Grade 5 Math Studio
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Chapter 2 Β· Lesson 2.1 Β· Grade 5

From Raw Data to a Frequency Table

Learn how to turn a messy list of scores or labels into a clear table that answers useful questions. Everything you need is inside this page.

One item. One group. Count it once.
35–50 minutes No book required Numerical and categorical data Progress saves on this device
1

Begin with a purpose

A useful question needs data

Statistics is not just making tables. It begins with a question, uses data to answer it, and ends with a conclusion.

Data are collected facts, measurements, or labels. Raw data are the original observations before we organize them.
1. CollectGather accurate observations.
2. OrganizePut the observations into a table or graph.
3. AnalyzeRead the display and explain what it tells us.

Which question is best answered by collecting data from many fifth graders?

Tap the stages in the correct order.

2

Know what you collected

Raw data can be numerical or categorical

The kind of data determines how we organize it.

Numerical data
Values that measure or count something, such as 96 points, 1.82 metres, or 14 stalls.
Categorical data
Labels that name a group, such as Good, Choir, or Needs improvement.
96 points
Good
1.82 metres
Orchestra
14 market stalls
Needs improvement
3

Give every number one home

Intervals are labeled boxes on a number line

For this score activity, every score is a whole number from 60 through 100. Each interval includes both written endpoints. Every allowed score must belong to exactly one group.

Boundary rule: Each value must belong to exactly one interval. In this lesson, a score of 90 belongs to 90–99, while 89 belongs to 80–89.
60–69
70–79
80–89
90–99
100
60 70 80 90 100

Where does 90 belong?

Where does 89 belong?

Where does 100 belong?

Where does 69 belong?

Which grouping includes every whole-number score from 60 through 100 exactly once?

4

Guided numerical example

Sort 15 quiz scores

These are the whole-number quiz scores of 15 students, with one score per student. Select a score card, then select its interval.

9683958698100846772817688658590

Unsorted score cards

Tap one card. You can change your selection before choosing a bin.

Selected: none0 of 15 placed

Interval bins

5

Compress the list

Tally first, then write the frequency

A tally is a quick counting mark. The frequency is the number of observations in a group. Count every observation, including repeated values. Here each observation is one student’s score.

Tally example: Six observations can be marked as one group of five and one extra mark: ||||/|. The frequency is 6.
Score intervalTallyFrequency
Exactly 100|
90–99||||
80–89||||/|
70–79||
60–69||
TotalAll observations0

Why should the five frequencies add to 15?

76543210
0
100
0
90–99
0
80–89
0
70–79
0
60–69
6

Turn counts into conclusions

Read what the frequency table says

Use the 15 quiz scores from Mission 4 and their frequency table in Mission 5 for all five questions below.

Do not confuse a value with a frequency. In the row 80–89 β†’ 6, the interval is a set of score values. The frequency 6 is the number of students in that interval.

Which interval has the greatest frequency?

How many students scored 90 or above?

How many students scored below 80?

How many more are in 80–89 than in 70–79?

What does the frequency 6 mean?

7

Categorical raw data

Categories use names instead of number intervals

Forty students received one of four music results. Count each label and build a categorical frequency table.

Excellent
Good
Satisfactory
Needs improvement
Counting strategy: Scan one category at a time. Mark or tally every matching label, then check that all four frequencies add to 40.
ResultExcellentGoodSatisfactoryNeeds improvementTotal
Frequency0

Which result occurs most often?

How many times as many students received Good as received Excellent? Divide the Good count by the Excellent count.

8

Decimal measurements

Choose a sensible grouping for long-jump results

These are one recorded jump from each of 20 students. Distances are recorded to the nearest 0.01 m. Use the four displayed groups: each covers 20 successive hundredth-metre values, such as 1.40, 1.41, …, 1.59. Count repeated distances separately: each entry belongs to a different student.

1.801.571.611.761.901.771.521.731.751.931.711.851.791.821.642.001.651.851.821.40
2.00–2.19 m
Includes 2.00.
1.80–1.99 m
Includes 1.80 but not 2.00.
1.60–1.79 m
Includes 1.60 through 1.79.
1.40–1.59 m
Includes 1.40 through 1.59.
Jump intervalFrequency
2.00–2.19 m
1.80–1.99 m
1.60–1.79 m
1.40–1.59 m
Total0

Which interval has the most students?

How many students are in that interval?

9

Independent practice

Frequency Table Workshop

Solve at least five of the six questions. You may change an answer and check again.

1. In 7, 9, 8, 6, 7, 8, 9, 7, 6, 8, how many values are in 6–7?

2. With intervals 40–49, 50–59, 60–69, where does 50 belong?

3. Cat, Dog, Cat, Fish, Dog, Cat, Dog, Cat: what is the frequency of Cat?

4. For whole-number observations, which grouping counts a boundary number in more than one interval? All written endpoints are included.

5. Exactly 24 observations are divided into four groups, with each observation counted once. Their frequencies are 5, 9, ?, and 4. Find the missing frequency.

6. A table shows 30–39: 3, 40–49: 7, 50–59: 5. How many values are 40 or above?

10

Mastery check

Exit Ticket

Try all five questions using what you learned about observations, groups and frequencies. Check your answers when you are ready.

1. What are raw data?

2. Using score groups 60–69, 70–79, 80–89, 90–99, and exactly 100, where does 90 belong?

3. In 12, 10, 11, 12, 13, 10, what is the frequency of 12?

4. A table has Good: 18 and Excellent: 6. How many times as many students received Good as received Excellent? Divide the Good count by the Excellent count.

5. Double-check the frequency total

Part A. Which rule makes the sum of all frequencies match the number of raw observations?

Part B. If exactly one observation is missed or counted twice, and all others are counted correctly, what happens to the frequency total?

Certificate of Mastery

Frequency Table Builder

This certifies that

can organize numerical and categorical raw data, use non-overlapping intervals, build accurate frequency tables, and read conclusions from them.

Lesson 2.1 Β· Grade 5 Math Studio

Teaching notes β€” adults

The worked example’s listed score data begin with 96, while its printed mean calculation substitutes 90. This rebuilt student lesson keeps the listed raw data consistently. Mean is intentionally reserved for Lesson 2.3, so the printing discrepancy is not turned into a child-facing activity.

The equal-width 0.20 m long-jump intervals are an instructional choice. The worked example asks for a reasonable grouping but does not specify a unique grouping.