Average
Find a mean by combining all values and sharing the total equally.
Lesson 2.3 · Simple Statistics
Find Information That Is Not Written Directly
Tables often give clues instead of final answers. Learn to combine the clues, choose the right operation, and check whether your result makes sense.
Read the clues. Choose the operation. Derive the answer. Check the meaning.
Rate
Find a percentage by comparing a part with the correct whole.
Missing value
Use totals, differences, and inverse operations to uncover information that is not shown.
derived information
mean / average
percentage
total
difference
missing value
weighted average
1
Data detective · What is already shown?
Read directly—or derive it?
A table can show some facts directly. Other facts must be calculated from the facts that are shown. Those calculated facts are called derived information.
| Grade | Total students | Absent |
|---|---|---|
| Grade 1 | 168 | 4 |
Total students
The table states 168. No calculation is needed.
Absent students
The table states 4. No calculation is needed.
Students present
Use the two shown facts: 168 − 4 = 164.
Attendance rate
Compare the number present with the total expected.
A. Present students
Is “164 students present” read directly or derived?
B. Total students
Is “168 students in Grade 1” read directly or derived?
C. Operation
Which operation finds the number present?
D. A useful habit
What should you ask before calculating?
2
Average · Equal sharing
The mean is a fair-share value
Imagine three teams collected 5, 7, and 12 tokens. If all 24 tokens were shared equally, each team would receive 8. That equal-share value is the mean.
The mean formula
Mean = total of all values ÷ number of values
(5 + 7 + 12) ÷ 3 = 24 ÷ 3 = 8
The mean does not have to be one of the original values. Here, 8 was not in the original list.
1Add every value5 + 7 + 12 = 24
2Count the valuesThere are 3 values.
3Divide24 ÷ 3 = 8
1. Total
What is 5 + 7 + 12?
2. Count
How many values are being averaged?
3. Mean
What is the fair-share value?
4. Meaning
Which statement is correct?
3
A complete worked example · 15 competition scores
Find a mean from raw data
The scores below are already collected. To find the mean, we must use every score exactly once.
9683958698
10084677281
7688658590
Accuracy strategy: Instead of adding all 15 numbers in one long line, make three groups of five. Then add the three subtotals.
96 + 83 + 95 + 86 + 98
100 + 84 + 67 + 72 + 81
76 + 88 + 65 + 85 + 90
Combine the subtotals
Count the scores
Divide total by count
Interpret the result
See a worked explanation
Percentages and rounding: a quick refresher
Percent means “out of 100.” If 18 of 20 students are present, 18 ÷ 20 = 0.9, and 0.9 × 100 = 90. Write 90% in words as “90 out of every 100.”
To round to one decimal place, look at the hundredths digit. In 97.619…, the hundredths digit is 1, so keep 97.6. In 93.75, it is 5, so round up to 93.8. Round only after the calculation.
For an attendance answer box followed by %, enter 93.8. Counts of students are whole numbers.
4
Percentage · Compare a part with the whole
Build an attendance rate
An attendance rate answers: “What percent of the students who were expected are present?” First find the number present, then compare it with the total enrollment.
Grade 1 information
Total = 168 students
Absent = 4 students
1Find present168 − 4
2Make a fractionpresent ÷ 168
3Make a percent× 100%
What the percentage looks like
97.6%
0%100%
A rate lets us compare grades of different sizes fairly.
Attendance rate = (number present ÷ total enrollment) × 100%
1. Present
168 total − 4 absent = ?
2. Correct denominator
Which number belongs under the fraction bar?
3. Attendance percentage
164 ÷ 168 × 100% ≈ ? Round to one decimal place.
4. Meaning
What does 97.6% describe?
5
Table lab · Complete and compare
Calculate five attendance rates
For each grade, calculate students present and then the attendance rate. Round each rate to one decimal place.
| Grade | Total | Absent | Present | Attendance rate |
|---|---|---|---|---|
| 1 | 168 | 4 | 164 | 97.6% |
| 2 | 84 | 0 | % | |
| 3 | 108 | 3 | % | |
| 4 | 154 | 2 | % | |
| 5 | 202 | 1 | % |
Highest rate
Which grade has the highest attendance rate?
Lowest rate
Which grade has the lowest attendance rate?
Why use a rate?
Grade 1 had 4 of 168 students absent; Grade 3 had 3 of 108 absent. Why does Grade 1 have the higher attendance rate?
6
Part–whole reasoning · Use inverse operations
Recover a missing value from a total
When a total and all but one of its parts are known, subtract all known parts from the total. When every part is known but the total is missing, add the parts.
Children’s books
Three-month total = 7,671 books sold
October = 7,671 − 1,538 − 1,895
Adult books
Three-month total = ?
Total = 890 + 723 + 1,046
| Type | Three-month total | October | November | December |
|---|---|---|---|---|
| Children’s books | 7,671 | 1,538 | 1,895 | |
| Adult books | 890 | 723 | 1,046 |
Missing part
Which rule finds a missing part?
Missing total
Which rule finds a missing total?
7
One table, several questions
Totals, differences, and averages
The same table can answer different questions. The wording tells us which operation to use.
Build the table from these facts. The factory planned 410 machines in each month. It actually made 416 in January, 430 in February, and 414 in March. Copy each fact into the correct cell before calculating totals.
| Month | Planned | Actual |
|---|---|---|
| January | ||
| February | ||
| March |
PlannedActual
Altogether
Add the months.
Add the months.
How many more?
Subtract plan from actual.
Subtract plan from actual.
Average per month
Divide the total by 3.
Divide the total by 3.
1. Planned total for January–March (machines)
2. Actual total for January–March (machines)
3. Actual total − planned total (machines)
4. Planned mean per month (machines)
5. Actual mean per month (machines)
6. Actual monthly mean − planned monthly mean (machines)
7. In which month was actual output minus planned output greatest?
8
Different group sizes · A weighted average
Combine villages fairly
A village average is “savings per household.” To combine villages of different sizes, first combine their total savings and their household counts.
Village A
Households120
Total savings38,400 yuan
Average per household
Village B
Households
Total savings39,000 yuan
Average per household300 yuan
Village C
Households110
Total savings
Average per household340 yuan
All three villages
Total households
Total savings
Average per household
Three steps: recover each village, combine totals, then divide The villages do not have the same number of households. Each household should count equally, so use total savings ÷ total households.
Correct combined method
Rounding
The combined result is 318.888…. What is it to the nearest yuan?
Enter 318.9 for the combined average to one decimal place, and 319 for the nearest whole yuan.
9
Independent practice · Match words to operations
Data Reasoning Workshop
Answer at least 5 of the 6 cards correctly. Each card gives enough information; your job is to decide what must be derived.
Solved: 0 of 6. Reach 5 to complete the workshop.
original challenge · Put your skills together
From quarterly sales to a complete table
A bookstore recorded the number of books sold in four quarters of one year (each quarter is three months). These are counts of books, not amounts of money. Organize these facts, then use your table to answer the questions.
- Quarter 1: 7,200 storybooks, 2,400 science and technology books, and 3,120 comics.
- Quarter 2: 6,000 storybooks, 1,980 science and technology books, and 3,020 comics.
- Quarter 3: 4,800 storybooks, 3,300 science and technology books, and 3,800 comics.
- Quarter 4: 6,600 storybooks, 4,800 science and technology books, and 2,560 comics.
| Book type | Q1 | Q2 | Q3 | Q4 | Annual total |
|---|---|---|---|---|---|
| Stories | |||||
| Science and technology | |||||
| Comics | |||||
| Total |
See a worked explanation
Annual totals: stories 24,600; science/technology 12,480; comics 12,500. Quarter totals: 12,720; 11,000; 11,900; 13,960. Both routes give 49,580 books.
Stories have the greatest annual sales. Q4 has the greatest quarterly total. Science/technology mean = 12,480 ÷ 4 = 3,120 books per quarter.
11
Mastery check · No lesson hints
Exit Ticket
Earn at least 4 of 5 to unlock the Hidden Information Detective certificate.
1. Mean
Find the mean of 18, 22, 20, and 24.
2. Attendance
A grade has 96 students and 6 are absent. Enter the number present and the attendance rate to one decimal place.
3. Missing part
A total of 1,200 is split into 380, 415, and one missing part. Find the missing part.
4. Plan versus actual
The plan is 400 items per month for 3 months. Actual outputs are 398, 420, and 422. How many items above plan is the actual total?
5. Choose the rule
Part A. Which formula finds a mean?
Part B. What is the denominator in an attendance rate?
Part C. How do you find one missing part when the total and all other parts are known?
Question 5 is scored from Parts A, B, and C. The optional reflection is for a teacher, parent, or student journal.
Mastery result
Certificate of Mastery
Hidden Information Detective
This certifies that
can derive means, percentages, missing values, totals, differences, and combined averages from statistical data.
Read the clues → choose the operation → check the meaning
Teaching notes
The visual models, vocabulary sequence, workshop, feedback, and exit ticket are instructional adaptations.
original consistency: The raw score list begins with 96, while the worked example’s printed mean calculation substitutes 90. This student page consistently uses the listed value 96, giving a total of 1,266 and a mean of 84.4. The discrepancy is kept here for teacher awareness rather than presented as a child task.
The quarterly bookstore challenge uses this lesson Exercise 3 ().