Data Collection, Graphing, and Analysis

Formulating Questions and Collecting Data

We can learn about a group by asking a question and collecting data. Data is information that we gather. In 2nd grade, we often collect data by doing a simple survey.

A survey is when we ask people the same question and record their answers. Then we can count the answers and learn something from the data.

For example, we might ask, “What is your favorite fruit?” Then we listen to each person’s answer and write it down. After that, we count how many chose apples, bananas, grapes, or oranges.

Step 1: Ask a good question

A good survey question is clear and easy to answer. It should be a question that many people can answer in just one way.

• Good question: What is your favorite recess game?
• Good question: How do you get to school?
• Not as good: Tell me everything you like.

The best questions for collecting data often have answer choices. These are called categories. Categories are groups for the answers.

Here are some examples of categories:

• Favorite pet: dog, cat, fish, bird
• Favorite color: red, blue, green, yellow
• How you get to school: bus, car, walk, bike

Step 2: Choose categories

Before you ask people, think about the answer groups you will use. The categories should match the question.

For the question “What is your favorite snack?”, the categories might be:

• Fruit
• Crackers
• Yogurt
• Pretzels

It is important that the categories are easy to understand. Each answer should fit into just one group.

Step 3: Ask the same question to each person

When collecting data, ask the same question every time. This helps make the data fair.

If one student is asked, “Do you like apples?” and another student is asked, “What fruit do you like best?”, the answers may not match. To collect good data, keep the question the same for everyone.

Step 4: Record the answers carefully

As people answer, write their answers down. One easy way is to use tally marks.

Tally marks help us count quickly:

• One tally: \( | \)
• Two tallies: \( || \)
• Three tallies: \( ||| \)
• Four tallies: \( |||| \)
• Five tallies: the 5th tally goes across the first four

We can show 5 tallies like this:

$$||||\! /$$

Every time a person gives an answer, add one tally to that category.

Worked Example 1: Picking a survey question

Let’s say we want to collect data from our class.

Which question is best?

1. What is your favorite school subject?
2. What do you think about school and home and games?
3. Tell me many things you like.

The best question is 1. What is your favorite school subject?

Why?

• It is clear.
• It asks one thing.
• It is easy to answer.
• The answers can be sorted into categories.

Worked Example 2: Collecting data with tallies

Question: How do you get to school?

Categories:

• Bus
• Car
• Walk
• Bike

Suppose 10 students answer like this:

• Bus, Car, Bus, Walk, Car, Bus, Bike, Car, Walk, Bus

Now let’s record the data.

• Bus: 4 students
• Car: 3 students
• Walk: 2 students
• Bike: 1 student

With tally marks, it looks like this:

• Bus: ||||
• Car: |||
• Walk: ||
• Bike: |

We can also write the counts as numbers:

$$4 + 3 + 2 + 1 = 10$$

That tells us all 10 answers were recorded.

Worked Example 3: Making sure data is correct

Question: What is your favorite pet?

Categories: dog, cat, fish

A student recorded these tallies:

• Dog: |||
• Cat: ||||
• Fish: ||

How many students answered the survey?

Let’s count:

$$3 + 4 + 2 = 9$$

9 students answered the survey.

This is a good way to check your work. Add all the category counts to find the total number of answers.

Worked Example 4: Choosing matching categories

Question: What do you like to drink at lunch?

Which set of categories makes sense?

1. Milk, water, juice
2. Blue, happy, running

The correct categories are Milk, water, juice.

Why? Because those are all possible answers to the question about drinks. The other choices do not match the question.

How to collect data step by step

1. Think of one clear question.
2. Choose answer categories that match the question.
3. Ask each person the same question.
4. Record each answer with a tally mark or check mark.
5. Count the tallies.
6. Check that the total matches the number of people you asked.

Tips for doing a good survey

• Ask only one question at a time.
• Use categories that are easy to understand.
• Listen carefully to each answer.
• Record answers right away so you do not forget.
• Count carefully at the end.

Why do we collect data?

Collecting data helps us learn about a class or a group. We can find out things like:

• Which choice is the most popular
• Which choice is the least popular
• How many people chose each category

Later, we can use the data to make a bar graph, picture graph, or other chart. But first, we need a good question and careful data collection.

Summary

To formulate questions and collect data, start with a clear survey question. Next, choose categories that match the question. Then ask everyone the same question and record each answer carefully, often with tally marks. Last, count the answers and check your total.

When you ask good questions and record answers carefully, your data will be helpful and easy to understand.

Tally Charts and Frequency Tables

Sometimes we collect data. Data is information we gather. We might collect data about favorite fruits, pets, or colors in our class.

When we have a lot of answers, we need a neat way to organize them. Two helpful tools are called a tally chart and a frequency table.

A tally chart uses quick marks to count. A frequency table shows how many there are using numbers.

These tools help us answer questions like:

• Which choice got the most votes?
• Which choice got the fewest votes?
• How many votes were there altogether?

1. What is a tally mark?

A tally mark is a counting mark. We draw one tally for each thing we count.

The first four tally marks are straight lines:

\(1 = |\)

\(2 = ||\)

\(3 = |||\)

\(4 = ||||\)

The fifth tally is special. We draw a line across the first four marks to make a group of five.

That looks like this: ||||/

Grouping by five helps us count faster.

Here is what tally groups can look like:

• \(5 = ||||/\)
• \(6 = ||||/ |\)
• \(7 = ||||/ ||\)
• \(8 = ||||/ |||\)
• \(9 = ||||/ ||||\)
• \(10 = ||||/ ||||/\)

2. What is a tally chart?

A tally chart helps us organize answers into categories. A category is a group name, like apples, bananas, and grapes.

A tally chart usually has:

• the category name
• the tally marks

For example, if students vote for their favorite pet, we can keep track with tally marks.

3. What is a frequency table?

The word frequency means how many.

A frequency table shows the total number for each category using numerals such as \(2\), \(5\), or \(9\).

We can make a frequency table from a tally chart by counting the tally marks in each row.

4. How to make a tally chart

1. Choose the categories.
2. Look at each piece of data one at a time.
3. Add one tally mark in the correct row.
4. Every fifth tally becomes a group of five.
5. Count the tallies to find the total.

5. How to make a frequency table from a tally chart

1. Look at one row.
2. Count the tally marks carefully.
3. Write the number in the frequency column.
4. Do the same for every row.

Worked Example 1: Counting classroom pets

A class is asked, “What pet do you have?” The answers are: dog, cat, dog, fish, dog, cat.

Let’s sort the data into categories: Dog, Cat, and Fish.

Tally Chart

Dog: |||

Cat: ||

Fish: |

Now let’s write the frequency table.

Frequency Table

Dog: \(3\)

Cat: \(2\)

Fish: \(1\)

What can we learn?

• Dogs are the most common pet.
• Fish is the least common pet.
• There are $$3+2+1=6$$ answers altogether.

Worked Example 2: Favorite fruit

Students choose their favorite fruit: apple, banana, apple, orange, banana, apple, banana, banana.

Categories: Apple, Banana, Orange

Let’s count carefully.

• Apple appears \(3\) times.
• Banana appears \(4\) times.
• Orange appears \(1\) time.

Tally Chart

Apple: |||

Banana: ||||

Orange: |

Frequency Table

Apple: \(3\)

Banana: \(4\)

Orange: \(1\)

What can we learn?

• Banana got the most votes.
• Orange got the fewest votes.
• There are $$3+4+1=8$$ votes in all.

Worked Example 3: Using groups of five

Now let’s try a bigger set of data. A class chooses a favorite color. The results show:

• Blue: \(7\)
• Red: \(5\)
• Green: \(3\)

We will write tallies using groups of five.

Tally Chart

Blue: ||||/ ||

Red: ||||/

Green: |||

Now write the numbers in a frequency table.

Frequency Table

Blue: \(7\)

Red: \(5\)

Green: \(3\)

How did we know Blue is 7?

Blue has one group of five and two more tallies.

So, $$5+2=7$$

How did we know Red is 5?

Red has one full group of five.

So, $$5=5$$

What can we learn?

• Blue is the most popular color.
• Green is the least popular color.
• There are $$7+5+3=15$$ students in all.

Worked Example 4: From raw data to a tally chart and frequency table

Here is some raw data about favorite ice cream flavors:

vanilla, chocolate, vanilla, strawberry, chocolate, vanilla, chocolate, chocolate, strawberry, vanilla

Step 1: Choose the categories.

• Vanilla
• Chocolate
• Strawberry

Step 2: Count each answer.

• Vanilla = \(4\)
• Chocolate = \(4\)
• Strawberry = \(2\)

Step 3: Make the tally chart.

Tally Chart

Vanilla: ||||

Chocolate: ||||

Strawberry: ||

Step 4: Make the frequency table.

Frequency Table

Vanilla: \(4\)

Chocolate: \(4\)

Strawberry: \(2\)

What can we learn?

• Vanilla and chocolate are tied. That means they have the same number.
• Strawberry has the fewest votes.
• There are $$4+4+2=10$$ votes altogether.

6. Tips for reading tally charts

• Count by fives first.
• Then add the extra tally marks.
• Be careful not to count the same tally twice.
• Check that each piece of data is counted once.

For example, if you see ||||/ |||, think:

one group of five and three more

$$5+3=8$$

7. Common mistakes to watch for

• Forgetting the group of five: The fifth tally should cross the first four.
• Counting too fast: Touch or point to each tally as you count.
• Writing the wrong number: Check that the frequency matches the tallies.
• Skipping data: Make sure every answer is counted.

8. Let’s think about questions we can answer

After making a tally chart or frequency table, we can answer questions like:

• Which category has the most?
• Which category has the least?
• How many are in one category?
• How many are there altogether?

If a table shows:

• Soccer: \(6\)
• Basketball: \(2\)
• Tennis: \(4\)

Then:

• Soccer has the most.
• Basketball has the least.
• Tennis has \(4\).
• Altogether there are $$6+2+4=12$$ choices.

9. Why tally charts and frequency tables are useful

These tools help us keep data neat and easy to read.

Tally charts are great when we are still counting. Frequency tables are great when we want to know the total quickly.

We can also use them before making a bar graph or picture graph.

Summary

A tally chart uses marks to count data. Every fifth tally makes a group of five.

A frequency table shows the same data with numbers. It tells how many are in each category.

When you see a set of answers, count each one carefully, make tally marks, group by five, and then write the total in a frequency table.

With tally charts and frequency tables, we can organize data and understand it more easily.

Picture Graphs with Scales help us show data with pictures or symbols.

A picture graph uses small pictures to stand for numbers. A scale, or key, tells us how many each picture means.

For example, a key might say:

Key: ★ = 2

This means each star stands for 2 things, not just 1 thing.

When we read a picture graph, we must always look at the key first. The key tells us the value of each picture.

Why the Key Matters

If you see 3 stars, you do not just say 3. You use the key.

If ★ = 2, then:

$$3 \text{ stars} = 3 \times 2 = 6$$

So 3 stars mean 6 things.

The key helps us count faster, especially when there is a lot of data.

Parts of a Picture Graph

• Title tells what the graph is about.
• Categories tell the groups being counted.
• Pictures or symbols show the data.
• Key tells how much each picture is worth.

Here is what you should do when reading a picture graph:

1. Read the title.
2. Look at the key.
3. Count the pictures in each row.
4. Use the key to find the total.
5. Compare the totals.

Example 1: Reading a Simple Picture Graph

Title: Pets in Our Class

Key: ● = 2 pets

• Dogs: ●●●
• Cats: ●●
• Fish: ●

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

Dogs: There are 3 circles. Each circle means 2 pets.

$$3 \times 2 = 6$$

So there are 6 dogs.

Cats: There are 2 circles.

$$2 \times 2 = 4$$

So there are 4 cats.

Fish: There is 1 circle.

$$1 \times 2 = 2$$

So there are 2 fish.

What group has the most? Dogs, because 6 is the greatest number.

What group has the least? Fish, because 2 is the smallest number.

Example 2: Finding How Many Altogether

Title: Books Read This Week

Key: 📘 = 5 books

• Anna: 📘📘
• Ben: 📘📘📘
• Cara: 📘

Let’s find each total.

Anna:

$$2 \times 5 = 10$$

Anna read 10 books.

Ben:

$$3 \times 5 = 15$$

Ben read 15 books.

Cara:

$$1 \times 5 = 5$$

Cara read 5 books.

How many books did they read altogether?

$$10 + 15 + 5 = 30$$

They read 30 books altogether.

Example 3: Making a Picture Graph

Now let’s make a picture graph from data.

Suppose students voted for their favorite fruit:

• Apples: 6
• Bananas: 4
• Grapes: 2

We want to use a key where each picture stands for 2 votes.

Key: 🍎 = 2 votes

Now we change each number into pictures.

Apples: 6 votes means:

$$6 \div 2 = 3$$

So Apples gets 3 pictures: 🍎🍎🍎

Bananas: 4 votes means:

$$4 \div 2 = 2$$

So Bananas gets 2 pictures: 🍎🍎

Grapes: 2 votes means:

$$2 \div 2 = 1$$

So Grapes gets 1 picture: 🍎

The picture graph would look like this:

• Apples: 🍎🍎🍎
• Bananas: 🍎🍎
• Grapes: 🍎

Remember, the pictures do not mean apples here. They are just symbols. The key tells their value.

Example 4: Be Careful with the Scale

Title: Toys Collected

Key: ▲ = 3 toys

• Cars: ▲▲
• Balls: ▲▲▲
• Dolls: ▲

Let’s read the graph carefully.

Cars:

$$2 \times 3 = 6$$

Cars = 6 toys

Balls:

$$3 \times 3 = 9$$

Balls = 9 toys

Dolls:

$$1 \times 3 = 3$$

Dolls = 3 toys

A common mistake is to count only the pictures. If you say Balls = 3, that is not correct. The key says each triangle means 3 toys.

You must use the key every time.

Tips for Success

• Always read the key first.
• Count the number of pictures.
• Multiply by the number in the key.
• Check if you need to find most, least, or total.
• Do not forget that one picture may stand for more than 1.

Let’s Practice Thinking

If a graph has the key ☀ = 4 and you see 2 suns, how many does that mean?

$$2 \times 4 = 8$$

So 2 suns mean 8.

If you see 5 suns, then:

$$5 \times 4 = 20$$

So 5 suns mean 20.

Summary

A picture graph uses symbols to show data.

The key tells how much each symbol stands for. To read the graph, count the pictures and use the key to find the real total.

When making a picture graph, choose a key and match the right number of pictures to each category.

If you remember to look at the key first, you can read picture graphs with scales correctly.

Creating and Reading Bar Graphs

A bar graph is a picture that helps us show and compare data. Data is information we collect, like favorite fruits, pets, or colors.

In a bar graph, each category has a bar. The length or height of the bar tells how many there are.

Bar graphs help us answer questions like:

• Which group has the most?
• Which group has the least?
• How many more or fewer?

Parts of a Bar Graph

• Title tells what the graph is about.
• Categories are the groups we are counting, like apples, bananas, and grapes.
• Scale shows the numbers, such as 0, 1, 2, 3, 4, 5.
• Bars show how many are in each category.

A bar graph can be vertical or horizontal.

• In a vertical bar graph, the bars go up and down.
• In a horizontal bar graph, the bars go side to side.

No matter which way the bars go, we read the graph the same way: we look at each bar and match it to the number it reaches.

How to Read a Bar Graph

1. Read the title.
2. Look at the categories.
3. Look at the numbers on the graph.
4. See where each bar ends.
5. Compare the bars.

If one bar reaches 6, that means there are 6 in that category.

If one bar is taller or longer than another bar, then that category has more.

How to Create a Bar Graph

1. Collect data.
2. Count how many are in each category.
3. Write a title.
4. Write the categories.
5. Write numbers in order, starting at 0.
6. Draw one bar for each category.

The bars should be neat and should stop at the correct number.

There should be space between the bars. Bar graphs use separate bars because each bar stands for a different category.

Worked Example 1: Reading a Simple Bar Graph

Here is a bar graph called Favorite Pets.

• Dogs: 5
• Cats: 3
• Fish: 2

Let’s answer some questions.

Question 1: Which pet is liked by the most students?

The bar for dogs reaches 5. That is the greatest number. So, dogs are liked by the most students.

Question 2: Which pet is liked by the fewest students?

The bar for fish reaches 2. That is the smallest number. So, fish are liked by the fewest students.

Question 3: How many students like cats?

The cats bar reaches 3. So, 3 students like cats.

Question 4: How many more students like dogs than fish?

Dogs = 5 and Fish = 2.

We subtract:

$$5 - 2 = 3$$

So, 3 more students like dogs than fish.

Worked Example 2: Making a Vertical Bar Graph

A class counts favorite snacks.

• Apples: 4
• Crackers: 6
• Yogurt: 2

Let’s make a bar graph.

1. Title: Favorite Snacks
2. Categories: Apples, Crackers, Yogurt
3. Numbers on the side: 0, 1, 2, 3, 4, 5, 6
4. Draw bars up to the right numbers.

The apples bar goes to 4.

The crackers bar goes to 6.

The yogurt bar goes to 2.

Now we can read the graph.

• The tallest bar is crackers, so crackers are the most popular.
• The shortest bar is yogurt, so yogurt is the least popular.
• Apples have 4 votes.

Worked Example 3: Making a Horizontal Bar Graph

Now let’s use a horizontal bar graph. A class counts how many books students read this week in different groups.

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

To make the graph:

1. Write the title: Books Read This Week
2. Write the categories on the side: Group A, Group B, Group C
3. Write numbers across the bottom: 0, 1, 2, 3, 4, 5
4. Draw bars going side to side.

The bar for Group A goes to 3.

The bar for Group B goes to 5.

The bar for Group C goes to 4.

Question: How many fewer books did Group A read than Group B?

Group A = 3 and Group B = 5.

$$5 - 3 = 2$$

So, Group A read 2 fewer books than Group B.

Worked Example 4: Reading and Comparing Carefully

A teacher asks students for their favorite color.

• Red: 2
• Blue: 7
• Green: 5
• Yellow: 4

Question 1: Which color is the most popular?

Blue has 7. That is the greatest number. So, blue is the most popular.

Question 2: Which color is the least popular?

Red has 2. That is the smallest number. So, red is the least popular.

Question 3: How many students chose green and yellow altogether?

Green = 5 and Yellow = 4.

$$5 + 4 = 9$$

So, 9 students chose green and yellow altogether.

Question 4: How many more students chose blue than red?

Blue = 7 and Red = 2.

$$7 - 2 = 5$$

So, 5 more students chose blue than red.

Tips for Success

• Always read the title first.
• Check what each bar stands for.
• Look carefully at the numbers.
• Make sure the bar stops at the correct number.
• Use counting, adding, and subtracting to answer questions.

Common Mistakes to Watch Out For

• Forgetting to start the numbers at 0.
• Drawing a bar too short or too long.
• Reading the wrong category.
• Mixing up most and least.

Let’s Remember

A bar graph shows data with bars. Each bar tells how many are in a category.

We can read a bar graph by looking at the title, categories, numbers, and bars. We can also create a bar graph by collecting data, counting it, and drawing bars to match the numbers.

Bar graphs help us compare groups quickly and clearly.

Comparing Graph Categories means looking at two parts of a graph and deciding which has more, which has fewer, and how many more or how many fewer.

We can answer these questions by using subtraction. Sometimes we may also use addition to check our answer.

Graphs help us organize information so it is easy to compare. You may see this in a bar graph, a picture graph, or a line plot.

When you compare categories on a graph, follow these steps:

1. Look carefully at the graph.
2. Find the two categories you need to compare.
3. Count or read the numbers for each category.
4. Subtract to find the difference.
5. Answer with words, like “___ has 3 more than ___.”

Important words:

• Category: a group on the graph, like apples, bananas, or cats.
• More: the bigger number.
• Fewer: the smaller number.
• Difference: how far apart two numbers are.

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

$$\text{Difference} = \text{bigger number} - \text{smaller number}$$

For example, if one category has 8 and another has 5, then:

$$8 - 5 = 3$$

This means one category has 3 more, and the other category has 3 fewer.

Main idea: “How many more?” and “How many fewer?” both ask for the difference.

Let’s practice with some worked examples.

Example 1: Bar Graph

A class makes a bar graph of favorite pets.

• Dogs: 7
• Cats: 4

Question: How many more students chose dogs than cats?

First, find the two numbers: dogs = 7 and cats = 4.

Now subtract:

$$7 - 4 = 3$$

Answer: 3 more students chose dogs than cats.

You can also say: cats have 3 fewer than dogs.

Example 2: Picture Graph

A picture graph shows favorite snacks. Each picture stands for 1 snack.

• Apples: 6
• Crackers: 9

Question: How many fewer apples than crackers?

Find the numbers: apples = 6 and crackers = 9.

Subtract the smaller number from the bigger number:

$$9 - 6 = 3$$

Answer: There are 3 fewer apples than crackers.

We can turn it around and say: there are 3 more crackers than apples.

Example 3: Comparing Three Categories

A bar graph shows favorite colors.

• Blue: 10
• Red: 8
• Green: 5

Question 1: How many more chose blue than green?

Blue is 10. Green is 5.

$$10 - 5 = 5$$

Answer: 5 more students chose blue than green.

Question 2: How many fewer chose green than red?

Red is 8. Green is 5.

$$8 - 5 = 3$$

Answer: 3 fewer students chose green than red.

This example shows that on a graph, you can compare any two categories.

Example 4: Line Plot

A line plot shows how many books students read.

• 2 books: 3 students
• 3 books: 6 students

Question: How many more students read 3 books than 2 books?

Read the counts from the line plot: 6 students and 3 students.

Now subtract:

$$6 - 3 = 3$$

Answer: 3 more students read 3 books than 2 books.

Helpful tips:

• Always compare the two categories named in the question.
• Find the bigger number and the smaller number.
• Subtract to find the difference.
• Read the question carefully. It may ask for more or fewer.
• Your answer should name the categories, not just the number.

Let’s think about mistakes to avoid.

• Do not add when the question asks “how many more” or “how many fewer.”
• Do not forget to read the graph correctly.
• Do not compare the wrong categories.

Quick practice thinking:

If a graph shows birds = 8 and fish = 2, then:

$$8 - 2 = 6$$

Birds have 6 more than fish.

Fish have 6 fewer than birds.

Same difference, different words.

Summary

When we compare graph categories, we look at two groups and find how many more or how many fewer. We do this by reading the numbers on the graph and subtracting the smaller number from the bigger number. If you can find the difference, you can answer both kinds of questions.

Generating Measurement Data means measuring things and writing down the numbers you get.

In 2nd grade, we can measure classroom objects to the nearest whole unit. Then we can use those measurements as data.

Data is a collection of information. If we measure many objects, we make a data set.

This lesson will help you learn how to measure objects, record the measurements, and make a data set you can use for graphs and questions.

What does “nearest whole unit” mean?

A unit is what we measure with, like inches or centimeters.

A whole unit is a full inch or a full centimeter: \(1\), \(2\), \(3\), and so on.

Sometimes an object is not exactly on a number. Then we choose the nearest whole unit.

• If the object is very close to \(5\), we write \(5\).
• If the object is very close to \(6\), we write \(6\).

We do not need tiny parts for this lesson. We just use the closest whole number.

Tools we can use

• a ruler
• a yardstick
• a measuring tape

We can measure things like:

• crayons
• books
• pencils
• erasers
• paper clips
• notebooks

How to measure correctly

1. Put one end of the object at \(0\) on the ruler.
2. Keep the object straight.
3. Look at where the other end stops.
4. Choose the nearest whole unit.
5. Write the measurement down.

Important measuring tip: Start at \(0\), not at the edge of the ruler if the edge is before \(0\).

Why do we measure more than one object?

When we measure many objects, we get many numbers. Those numbers become a data set.

A data set can help us answer questions like:

• Which object is longest?
• Which object is shortest?
• How many objects are \(6\) inches long?
• What lengths do we see most often?

Example 1: Measuring one object

Lena measures a pencil.

The pencil ends a little past \(7\) inches and is closest to \(7\) inches.

So Lena records:

$$7\text{ inches}$$

Now she has one piece of measurement data: \(7\).

Example 2: Measuring several objects

Now Lena measures \(5\) classroom objects to the nearest inch.

• eraser: \(2\) inches
• glue stick: \(4\) inches
• pencil: \(7\) inches
• book: \(9\) inches
• crayon: \(3\) inches

Her data set is:

$$2,\ 4,\ 7,\ 9,\ 3$$

This is called a measurement data set because the numbers came from measuring.

We can put the data in order from least to greatest:

$$2,\ 3,\ 4,\ 7,\ 9$$

Now it is easier to see the shortest and longest lengths.

• Shortest: \(2\) inches
• Longest: \(9\) inches

Example 3: Some measurements are the same

A class measures \(6\) pencils. Their lengths to the nearest inch are:

$$6,\ 7,\ 7,\ 6,\ 5,\ 7$$

Let’s answer some questions.

How many pencils were measured?

Count the numbers:

$$6\text{ pencils}$$

How many pencils are \(7\) inches long?

Count the \(7\)s:

$$3\text{ pencils}$$

What length appears most often?

The number \(7\) appears most often, so \(7\) inches is the length we see the most.

Example 4: Making a simple table of measurement data

Jose measures \(8\) objects in centimeters.

His measurements are:

$$5,\ 5,\ 6,\ 4,\ 8,\ 6,\ 5,\ 7$$

He organizes them in a table.

Now we can see the data clearly.

• There is \(1\) object that is \(4\) cm long.
• There are \(3\) objects that are \(5\) cm long.
• There are \(2\) objects that are \(6\) cm long.

This is helpful when making a bar graph, picture graph, or line plot later.

How to create your own measurement data set

1. Choose a group of objects.
2. Pick one unit, like inches or centimeters.
3. Measure each object carefully.
4. Round to the nearest whole unit.
5. Write each number down.
6. Check that all measurements use the same unit.

Things to remember

• Measure from \(0\).
• Use the same unit for all objects in one data set.
• Write every measurement carefully.
• If two objects have the same length, both numbers belong in the data set.
• A data set can have repeated numbers.

Let’s think about a classroom example

Suppose you measure these objects to the nearest inch:

• marker: \(5\)
• scissors: \(6\)
• glue stick: \(4\)
• paintbrush: \(6\)
• small book: \(8\)

Your data set is:

$$5,\ 6,\ 4,\ 6,\ 8$$

We can answer questions from the data.

• How many objects were measured? \(5\)
• What is the shortest length? \(4\) inches
• What is the longest length? \(8\) inches
• Which length appears more than once? \(6\) inches

Why this skill matters

Generating measurement data helps us collect real information from the world around us.

After we collect the data, we can sort it, count it, graph it, and answer questions about it.

This is how measurement and graphing work together.

Summary

Generating measurement data means measuring objects and recording the lengths as numbers.

We measure to the nearest whole unit, such as the nearest inch or centimeter.

When we measure several objects, the numbers we record make a data set.

Then we can use the data set to compare lengths, count how many of each length there are, and make graphs.

Creating Line Plots

A line plot is a simple graph that helps us show data on a number line.

When we make a line plot, we put numbers in order from least to greatest. Then we use X marks above the numbers to show how many times each number appears.

Line plots are helpful when we measure things, like pencil lengths, plant heights, or how many blocks tall something is.

What is a line plot?

A line plot has:

• a number line going across,
• numbers in order,
• an X above a number for each time that value appears.

If the number 4 shows up 3 times, we put 3 X marks above 4.

How to make a line plot

1. Look at the data. These are the numbers you collected.
2. Find the smallest and largest numbers. This tells you what numbers to put on the line.
3. Draw a number line. Write the numbers in order.
4. Count each value. Every time you see a number, put one X above it.
5. Check your work. Make sure the total number of X marks matches the total number of data pieces.

Important idea: Each X means one piece of data.

So if there are 8 measurements, there should be 8 X marks on the line plot.

Worked Example 1: Making a line plot

Suppose 7 students measured their crayons. The lengths were:

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

Step 1: Put the numbers in order on a number line.

We need numbers 2 through 5.

Step 2: Count how many times each number appears.

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

Step 3: Place X marks above each number.

It would look like this:

2: XXX
3: XX
4: X
5: X

That means:

• 3 crayons were length 2
• 2 crayons were length 3
• 1 crayon was length 4
• 1 crayon was length 5

Now check the total number of X marks:

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

There are 7 X marks, so the line plot matches the 7 measurements.

Worked Example 2: Reading a line plot

Here is a line plot about how many flowers are in small pots:

1: X
2: XX
3: XXXX
4: XX

Let’s answer some questions.

How many pots had 3 flowers?

There are 4 X marks above 3, so 4 pots had 3 flowers.

Which number has the most X marks?

The number 3 has the most X marks. So, 3 flowers happened the most.

How many pots were counted in all?

Add all the X marks:

$$1+2+4+2=9$$

So, there were 9 pots in all.

Worked Example 3: Making a line plot with more data

A class counted how many books each student read last week:

$$4,\ 5,\ 3,\ 4,\ 6,\ 5,\ 4,\ 3,\ 5,\ 4$$

Step 1: Find the smallest and largest numbers.

• Smallest number: 3
• Largest number: 6

So our number line goes from 3 to 6.

Step 2: Count each number.

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

Step 3: Make the line plot.

3: XX
4: XXXX
5: XXX
6: X

What can we learn?

• The most common number is 4 books.
• Only 1 student read 6 books.
• There were 10 students because there are 10 X marks.

Check the total:

$$2+4+3+1=10$$

Worked Example 4: Fixing a mistake

Here is the data:

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

A student made this line plot:

1: X
2: X
3: XXX

Is it correct?

Let’s count carefully:

• 1 appears 1 time
• 2 appears 2 times
• 3 appears 3 times

The student only put 1 X above 2, but there should be 2 X marks above 2.

The correct line plot is:

1: X
2: XX
3: XXX

This is why counting carefully is important.

Tips for success

• Write the numbers in order.
• Give each piece of data only one X.
• Count slowly and carefully.
• Check that the total number of X marks matches the total data.
• Look for which number has the most or fewest X marks.

What line plots help us see

Line plots help us answer questions like:

• Which number happens the most?
• Which number happens the least?
• How many data pieces are there in all?
• How many times does one value appear?

They help us organize information so it is easy to read.

Summary

A line plot shows data on a number line using X marks.

To make one, put the numbers in order, then place one X above a number each time it appears.

After you finish, count the X marks to make sure they match the number of data pieces. If they match, your line plot is ready to use.