Data Representation and Statistical Analysis

Distinguishing Categorical and Numerical Data

When we collect information, we call it data. Data helps us answer questions, compare things, and learn about the world.

There are two important kinds of data you need to know: categorical data and numerical data.

Learning the difference helps you know how to sort information, make charts, and understand what the data is telling you.

1. What is categorical data?

Categorical data tells about a group, type, or label. It names what something is like, but it does not tell a number you can count or measure.

These are often answers such as:

• favorite fruit
• eye color
• pet type
• day of the week
• shirt color

For example, if students say their favorite sport is soccer, basketball, or tennis, those answers are categories. They are names of groups.

2. What is numerical data?

Numerical data is data shown with numbers. It tells how many, how much, or how long.

Numerical data can be counted or measured.

These are often answers such as:

• age
• height
• number of books
• minutes spent reading
• how many pets

For example, if students say they have 2 pets, 1 pet, or 0 pets, those answers are numerical data because they are numbers.

3. A simple way to tell the difference

Ask yourself this question:

Is the answer a label or a number?

• If it is a label, type, or name, it is categorical data.
• If it is a number that tells how many or how much, it is numerical data.

Here is another helpful check:

• Categorical data: You sort it into groups.
• Numerical data: You can count it, compare the numbers, or put the numbers in order.

4. Words and numbers together

Sometimes data may include words and numbers, so you have to think carefully.

For example, a jersey number like 8 is written as a number, but it is really a label for a player. It is not telling how many or how much. So in that case, it acts like categorical data.

But if you ask, “How many goals did the player score?” and the answer is 8, then it is numerical data because it tells an amount.

5. Why this matters

Different kinds of data are shown in different ways.

• Categorical data is often shown in bar graphs or picture graphs because it compares groups.
• Numerical data is often listed, ordered, or shown on graphs that help us compare numbers.

If you know what kind of data you have, you can choose the best way to organize it.

Worked Example 1: Favorite ice cream flavor

A class is asked, “What is your favorite ice cream flavor?”

Some answers are:

• chocolate
• vanilla
• strawberry

Step 1: Look at the answers. Are they labels or numbers?

They are labels.

Step 2: Decide the type of data.

This is categorical data because the answers are groups or names of flavors.

Worked Example 2: Number of siblings

A class is asked, “How many siblings do you have?”

Some answers are:

• 0
• 1
• 3
• 2

Step 1: Look at the answers. Are they labels or numbers?

They are numbers.

Step 2: Ask what the numbers mean.

They tell how many siblings each student has.

Step 3: Decide the type of data.

This is numerical data because it tells an amount.

Worked Example 3: Sorting different data

Decide whether each one is categorical or numerical data.

1. favorite school subject
2. height in centimeters
3. type of pet
4. minutes to clean your room

1. favorite school subject

Possible answers: math, reading, science

These are names of groups, so this is categorical data.

2. height in centimeters

Possible answers: 130, 142, 138

These are numbers that measure height, so this is numerical data.

3. type of pet

Possible answers: dog, cat, fish

These are labels, so this is categorical data.

4. minutes to clean your room

Possible answers: 10, 15, 20

These are numbers that measure time, so this is numerical data.

Worked Example 4: A tricky one

Question: “What bus do you ride home?”

Answers: 12, 7, 3

These answers are written as numbers, but here they name which bus each student rides. They do not tell how many buses or how long.

So this is categorical data because the numbers are being used as labels.

6. Practice thinking

Try asking these questions to yourself:

• Does this answer tell what kind of thing it is?
• Or does it tell how many or how much?

If it tells what kind, it is probably categorical.

If it tells how many or how much, it is probably numerical.

7. Quick comparison chart

• Categorical data
  • shows labels or groups
  • answers questions like “Which kind?”
  • examples: color, animal, month, food
• Numerical data
  • shows numbers
  • answers questions like “How many?” or “How much?”
  • examples: 4 pencils, 12 minutes, 135 centimeters

8. Let’s check a few more

• shoe size → numerical
• favorite animal → categorical
• number of students in a class → numerical
• birthday month → categorical
• length of a pencil in centimeters → numerical
• kind of sandwich → categorical

Summary

Data is information we collect.

Categorical data tells about groups, labels, or types. Examples are favorite color, type of pet, and birthday month.

Numerical data tells numbers that show how many or how much. Examples are age, height, number of books, and time in minutes.

To decide which kind of data you have, ask: Is it a label or a number that tells an amount? That one question can help you choose the correct type of data.

Constructing Frequency Tables and Tally Charts

Sometimes we collect data by looking at answers, objects, colors, animals, or choices. When we have a lot of information, it helps to organize it so it is easy to read.

Two helpful tools for organizing data are a tally chart and a frequency table.

A tally chart uses tally marks to keep track of how many times something happens.

A frequency table shows each category and the number of times it appears. The number of times something appears is called its frequency.

In this lesson, you will learn how to make tally marks correctly, how to count by groups of five, and how to turn data into a neat chart or table.

1. What is a tally mark?

A tally mark is a quick way to count. We usually draw straight lines.

• 
  = 1 tally
• 
  
  = 2 tallies
• 
  
  
  = 3 tallies
• 
  
  
  
  = 4 tallies
• For 5 tallies, we draw a line across the first four: ||||/

This is important because tally marks are easier to count when they are grouped in fives.

So the number 8 would look like this:

||||/ |||

That means \(5 + 3 = 8\).

The number 12 would look like this:

||||/ ||||/ ||

That means \(5 + 5 + 2 = 12\).

2. What is a category?

A category is a group name in your data.

For example, if students choose a favorite fruit, the categories might be:

• Apple
• Banana
• Orange
• Grapes

Each time a fruit is chosen, you add one tally mark to that category.

3. How to make a tally chart

Follow these steps:

1. Look at the data.
2. Find the categories.
3. Make a row for each category.
4. Go through the data one item at a time.
5. Add one tally mark to the correct category for each item.
6. Count the tally marks to find the total for each category.

A tally chart often has columns like these:

• Category
• Tally
• Frequency

4. What is frequency?

The frequency is just the total number of tally marks in a category.

If the tally marks for cats are ||||/ ||, then the frequency is:

$$5 + 2 = 7$$

So 7 cats were counted.

Worked Example 1: Reading a small list

A class is asked to choose a favorite pet. The answers are:

Dog, Cat, Dog, Fish, Cat, Dog, Dog, Fish

Step 1: Find the categories.

• Dog
• Cat
• Fish

Step 2: Add tally marks.

Dog appears 4 times, Cat appears 2 times, and Fish appears 2 times.

Here is the tally chart:

Category - Tally - Frequency

• Dog - |||| - 4
• Cat - || - 2
• Fish - || - 2

This chart helps us see that Dog was the most popular pet.

Worked Example 2: Using groups of five

A teacher counts favorite ice cream flavors:

Chocolate = 7, Vanilla = 5, Strawberry = 3

Let us write tally marks for each number.

• Chocolate: \(7 = 5 + 2\) so the tally is ||||/ ||
• Vanilla: \(5\) so the tally is ||||/
• Strawberry: \(3\) so the tally is |||

Now we can write the frequency table.

Category - Tally - Frequency

• Chocolate - ||||/ || - 7
• Vanilla - ||||/ - 5
• Strawberry - ||| - 3

Because the tally marks are grouped, it is easy to count the totals.

Worked Example 3: Making a chart from raw data

Look at this data about favorite school subjects:

Math, Reading, Math, Science, Reading, Math, Science, Math, Reading, Art, Math, Art

Step 1: List the categories.

• Math
• Reading
• Science
• Art

Step 2: Count each category.

• Math = 5
• Reading = 3
• Science = 2
• Art = 2

Step 3: Write the tally chart.

Category - Tally - Frequency

• Math - ||||/ - 5
• Reading - ||| - 3
• Science - || - 2
• Art - || - 2

We can answer questions from the table.

• Which subject was chosen most? Math
• How many students chose Art? 2
• How many students answered in all?

Add the frequencies:

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

So 12 students answered.

Worked Example 4: Checking your work carefully

A box has colored counters. The colors are:

Red, Blue, Red, Green, Blue, Red, Yellow, Blue, Green, Red, Blue

Step 1: Categories

• Red
• Blue
• Green
• Yellow

Step 2: Count each one

• Red = 4
• Blue = 4
• Green = 2
• Yellow = 1

Step 3: Make the chart

Category - Tally - Frequency

• Red - |||| - 4
• Blue - |||| - 4
• Green - || - 2
• Yellow - | - 1

Step 4: Check the total

Add the frequencies:

$$4 + 4 + 2 + 1 = 11$$

Now count the data items in the list. There are 11 items.

The totals match, so the chart is correct.

5. Tips for making tally charts and frequency tables

• Read one item at a time. Do not rush.
• Put each tally in the correct row.
• Group tallies in fives. This makes counting easier.
• Count again to check.
• Add all frequencies to make sure they match the total number of data items.

6. Common mistakes to avoid

• Forgetting a category
• Putting a tally mark in the wrong row
• Not grouping the fifth tally correctly
• Counting the tally marks incorrectly
• Forgetting to check the total number of data items

7. How tally charts help us

Tally charts and frequency tables help us organize information clearly.

They help us answer questions like:

• Which category has the most?
• Which category has the least?
• How many are there altogether?
• How many more does one category have than another?

When data is organized, it is much easier to understand.

Summary

A tally chart uses marks to keep track of data, and a frequency table shows the total for each category.

Remember to group tally marks in sets of five, count carefully, and check that all the frequencies add up to the total number of data items.

When you build charts step by step, you can organize data neatly and answer questions with confidence.

Analyzing Scaled Picture Graphs means reading a picture graph where each picture stands for more than 1 item.

A regular picture graph might use 1 picture to mean 1 thing. But in a scaled picture graph, 1 picture might mean 2, 5, 10, or another number. This helps us show data more quickly and neatly.

When you analyze a scaled picture graph, you are figuring out what the graph tells you. You might answer questions like:

• How many are in each category?
• Which category has the most or least?
• How many more or fewer?
• How many altogether?

To read a scaled picture graph correctly, always look for the key first.

The key tells what each picture is worth. For example, if the key says 1 star = 4 votes, then every star stands for 4 votes.

Steps for Reading a Scaled Picture Graph

1. Read the title to find out what the graph is about.
2. Look at the categories on the side or bottom.
3. Check the key to see how much each picture represents.
4. Count the pictures in each category.
5. Multiply the number of pictures by the value in the key.
6. Compare the totals to answer questions.

Here is the math idea:

If 1 picture = 3 items, and a row has 4 pictures, then:

$$4 \times 3 = 12$$

So that category has 12 items.

Important Tip: Do not just count the pictures. You must use the key. If you forget the key, your answer will be too small.

Example 1: Reading a Simple Scaled Picture Graph

A class voted for their favorite fruit.

Key: 1 apple picture = 2 students

• Apples: 4 pictures
• Bananas: 3 pictures
• Grapes: 5 pictures

Let’s find how many students chose each fruit.

• Apples: \(4 \times 2 = 8\)
• Bananas: \(3 \times 2 = 6\)
• Grapes: \(5 \times 2 = 10\)

So:

• 8 students chose apples.
• 6 students chose bananas.
• 10 students chose grapes.

What fruit was most popular? Grapes, because 10 is the greatest number.

What fruit was least popular? Bananas, because 6 is the smallest number.

Example 2: Finding How Many More

A picture graph shows books read by 4 students.

Key: 1 book picture = 5 books

• Lena: 2 pictures
• Marco: 4 pictures
• Ava: 3 pictures
• Noah: 5 pictures

First, find each total.

• Lena: \(2 \times 5 = 10\) books
• Marco: \(4 \times 5 = 20\) books
• Ava: \(3 \times 5 = 15\) books
• Noah: \(5 \times 5 = 25\) books

How many more books did Noah read than Ava?

Subtract:

$$25 - 15 = 10$$

Noah read 10 more books than Ava.

How many fewer books did Lena read than Marco?

$$20 - 10 = 10$$

Lena read 10 fewer books than Marco.

Example 3: Finding the Total Altogether

A scaled picture graph shows how many cans were collected for recycling.

Key: 1 can picture = 4 cans

• Monday: 3 pictures
• Tuesday: 6 pictures
• Wednesday: 2 pictures
• Thursday: 5 pictures

Find each day's total:

• Monday: \(3 \times 4 = 12\)
• Tuesday: \(6 \times 4 = 24\)
• Wednesday: \(2 \times 4 = 8\)
• Thursday: \(5 \times 4 = 20\)

Now find the total collected altogether:

$$12 + 24 + 8 + 20 = 64$$

They collected 64 cans altogether.

Example 4: Watch Out for a Common Mistake

A graph shows pets owned by students.

Key: 1 paw print = 3 pets

• Cats: 4 pictures

Someone says there are 4 cats because they counted 4 pictures.

That is not correct, because each picture stands for 3 pets.

The correct answer is:

$$4 \times 3 = 12$$

There are 12 cats.

How to Answer Questions About Scaled Picture Graphs

• For how many questions, multiply pictures by the key.
• For how many more or how many fewer, find both totals first, then subtract.
• For how many altogether, find each total, then add.
• For most and least, compare the totals, not just the pictures.

Helpful Questions to Ask Yourself

• What does 1 picture stand for?
• How many pictures are in this category?
• Do I need to multiply, add, or subtract?
• Am I comparing the real totals?

Let’s Practice Thinking

If a graph has the key 1 smiley face = 10 stickers, and Mia has 3 smiley faces, then Mia has:

$$3 \times 10 = 30$$

Mia has 30 stickers.

If Jay has 5 smiley faces, then Jay has:

$$5 \times 10 = 50$$

Jay has 50 stickers.

How many more stickers does Jay have than Mia?

$$50 - 30 = 20$$

Jay has 20 more stickers than Mia.

Summary

A scaled picture graph uses a key to show that each picture stands for more than 1 item. To read it, count the pictures and multiply by the number in the key. Then you can compare categories, find differences, and find totals.

Always remember: the key is the most important part. If you use the key carefully, you can understand the graph correctly.

Constructing Line Plots with Fractional Data

A line plot is a simple graph that shows data on a number line. It helps us see how many times each value appears.

Sometimes the data are whole numbers, like 1, 2, and 3. But sometimes the data are fractions, like \(\frac{1}{2}\), \(\frac{3}{4}\), and \(1\frac{1}{4}\). In this lesson, you will learn how to make a line plot when the data include fractions.

When we use fractions on a line plot, we must be careful to mark the number line into equal parts. That way, each fraction goes in the correct place.

What is a line plot?

A line plot uses:

• a number line across the bottom,
• X marks above each value,
• and each X stands for one piece of data.

If a value happens more than once, we stack the X marks above that number.

Why use a line plot?

• It shows which values happen the most.
• It helps us compare data quickly.
• It helps us organize measurements clearly.

Fractions on a number line

Before making a line plot with fractions, we need to understand how fractions fit on a number line.

If the fractions are in halves, then each whole is split into 2 equal parts:

\(0, \frac{1}{2}, 1, 1\frac{1}{2}, 2\)

If the fractions are in fourths, then each whole is split into 4 equal parts:

\(0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\)

If the fractions are in eighths, then each whole is split into 8 equal parts.

The most important rule is this:

All spaces on the number line must be equal in size.

Steps for constructing a line plot with fractional data

1. Look at the data. Find out what fractions are used.
2. Choose the scale. Decide how the number line should be divided. For example, into halves, fourths, or eighths.
3. Draw the number line. Mark equal spaces and label them with the fraction values.
4. Read each data value carefully.
5. Put one X above the correct value for each piece of data.
6. Stack Xs when the same value appears more than once.

Worked Example 1: Fractions in halves

Mia measured the lengths of ribbons. The lengths were:

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

Step 1: Find the fractions used.

The data use halves: \(\frac{1}{2}\) and \(1\frac{1}{2}\).

Step 2: Draw a number line in halves.

We label it:

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

Step 3: Place the X marks.

• \(\frac{1}{2}\) appears 2 times
• \(1\) appears 2 times
• \(1\frac{1}{2}\) appears 1 time
• \(2\) appears 1 time

The line plot would show:

\(\frac{1}{2}\): XX
\(1\): XX
\(1\frac{1}{2}\): X
\(2\): X

This tells us that the most common lengths are \(\frac{1}{2}\) and \(1\).

Worked Example 2: Fractions in fourths

Some students measured the amount of water in cups. Their measurements were:

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

Step 1: Notice the fractions.

The data use fourths: \(\frac{1}{4}, \frac{1}{2}, \frac{3}{4}\). Remember that \(\frac{1}{2} = \frac{2}{4}\), so it belongs on a number line divided into fourths.

Step 2: Draw and label the number line.

Use equal spaces for:

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

Step 3: Count each value.

• \(\frac{1}{4}\) appears 2 times
• \(\frac{1}{2}\) appears 2 times
• \(\frac{3}{4}\) appears 2 times
• \(1\) appears 1 time

The line plot would show:

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

This line plot shows that three values tie for the most.

Worked Example 3: Mixed numbers with fourths

A gardener measured the heights of plants. The heights were:

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

Step 1: Decide the scale.

The fractions are in fourths, so the number line must be divided into fourths.

Step 2: Label the number line.

We need values from \(1\frac{1}{4}\) to \(2\):

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

Step 3: Count the data.

• \(1\frac{1}{4}\) appears 3 times
• \(1\frac{1}{2}\) appears 2 times
• \(1\frac{3}{4}\) appears 1 time
• \(2\) appears 1 time

The line plot would show:

\(1\frac{1}{4}\): XXX
\(1\frac{1}{2}\): XX
\(1\frac{3}{4}\): X
\(2\): X

This tells us that \(1\frac{1}{4}\) is the most common plant height.

How to avoid common mistakes

• Do not use uneven spaces. Every jump on the number line must be the same size.
• Make sure the scale matches the data. If the data use fourths, divide the number line into fourths.
• Read mixed numbers carefully. \(1\frac{1}{2}\) is not the same as \(\frac{1}{2}\).
• Put one X for each data value. Do not skip any values.
• Stack Xs straight up. This makes the plot easy to read.

Helpful thinking

When you see fractional data, ask yourself:

• What kind of fractions are these: halves, fourths, or eighths?
• What is the smallest value?
• What is the largest value?
• How should I label the number line so every value has a place?

Try this thinking on your own

Suppose the data are:

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

You would:

1. Notice the data use fourths.
2. Draw a number line with \(\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\).
3. Place one X for each measurement.

Then you would see:

• \(\frac{1}{4}\): 1 X
• \(\frac{1}{2}\): 2 Xs
• \(\frac{3}{4}\): 2 Xs
• \(1\): 1 X

What line plots help us notice

After making a line plot, we can answer questions like:

• Which value appears the most?
• Which value appears the least?
• How many total data points are there?
• Are some values the same amount?

For example, if \(\frac{1}{2}\) has 3 Xs, that means 3 measurements were \(\frac{1}{2}\).

Summary

A line plot shows data on a number line. When the data include fractions, the number line must be divided into equal parts that match the fractions in the data.

To construct a line plot with fractional data, first study the fractions, then choose the correct scale, label the number line, and place one X for each data value. Stacked Xs show repeated values.

With practice, line plots make it easy to organize fractional measurements and understand the data quickly.

Interpreting Line Plots

A line plot is a simple graph that shows data on a number line. It helps us see how many times each value appears.

On a line plot, each X stands for one piece of data. If more than one item has the same value, the Xs are stacked above that number.

Line plots are useful because they help us answer questions like:

• Which value happens the most?
• Which value happens the least?
• How many data points are there in all?
• What values are in the middle, or close together?

Parts of a line plot

• A number line across the bottom
• Labels that tell what is being measured
• X marks above the numbers

To interpret a line plot means to read it carefully and understand what the data is showing.

How to read a line plot

1. Look at the title or label to see what the data is about.
2. Look at the numbers on the number line.
3. Count the Xs above each number.
4. Use the counts to answer questions.

Important idea: stacked Xs

If 3 Xs are above the number 5, that means the value 5 appears 3 times.

The more Xs stacked above a number, the more often that value appears. This can show a cluster, which means several data points are close together in one part of the line plot.

If most of the Xs are around the middle numbers, that tells us the data has a central concentration. That means many values are gathered near the center.

Example 1: Reading values

This line plot shows the number of books read by students in one month.

0: X
1: XX
2: XXX
3: X
4: XX

Let’s interpret it.

• Above 0, there is 1 X, so 1 student read 0 books.
• Above 1, there are 2 Xs, so 2 students read 1 book.
• Above 2, there are 3 Xs, so 3 students read 2 books.
• Above 3, there is 1 X, so 1 student read 3 books.
• Above 4, there are 2 Xs, so 2 students read 4 books.

How many students are there altogether?

Add all the Xs:

$$1 + 2 + 3 + 1 + 2 = 9$$

So, there are 9 students.

Which number of books was read the most?

The greatest stack is above 2, with 3 Xs. So, 2 books was the most common number.

Example 2: Finding a cluster

This line plot shows the lengths of pencils in inches.

4: X
5: XX
6: XXXX
7: XXX
8: X

Let’s look for where the data is grouped.

• 4 inches has 1 pencil
• 5 inches has 2 pencils
• 6 inches has 4 pencils
• 7 inches has 3 pencils
• 8 inches has 1 pencil

Most of the Xs are above 6 and 7. That means the data is clustered around 6 and 7 inches.

The tallest stack is above 6, so 6 inches is the value that appears most often.

Also, the data is not spread out evenly. Most values are in the middle. This shows a central concentration near 6 and 7.

Example 3: Answering comparison questions

This line plot shows how many minutes students read at home.

10: XX
15: XXX
20: XXXX
25: XX
30: X

Question 1: How many students read for 20 minutes?

There are 4 Xs above 20, so 4 students read for 20 minutes.

Question 2: How many more students read for 20 minutes than 30 minutes?

20 minutes has 4 students. 30 minutes has 1 student.

$$4 - 1 = 3$$

So, 3 more students read for 20 minutes than 30 minutes.

Question 3: How many students are shown in all?

$$2 + 3 + 4 + 2 + 1 = 12$$

So, there are 12 students in all.

Question 4: Where is the data mostly concentrated?

The largest stacks are around 15, 20, and 25. The biggest one is at 20. So the data is mostly concentrated near 20 minutes.

Example 4: Reading a line plot with halves

Sometimes a line plot uses values like halves. You can still read it the same way.

This line plot shows ribbon lengths in inches.

1: X
1.5: XX
2: XXX
2.5: XX
3: X

Question 1: How many ribbons are 2 inches long?

There are 3 Xs above 2, so 3 ribbons are 2 inches long.

Question 2: How many ribbons are longer than 2 inches?

Longer than 2 means 2.5 and 3.

At 2.5, there are 2 ribbons. At 3, there is 1 ribbon.

$$2 + 1 = 3$$

So, 3 ribbons are longer than 2 inches.

Question 3: Where is the center of the data?

The tallest stack is above 2. The values on both sides get smaller. So the data is centered around 2 inches.

Tips for success

• Count each X carefully. Each X stands for 1 data point.
• Do not skip numbers on the number line.
• If Xs are stacked, count from bottom to top.
• When asked for a total, add all the Xs.
• When asked to compare, subtract the two counts.
• Look for the tallest stacks to find what appears most often.
• Look for groups of Xs close together to find a cluster.

Common mistakes to avoid

• Mistake: Counting the numbers on the line instead of the Xs.
  Remember: The Xs show the data.
• Mistake: Forgetting to count all stacked Xs.
  Remember: Every X matters.
• Mistake: Answering with the wrong unit.
  Remember: Check if the plot is about books, inches, minutes, or something else.

Summary

A line plot shows data on a number line using Xs. Each X stands for one piece of data.

To interpret a line plot, count the Xs above each value, find totals, compare counts, and look for clusters or values near the center.

If many Xs are grouped around the same value or nearby values, that shows where the data is concentrated. With careful counting, line plots are a great way to understand data quickly.

Comparing Data Sets across Categories means looking at information in different groups and deciding how the groups are alike or different.

For example, you might compare favorite fruits in two classrooms, or compare how many books students read in different months. Each group, like apples, bananas, or oranges, is called a category.

When we compare data sets, we ask questions like:

• Which category has the greatest number?
• Which category has the least number?
• How many more or fewer are in one category than another?
• Which data set has larger amounts overall?

Learning to compare data helps us understand charts, make good observations, and solve real-world problems.

Main Idea 1: Read the categories carefully.

Before comparing, look at the names of the categories. These tell you what each number stands for. If a chart shows dogs, cats, fish, and birds, do not compare the wrong groups by mistake.

Main Idea 2: Read the scale.

Some graphs count by 1s. Others count by 2s, 5s, or 10s. The scale tells how much each mark or square is worth.

If one line on a graph means 2, then a bar reaching 5 lines high means:

$$5 \times 2 = 10$$

This is important because you must know the real value before you compare.

Main Idea 3: Compare numbers by subtracting.

To find how many more or how many fewer, subtract the smaller number from the larger number.

If one category has 14 and another has 9, the difference is:

$$14 - 9 = 5$$

So one category has 5 more than the other.

Main Idea 4: Compare two data sets category by category.

Sometimes you will see two sets of data about the same categories. For example, Class A and Class B may both vote for favorite snacks. To compare them, look at the same category in both sets.

• Compare apples in Class A to apples in Class B
• Compare bananas in Class A to bananas in Class B
• Compare crackers in Class A to crackers in Class B

This helps you see which class had more in each category.

Main Idea 5: Look for patterns.

After comparing categories, ask yourself:

• Is one data set usually larger?
• Are the differences small or large?
• Are some categories equal?
• Which category has the biggest difference?

These questions help you understand the whole picture, not just one number.

Steps for Comparing Data Sets

1. Read the title to know what the data is about.
2. Read the category names.
3. Check the scale.
4. Find the value for each category.
5. Subtract to find how many more or fewer.
6. Look for the greatest, least, and any patterns.

Worked Example 1: Comparing simple category counts

A survey shows how many students chose each pet.

• Dogs: 12
• Cats: 9
• Fish: 6
• Birds: 9

Question 1: Which category has the greatest number?

Dogs has 12, which is more than 9, 6, and 9. So dogs is the greatest category.

Question 2: Which category has the least number?

Fish has 6, which is less than 12, 9, and 9. So fish is the least category.

Question 3: How many more students chose dogs than fish?

Subtract:

$$12 - 6 = 6$$

So 6 more students chose dogs than fish.

Worked Example 2: Comparing two data sets in the same categories

Two classes voted for their favorite fruit.

• Class A: Apples 10, Bananas 8, Grapes 6
• Class B: Apples 7, Bananas 12, Grapes 6

Let us compare each category.

Apples: Class A has 10 and Class B has 7.

$$10 - 7 = 3$$

Class A has 3 more students choosing apples.

Bananas: Class A has 8 and Class B has 12.

$$12 - 8 = 4$$

Class B has 4 more students choosing bananas.

Grapes: Class A has 6 and Class B has 6.

$$6 - 6 = 0$$

They are equal in the grapes category.

What pattern do we see?

• Class A is greater in apples.
• Class B is greater in bananas.
• The classes are equal in grapes.

Worked Example 3: Comparing data with a scale

A bar graph shows the number of books read in four genres. Each grid line stands for 2 books.

• Mystery: 5 lines
• Science: 3 lines
• History: 4 lines
• Poetry: 2 lines

First, change each number of lines into real values.

Mystery:

$$5 \times 2 = 10$$

Science:

$$3 \times 2 = 6$$

History:

$$4 \times 2 = 8$$

Poetry:

$$2 \times 2 = 4$$

Now compare the categories.

Question 1: Which genre has the greatest number of books?

Mystery has 10 books, which is the greatest.

Question 2: How many more mystery books than poetry books were read?

$$10 - 4 = 6$$

So 6 more mystery books than poetry books were read.

Question 3: How many fewer science books than history books were read?

$$8 - 6 = 2$$

So 2 fewer science books than history books were read.

Worked Example 4: Multi-step comparison across two scaled data sets

Two clubs tracked how many cans they recycled in different categories. Each grid line stands for 5 cans.

• Club Red: Soda 4 lines, Juice 6 lines, Soup 3 lines
• Club Blue: Soda 5 lines, Juice 4 lines, Soup 2 lines

Step 1: Find the real values.

Club Red

• Soda: $$4 \times 5 = 20$$
• Juice: $$6 \times 5 = 30$$
• Soup: $$3 \times 5 = 15$$

Club Blue

• Soda: $$5 \times 5 = 25$$
• Juice: $$4 \times 5 = 20$$
• Soup: $$2 \times 5 = 10$$

Step 2: Compare each category.

Soda:

$$25 - 20 = 5$$

Club Blue collected 5 more soda cans than Club Red.

Juice:

$$30 - 20 = 10$$

Club Red collected 10 more juice cans than Club Blue.

Soup:

$$15 - 10 = 5$$

Club Red collected 5 more soup cans than Club Blue.

Step 3: Find totals if needed.

Club Red total:

$$20 + 30 + 15 = 65$$

Club Blue total:

$$25 + 20 + 10 = 55$$

Question: Which club collected more cans altogether?

Club Red collected 65 cans. Club Blue collected 55 cans.

$$65 - 55 = 10$$

So Club Red collected 10 more cans altogether.

Helpful Tips

• Always read the scale first.
• Compare the same category in each data set.
• Use subtraction to find the difference.
• If needed, add to find totals.
• Check your work to make sure you did not mix up categories.

Common Mistakes to Avoid

• Forgetting that each grid line may stand for more than 1.
• Comparing different categories by accident.
• Subtracting in the wrong order and getting confused about who has more.
• Answering with only a number and not saying what it means.

Summary

Comparing data sets across categories means studying numbers in groups and deciding which groups have more, less, or the same amount.

To do this well, read the categories, check the scale, find the real values, and subtract to compare. You can also add totals to compare the whole data sets. These skills help you understand graphs, charts, and tables clearly.

Finding the Minimum, Maximum, and Range

When we look at a group of numbers, we can learn a lot by finding the smallest number, the largest number, and how far apart they are.

These ideas are called minimum, maximum, and range.

• Minimum = the smallest number in a set
• Maximum = the largest number in a set
• Range = the difference between the largest and smallest numbers

You can find the range with this math sentence:

$$\text{Range} = \text{Maximum} - \text{Minimum}$$

This lesson will help you learn how to find each one step by step.

Why is this useful?

Minimum, maximum, and range help us describe data. Data is a group of numbers we collect, like test scores, temperatures, heights of plants, or numbers of books read.

These three ideas help us answer questions like:

• What is the smallest value?
• What is the largest value?
• How spread out are the numbers?

Step 1: Put the numbers in order

It is often easiest to find the minimum, maximum, and range when the numbers are written from least to greatest.

For example, if the numbers are:

\(9, 3, 12, 7, 5\)

Put them in order:

\(3, 5, 7, 9, 12\)

Now it is easy to see:

• The minimum is \(3\)
• The maximum is \(12\)

Step 2: Find the range

Subtract the minimum from the maximum.

$$12 - 3 = 9$$

So the range is \(9\).

Important reminder: The range is not found by adding all the numbers. It is found by subtracting the smallest number from the largest number.

Worked Example 1

A class grew bean plants. The heights were:

\(4, 6, 5, 8, 7\)

Step A: Put the numbers in order.

\(4, 5, 6, 7, 8\)

Step B: Find the minimum and maximum.

• Minimum = \(4\)
• Maximum = \(8\)

Step C: Find the range.

$$8 - 4 = 4$$

Answer: The minimum is \(4\), the maximum is \(8\), and the range is \(4\).

Worked Example 2

These are the numbers of books read by students:

\(10, 2, 6, 9, 2, 7\)

Step A: Put the numbers in order.

\(2, 2, 6, 7, 9, 10\)

Notice that the number \(2\) appears twice. That is okay. We still use the smallest and largest numbers.

Step B: Find the minimum and maximum.

• Minimum = \(2\)
• Maximum = \(10\)

Step C: Find the range.

$$10 - 2 = 8$$

Answer: The minimum is \(2\), the maximum is \(10\), and the range is \(8\).

Worked Example 3

The temperatures this week were:

\(15, 18, 14, 20, 17, 16\)

Step A: Put the numbers in order.

\(14, 15, 16, 17, 18, 20\)

Step B: Find the minimum and maximum.

• Minimum = \(14\)
• Maximum = \(20\)

Step C: Find the range.

$$20 - 14 = 6$$

Answer: The minimum is \(14\), the maximum is \(20\), and the range is \(6\).

Worked Example 4

A soccer team scored these numbers of goals in games:

\(11, 5, 8, 13, 9, 5, 12\)

Step A: Put the numbers in order.

\(5, 5, 8, 9, 11, 12, 13\)

Step B: Find the minimum and maximum.

• Minimum = \(5\)
• Maximum = \(13\)

Step C: Find the range.

$$13 - 5 = 8$$

Answer: The minimum is \(5\), the maximum is \(13\), and the range is \(8\).

How to remember the words

• Minimum sounds like mini, which can help you remember it means the smallest.
• Maximum can help you remember the biggest or greatest number.
• Range tells how far the numbers go from the smallest to the largest.

Tips for solving problems

1. Read all the numbers carefully.
2. Put them in order from least to greatest.
3. Circle the first number. That is the minimum.
4. Circle the last number. That is the maximum.
5. Subtract: maximum minus minimum.

Common mistakes to avoid

• Mistake: Choosing a number in the middle as the minimum or maximum.
  Always check for the smallest and largest numbers.
• Mistake: Adding instead of subtracting.
  Remember: $$\text{Range} = \text{Maximum} - \text{Minimum}$$
• Mistake: Forgetting to put numbers in order.
  Ordering the numbers makes the job easier.

Try thinking through this one

The data set is:

\(21, 19, 25, 18, 22\)

Put the numbers in order:

\(18, 19, 21, 22, 25\)

Now find:

• Minimum = \(18\)
• Maximum = \(25\)

Then subtract:

$$25 - 18 = 7$$

So the range is \(7\).

Brief Summary

To find the minimum, look for the smallest number. To find the maximum, look for the largest number. To find the range, subtract the minimum from the maximum.

Remember this rule:

$$\text{Range} = \text{Maximum} - \text{Minimum}$$

If you put the numbers in order first, it becomes much easier to find all three.

Concept of the Median

When we collect data, we often want to find the middle value. That middle value is called the median.

The median helps us understand what number is in the center of a set of data when the numbers are put in order. It is a useful way to describe data because it shows the middle, not just the biggest or smallest value.

In this lesson, you will learn:

• what the median means,
• how to find the median,
• what to do when there is one middle number,
• and what to do when there are two middle numbers.

What is the median?

The median is the number in the middle of a data set after the numbers are arranged in order.

You can put the data in:

• least to greatest, or
• greatest to least.

Most of the time, it is easier to use least to greatest.

For example, look at these numbers:

8, 3, 5

They are not in order, so first we arrange them:

3, 5, 8

The middle number is 5, so the median is \(5\).

Important rule: You must always put the numbers in order first. If the data is not in order, you might choose the wrong middle number.

How to find the median

1. Write the numbers in order from least to greatest.
2. Find the middle number.
3. If there is one middle number, that number is the median.
4. If there are two middle numbers, add them and divide by 2.

When there is one middle number

If a data set has an odd number of values, there will be exactly one middle number.

You can also cross out one number from each end until you reach the middle.

Example:

2, 4, 6, 8, 10

Cross out from both ends:

• Cross out 2 and 10
• Cross out 4 and 8

The number left is 6. So the median is \(6\).

When there are two middle numbers

If a data set has an even number of values, there will be two middle numbers.

In that case, the median is the number halfway between those two middle numbers. To find it, add the two middle numbers and divide by 2.

For example:

1, 3, 5, 7

The two middle numbers are 3 and 5.

Add them:

$$3 + 5 = 8$$

Divide by 2:

$$8 \div 2 = 4$$

So the median is \(4\).

Worked Example 1: Find the median of a small data set

Find the median of: 7, 2, 9

Step 1: Put the numbers in order.

2, 7, 9

Step 2: Find the middle number.

The middle number is 7.

Answer: The median is \(7\).

Worked Example 2: Find the median with more numbers

Find the median of: 12, 5, 9, 3, 7

Step 1: Put the numbers in order.

3, 5, 7, 9, 12

Step 2: Find the middle number.

The middle number is 7.

Answer: The median is \(7\).

Worked Example 3: Find the median when there are two middle numbers

Find the median of: 4, 10, 6, 8

Step 1: Put the numbers in order.

4, 6, 8, 10

Step 2: Find the two middle numbers.

The two middle numbers are 6 and 8.

Step 3: Add them.

$$6 + 8 = 14$$

Step 4: Divide by 2.

$$14 \div 2 = 7$$

Answer: The median is \(7\).

Worked Example 4: Median in a real-life data set

A class recorded how many books 5 students read in one month: 6, 2, 4, 9, 3.

Find the median number of books.

Step 1: Put the data in order.

2, 3, 4, 6, 9

Step 2: Find the middle number.

The middle number is 4.

Answer: The median number of books is \(4\).

Things to remember

• The median is the middle value.
• Always put the numbers in order first.
• If there are an odd number of values, there is one middle number.
• If there are an even number of values, there are two middle numbers.
• For two middle numbers, add them and divide by 2.

Common mistakes

• Forgetting to sort the data. The numbers must be in order before finding the median.
• Choosing the middle from unsorted data. The middle place only matters after ordering.
• Not averaging the two middle numbers. When there are two middle numbers, you must find the number halfway between them.

Try thinking about these

• What is the median of 1, 5, 3? First order them: 1, 3, 5. The median is 3.
• What is the median of 2, 6, 4, 8? First order them: 2, 4, 6, 8. The middle numbers are 4 and 6. $$4 + 6 = 10$$ and $$10 \div 2 = 5$$, so the median is 5.

Summary

The median is the middle number in a data set after the numbers are placed in order. If there is one middle number, that number is the median. If there are two middle numbers, add them and divide by 2 to find the median.

Knowing how to find the median helps you understand data better. It shows the center of the data and helps you compare sets of numbers in a clear way.

Concept of the Mode

When we collect data, we often want to know which item shows up the most. The mode helps us do that.

The mode is the value, number, or category that appears most often in a set of data.

For example, if a class votes for their favorite fruit, the fruit with the most votes is the mode.

This is useful because the mode tells us what is the most common choice or number.

How to Find the Mode

1. Look at all the data.

2. Count how many times each number or category appears.

3. Find the one that appears the most.

4. That value or category is the mode.

Sometimes data can have:

• One mode if one value appears more than all the others.

• More than one mode if two or more values tie for appearing the most.

• No mode if every value appears the same number of times.

Important Idea

The mode does not have to be the biggest number. It is the number that appears the most often.

For example, in the data set \(2, 2, 9\), the mode is \(2\), not \(9\), because \(2\) appears twice.

Worked Example 1: Finding One Mode

Find the mode of:

$$3,\ 5,\ 5,\ 2,\ 4,\ 5,\ 1$$

Let us count:

• \(1\) appears 1 time

• \(2\) appears 1 time

• \(3\) appears 1 time

• \(4\) appears 1 time

• \(5\) appears 3 times

The number that appears the most is \(5\).

Mode = \(5\)

Worked Example 2: Mode with Categories

A group of students chooses their favorite pet:

dog, cat, dog, fish, dog, cat

Now count each choice:

• dog: 3

• cat: 2

• fish: 1

The category that appears the most is dog.

Mode = dog

This shows that the mode can be a word, not just a number.

Worked Example 3: Two Modes

Find the mode of:

$$4,\ 6,\ 4,\ 7,\ 6,\ 9$$

Count each number:

• \(4\) appears 2 times

• \(6\) appears 2 times

• \(7\) appears 1 time

• \(9\) appears 1 time

Both \(4\) and \(6\) appear the most.

Modes = \(4\) and \(6\)

When two values tie for the greatest number of times, the data has two modes.

Worked Example 4: No Mode

Find the mode of:

$$1,\ 2,\ 3,\ 4$$

Count each number:

• \(1\) appears 1 time

• \(2\) appears 1 time

• \(3\) appears 1 time

• \(4\) appears 1 time

Every number appears the same number of times.

There is no mode.

Using a Tally Chart to Find the Mode

A tally chart can make the mode easier to find. Here is an example with favorite colors:

• Red: |||

• Blue: ||||

• Green: ||

Blue has the most tallies.

Mode = Blue

Tips for Finding the Mode

• Be careful to count each value correctly.

• Check which value appears most often.

• The mode can be a number or a category.

• There can be one mode, more than one mode, or no mode.

Let’s Compare

Look at this data set:

$$8,\ 8,\ 3,\ 1,\ 3,\ 8,\ 5$$

Count the numbers:

• \(8\) appears 3 times

• \(3\) appears 2 times

• \(1\) appears 1 time

• \(5\) appears 1 time

Since \(8\) appears the most, the mode is \(8\).

Quick Check Questions

1. What is the mode of \(2, 2, 4, 6, 2, 7\)?

2. What is the mode of apple, orange, apple, banana, orange, apple?

3. What are the modes of \(9, 1, 9, 4, 4, 7\)?

4. Does \(5, 6, 7, 8\) have a mode?

Answers

1. Mode = \(2\)

2. Mode = apple

3. Modes = \(9\) and \(4\)

4. No mode

Summary

The mode is the value or category that appears the most often in a data set.

To find it, count how many times each item appears and choose the one with the greatest count.

A data set can have one mode, more than one mode, or no mode.

When you are asked for the mode, remember: look for what shows up the most.

Drawing Evidence-Based Conclusions means looking at data in a chart, table, picture graph, or bar graph and deciding what the data shows.

In math, a conclusion is an idea or statement you can make after studying the evidence. Evidence means the numbers, labels, and parts of the graph or chart that help prove your idea is true.

When we draw an evidence-based conclusion, we do not guess. We do not use our opinion. We use only what the data tells us.

For example, if a bar graph shows that 12 students like apples and 8 students like bananas, we can conclude that more students like apples than bananas. We cannot conclude that apples are the best fruit, because that is an opinion, not a fact from the graph.

Why is this important?

Charts and graphs help us organize information. When we read them carefully, we can answer questions, compare groups, and make true statements based on the data.

Steps for drawing evidence-based conclusions

1. Read the title. The title tells what the data is about.
2. Look at the labels. Check what each category means.
3. Read the scale. On some graphs, each line may stand for 1, 2, 5, or 10.
4. Find the numbers. Look at how many are in each category.
5. Compare the data. Ask: Which is more? Which is less? How many more? How many fewer?
6. Write a conclusion using the data. Tell what the numbers show.

Good conclusions often use words like:

• more than
• less than
• equal to
• most
• fewest
• altogether
• difference

Be careful! A good conclusion must match the evidence.

• If the graph shows 9 and 6, the difference is $$9-6=3$$.
• If two bars are the same height, the amounts are equal.
• If a picture graph key shows 1 picture = 2 items, you must count by 2s.

Worked Example 1: Reading a table

A class voted for their favorite recess game.

• Tag: 6 students
• Jump Rope: 4 students
• Soccer: 9 students
• Hide-and-Seek: 5 students

Question: What conclusion can we make?

Step 1: Find the greatest number. Soccer has 9 students.

Step 2: Compare the other numbers. Tag has 6, Hide-and-Seek has 5, and Jump Rope has 4.

Conclusion: Soccer is the most popular recess game in this class.

This conclusion is evidence-based because it uses the data number 9, which is greater than 6, 5, and 4.

Worked Example 2: Comparing amounts on a bar graph

A bar graph shows how many books 4 students read in one month.

• Ana: 7 books
• Ben: 5 books
• Cora: 7 books
• Diego: 3 books

Question: What are two true conclusions?

Look at the data:

• Ana and Cora both read 7 books.
• Ben read 5 books.
• Diego read 3 books.

Conclusion 1: Ana and Cora read the same number of books.

We know this because $$7=7$$.

Conclusion 2: Diego read fewer books than Ben.

We know this because $$3<5$$.

We can also find how many fewer:

$$5-3=2$$

So Diego read 2 fewer books than Ben.

Worked Example 3: Using a picture graph key

A picture graph shows how many cans students collected for a food drive.

Key: 1 picture = 2 cans

• Room 1: 4 pictures
• Room 2: 3 pictures
• Room 3: 5 pictures

Question: Which room collected the most cans, and how do you know?

Step 1: Use the key.

• Room 1: $$4\times 2=8$$ cans
• Room 2: $$3\times 2=6$$ cans
• Room 3: $$5\times 2=10$$ cans

Step 2: Compare the totals.

10 is greater than 8 and 6.

Conclusion: Room 3 collected the most cans.

This is evidence-based because we used the key and the numbers from the graph.

Worked Example 4: Finding a difference and making a conclusion

A chart shows the number of sunny days in 4 months.

• April: 12 days
• May: 15 days
• June: 11 days
• July: 15 days

Question: What conclusion can we make about May and July? What is another conclusion about June?

Step 1: Compare May and July.

Both have 15 sunny days, so $$15=15$$.

Conclusion 1: May and July had the same number of sunny days.

Step 2: Compare June to the others.

June has 11 sunny days. That is less than 12, 15, and 15.

Conclusion 2: June had the fewest sunny days.

How to tell if a conclusion is strong

Ask yourself these questions:

• Did I use the chart, graph, or table?
• Did I use the correct numbers?
• Did I remember the scale or key?
• Is my statement a fact, not an opinion?

If the answer is yes to all of these, your conclusion is probably strong.

Fact or opinion?

• Fact: “The red bar is taller than the blue bar.”
• Opinion: “Red is the best color.”
• Fact: “18 students chose pizza, and 10 chose salad.”
• Opinion: “Pizza is a better lunch.”

Only facts from the data can be used as evidence-based conclusions.

Helpful sentence starters

• The graph shows that...
• Based on the data...
• The category with the most is...
• The category with the fewest is...
• Both categories have the same number of...
• There are ___ more than ___.

Let’s practice thinking

If a graph shows 14 students walk to school, 9 ride the bus, and 14 come by car, what can we conclude?

We can conclude that walking and coming by car are tied for the greatest number, because both have 14.

We can also conclude that more students walk than ride the bus, because $$14-9=5$$, so 5 more students walk than ride the bus.

Summary

Drawing evidence-based conclusions means studying data carefully and making true statements that match the numbers. Always read the title, labels, scale, or key first. Then compare the data and write a conclusion based only on facts from the chart or graph.