multi-step comparison

A graph shows how many cans were collected for recycling.

• Class A = 7
• Class B = 5
• Class C = 15

Question: How many more cans did Class C collect than Class A and Class B together?

Step 1: Add Class A and Class B.

$$7 + 5 = 12$$

Step 2: Compare that total to Class C.

$$15 - 12 = 3$$

Answer: Class C collected 3 more cans than Class A and Class B together.

Helpful Tips

• Read the question last and first. First read it to know what to find. Read it again after looking at the graph.
• Circle or say the important numbers.
• Check the scale. If the graph counts by 2s, 5s, or 10s, be careful.
• Look for clue words.

Some clue words are:

• in all, altogether, total 1 add
• how many more, how many fewer, difference 1 subtract

Common Mistakes to Avoid

• Do not forget to read the title and labels.
• Do not use the wrong numbers from the graph.
• Do not forget to check whether the graph scale counts by 1s, 2s, 5s, or 10s.
• Do not add when the question asks how many more.
• For multi-step problems, do one step at a time.

Let’s Practice Thinking

If a graph shows:

• Red balloons = 9
• Blue balloons = 4
• Green balloons = 7

You can answer questions like these:

1. How many balloons are red and blue altogether?
   $$9 + 4 = 13$$
2. How many more red balloons than blue balloons are there?
   $$9 - 4 = 5$$
3. How many balloons are there in all?
   $$9 + 4 + 7 = 20$$
4. How many more green balloons than blue balloons are there?
   $$7 - 4 = 3$$

Summary

To solve graph-based word problems, first read the title, labels, and scale. Then find the numbers you need from the graph.

Next, decide whether to add to find a total, subtract to find how many more or fewer, or do more than one step. Work carefully, and always check that your answer matches the question.