BTT Mathematics / Primary Mathematics Learning Hub / Guide 8
Data Mathematics begins by asking what was counted or measured, how it was recorded and what the representation allows us to conclude. A table, bar graph or line graph can make a pattern easier to see, but it can also hide mistakes if the scale, units or categories are ignored.
An average compresses several values into one representative value by redistributing the total equally. A graph compresses data visually. Neither representation contains every detail of the original observations. Strong interpretation therefore combines calculation with questions about what the numbers actually represent.
This guide develops tables, picture and bar graphs, line graphs, average, proportional displays and evidence checks within the Statistics strand of the current MOE Primary Mathematics syllabus. Use the sections appropriate to the learner’s current school scope.
Collect and organise · Read graphs · Read change over time · Average · Compare fairly · Interpret evidence · 24 questions · Worked answers
1. Every data value belongs to a variable and a unit
Suppose a class survey records favourite fruit. Each response belongs to a category such as apple, orange or banana. A tally can record how many responses enter each category, and a frequency table can show the final count.
If a second survey measures the mass of school bags, the data are different in kind. The values might be recorded in kilograms and can take many numerical values. Both investigations use data, but one records categories while the other records measured quantities.
Before doing arithmetic, ask what one value represents. “12” might mean twelve pupils, twelve books borrowed or twelve degrees Celsius. A correct calculation with a missing label is harder to interpret and easier to misuse.
Categories should match the question
If pupils choose exactly one favourite fruit, the category counts can be added to obtain the number of respondents. If pupils may choose several fruits they like, adding category counts can count the same pupil several times.
The survey rule therefore matters. “Choose one” and “select all that apply” produce different kinds of totals. A table alone may not reveal the rule unless it is stated.
Use a tally to reduce recording errors
Tallies group marks in fives, making counts easier to scan. The tally is an intermediate recording method; the frequency is the final numerical count.
When checking a table, compare the number of original observations with the sum of frequencies if each observation must appear exactly once. A mismatch can reveal an omitted or double-counted item.
Example A: Read a frequency table
A class table shows eight pupils choosing apples, five choosing oranges and seven choosing bananas, with exactly one choice per pupil. The total number of pupils is 8 + 5 + 7 = 20.
Apples are the most frequent category with eight responses. The difference between apples and oranges is three pupils. These are different questions answered from the same table.
2. A graph is a representation with a scale
Picture graphs
A picture graph may use one symbol to represent several items. If one book icon represents four books, six icons represent 6 × 4 = 24 books. Counting icons without applying the key would understate the data.
Half-symbols or partial symbols should be interpreted according to the key when the graph uses them. For example, if one full icon represents four books, a half icon represents two books. Do not assume every picture is one item.
Bar graphs
Bar graphs use bar length or height to represent frequency or quantity. The axis scale decides what a particular height means. If each grid interval represents two pupils, a bar reaching seven intervals represents fourteen pupils.
The bars should be compared using the numerical scale, not merely visual impressions. A graph displayed on a narrow range may make a small difference look dramatic. The data values, not the apparent area of ink, control the comparison.
Example B: Compare bars with a non-unit scale
A graph uses a vertical scale marked 0, 5, 10, 15, 20. Team A reaches 15 and Team B reaches 10. Team A exceeds Team B by 5, not by one simply because the bar tops are one labelled step apart.
If the graph begins at zero, the bar lengths also carry a proportional visual meaning. If the axis is truncated, read the labels carefully before making claims about relative size.
Choose axes that match the data
In a bar graph of fruit choices, the horizontal axis can list fruit categories while the vertical axis shows number of pupils. In a line graph of temperature over time, time normally belongs on the horizontal axis because the line tracks how the measured quantity changes as time progresses.
Axis labels and units are part of the graph. A vertical value of twenty could mean twenty pupils, twenty dollars or twenty degrees. Without the label, the graph is incomplete for interpretation.
3. Line graphs show change across an ordered variable
A line graph connects observations in order, commonly across time. The connecting segments help the eye track change, but the meaning between recorded points depends on the context.
Example C: Read successive changes
Temperatures recorded at four times are 24°C, 26°C, 29°C and 27°C. The changes are +2°C, +3°C and −2°C. The largest rise between consecutive observations is three degrees.
The highest recorded temperature is 29°C. This is a different statement from “the largest rise is 3°C.” One concerns a level; the other concerns a change between levels.
Example D: Constant change
A tank reading is 10 L, 18 L, 26 L and 34 L at equal two-minute intervals. Each interval adds eight litres. If the same pattern continues for one more interval, the next reading is 42 L.
The average rate over each stated interval is eight litres per two minutes, or four litres per minute. Saying the pattern will continue is an assumption supplied by the question. Real data do not automatically continue linearly beyond the observed range.
Do not confuse interpolation with observation
If a line is drawn between a 9 a.m. reading and a 10 a.m. reading, the graph may visually suggest intermediate values. Whether those values were actually measured or are merely represented by the connecting line depends on the data collection method.
In school exercises, a continuous line may be intended to support reading between times. In real investigations, note whether an intermediate value is observed, estimated or assumed.
4. Average redistributes the total equally
For the arithmetic mean, add all values and divide by the number of values. This can be understood as pooling the total and sharing it equally across the observations.
Example E: Direct average
Find the average of 6, 8, 10 and 12. Their total is thirty-six. Divide by four values: 36 ÷ 4 = 9.
The average lies between the smallest and largest values. For ordinary positive school data, an answer of ninety would therefore signal a likely place-value or division error.
Example F: Find a missing value from the average
Four values have average fifteen. Their total must be 4 × 15 = 60. If three known values total forty-two, the missing value is 60 − 42 = 18.
Do not average the three known values and compare directly with fifteen. The stated average belongs to all four values, so reconstruct the required total first.
Example G: Target average
A learner has scores 72, 68, 75 and 85 and wants an average of 76 after a fifth score. Five scores averaging seventy-six require total 5 × 76 = 380. The existing four scores total 300. The required fifth score is 80.
Check: 72 + 68 + 75 + 85 + 80 = 380, and 380 ÷ 5 = 76.
Example H: Combining two groups
Group A has four values with average twelve, so its total is 4 × 12 = 48. Group B has six values with average fifteen, so its total is 6 × 15 = 90. Together there are ten values totaling 138.
The combined average is 138 ÷ 10 = 13.8. Simply averaging twelve and fifteen to get 13.5 would give the two groups equal weight despite their different sizes.
Averages can be combined by weighting each group according to the number of observations it contains.
Example I: Removing one observation
Five daily values have average twenty-four, so their total is 120. One day with value thirty is removed. The remaining four values total ninety and have average 22.5.
The new average falls because the removed value was above the old average. This direction check can help before exact calculation.
5. Compare data using a common basis
Different group sizes need percentages or rates
Class A has eighteen correct submissions out of thirty. Class B has twenty out of forty. Comparing eighteen with twenty alone suggests Class B has more correct submissions, which is true in count. But if the question asks which class has the higher correct proportion, use a common basis.
Class A: 18/30 = 60%. Class B: 20/40 = 50%. Class A has the higher percentage. Counts and proportions answer different questions.
Example J: A proportional display
If one-quarter of a survey’s 120 responses belong to Category A, the count is 120 ÷ 4 = 30. If a chart shows thirty per cent of two hundred responses in Category B, the count is sixty.
A sector or percentage describes a share of its own whole. Two equal-looking sectors from charts with different total sample sizes need not represent the same number of observations.
Frequency and total contribution are different
Suppose pupils report how many books they read: one book by three pupils, two books by five pupils and three books by two pupils. The number of pupils is 3 + 5 + 2 = ten.
The total number of books read is 1×3 + 2×5 + 3×2 = 19. The frequency counts pupils; multiplying each value by its frequency finds the total contribution.
The average books per pupil is 19 ÷ 10 = 1.9. Adding the category labels 1 + 2 + 3 would ignore how many pupils were in each category.
6. A graph can describe a pattern without proving its cause
Suppose library visitors rise from one hundred to one hundred twenty after new posters are displayed, then fall to ninety the next week. The data show the recorded visitor counts. They do not by themselves prove that posters caused the rise or the later fall.
Other factors may have changed: school events, weather, timetable, data collection or random variation. A causal claim needs a design that separates competing explanations.
Ask what is missing
A graph can omit the sample size, measurement method or period of collection. An average can hide variation: two classes can both average seventy marks while one has scores clustered near seventy and the other has very high and very low scores.
Primary Mathematics does not require advanced statistical inference to build this habit. Asking “What exactly does this graph show?” and “What does it not show?” is already useful evidence discipline.
Truncated axes can change visual impact
Two values, ninety-eight and one hundred, differ by two. On an axis from zero to one hundred, their bars look nearly equal. On an axis from ninety-seven to one hundred, the difference looks much larger.
The numerical difference remains two. A truncated axis is not automatically dishonest; it can reveal small variation. But the reader must inspect the scale before judging how large a change really is.
Correlation is a relationship, not a reason
If two quantities rise together, the data may show an association. It does not automatically establish that one caused the other. This distinction protects the learner from reading a visual pattern as a complete explanation.
For school questions, answer what the stated data support. For real-world claims, preserve uncertainty where the data do not settle the cause.
7. Practice: 24 questions
Keep the worked answers covered. State what each number represents and inspect graph keys or scales before calculating. Some questions are described in words so they can be completed without a printed image.
Questions 1–8: Tables and graph scales
1. A class table shows 8 pupils choosing apples, 5 choosing oranges and 7 choosing bananas. Each pupil chooses exactly one fruit. How many pupils were surveyed?
2. In the same table, how many more pupils chose apples than oranges?
3. A picture graph uses one book icon to represent 4 books. Six icons are shown. How many books do they represent?
4. A bar graph scale increases by 2 pupils per grid interval. A bar reaches 7 intervals above zero. What frequency does it represent?
5. A bar graph has labelled values 0, 5, 10, 15 and 20. Team A reaches 15 and Team B reaches 10. By how much does A exceed B?
6. Temperatures recorded at four successive times are 24°C, 26°C, 29°C and 27°C. What is the highest recorded temperature?
7. For the temperatures in question 6, what is the largest rise between consecutive observations?
8. A graph of temperature over a day should place time on one axis and temperature on the other. Which variable is normally placed on the horizontal axis?
Questions 9–16: Average and frequency
9. Find the average of 6, 8, 10 and 12.
10. Four values have average 15. Three of the values total 42. Find the fourth value.
11. Scores are 72, 68, 75 and 85. What fifth score is needed for an average of 76?
12. Group A has 4 values with average 12. Group B has 6 values with average 15. Find the combined average.
13. Five daily values have average 24. One value of 30 is removed. Find the average of the remaining four values.
14. Three pupils read 1 book each, five pupils read 2 books each and two pupils read 3 books each. How many pupils are represented?
15. For question 14, how many books were read altogether?
16. Find the average number of books read per pupil in questions 14–15.
Questions 17–24: Proportion, trend and evidence
17. One-quarter of 120 survey responses belong to Category A. How many responses is that?
18. Thirty per cent of 200 responses belong to Category B. How many responses is that?
19. Class A has 18 correct submissions out of 30. Class B has 20 out of 40. Which class has the higher percentage of correct submissions?
20. Tank readings at equal two-minute intervals are 10 L, 18 L, 26 L and 34 L. If the same pattern continues, what is the next reading?
21. For question 20, what is the rate of increase in litres per minute?
22. A bar graph displays values 98 and 100 using an axis that begins at 97. Has the numerical difference between the values changed because the axis was truncated?
23. Two classes both have average score 70. Can you conclude that the score distributions in the two classes are identical? Explain.
24. Library visitors are 100 one week, 120 the week posters are introduced and 90 the following week. Does this information alone prove the posters caused the increase to 120? Explain.
8. Worked answers
Answers 1–8
1. 20 pupils. Because each pupil chooses exactly one fruit, add the three non-overlapping category frequencies: 8 + 5 + 7 = 20.
2. 3 pupils. Compare the apple frequency eight with the orange frequency five: 8 − 5 = 3.
3. 24 books. Each icon represents four books. Six icons represent 6 × 4 = 24 books.
4. 14 pupils. Seven intervals at two pupils per interval give 7 × 2 = 14. Counting intervals as individual pupils would ignore the scale.
5. 5. Read the axis values directly: 15 − 10 = 5. One labelled step represents five units.
6. 29°C. Compare the four recorded levels. Twenty-nine is the greatest.
7. 3°C. Successive changes are +2, +3 and −2 degrees. The largest rise is therefore three degrees.
8. Time. In a usual line graph of change over time, time is placed on the horizontal axis and the measured quantity on the vertical axis.
Answers 9–16
9. 9. The total is 6 + 8 + 10 + 12 = 36. Divide by four values: 36 ÷ 4 = 9.
10. 18. Four values averaging fifteen require total sixty. The known three total forty-two, so the missing value is 60 − 42 = 18.
11. 80. Five scores averaging seventy-six require total 380. The existing four total 300. The fifth score must supply the remaining eighty.
12. 13.8. Group A totals 4 × 12 = 48. Group B totals 6 × 15 = 90. Combined total 138 across ten values gives 13.8.
13. 22.5. The five-day total is 5 × 24 = 120. Remove thirty to leave ninety across four days. 90 ÷ 4 = 22.5.
14. 10 pupils. Add the frequencies: 3 + 5 + 2 = 10. The category values 1, 2 and 3 describe books per pupil, not numbers of pupils.
15. 19 books. Multiply each book count by its frequency: 1×3 + 2×5 + 3×2 = 3 + 10 + 6 = 19.
16. 1.9 books per pupil. Divide the total nineteen books by ten pupils. The mean can be non-whole even though individual pupils read whole numbers of books.
Answers 17–24
17. 30 responses. One-quarter of 120 is 120 ÷ 4 = 30.
18. 60 responses. Thirty per cent of two hundred is 0.30 × 200 = 60.
19. Class A. Class A has 18/30 = 60%. Class B has 20/40 = 50%. Class A has the higher proportion even though Class B has a larger raw count.
20. 42 L. Each equal interval adds eight litres: 10, 18, 26, 34, 42. The continuation depends on the stated assumption that the same pattern persists.
21. 4 L per minute. Eight litres are added in two minutes, so divide by two to obtain four litres per minute.
22. No. The numerical difference remains 100 − 98 = 2. The truncated axis changes visual emphasis, not the underlying values.
23. No. The same average can arise from different collections of scores. One class might cluster near seventy while another has a wider spread. The mean alone does not determine every observation.
24. No. The data show that the recorded count was higher in the week posters were introduced, but they do not isolate the posters from other possible causes. Association in this short sequence is not proof of causation.
9. Diagnose the reading before adding more arithmetic
If a learner answers six books for a picture graph with six icons where each icon represents four, the multiplication skill may not be the problem. Ask what one icon stands for. The missing step is the graph key.
If bar comparisons fail, inspect the scale and axis labels before revising subtraction. A correct difference calculation attached to the wrong bar values will still be wrong.
For average, rebuild the total
When reverse-average questions cause difficulty, write “number of values × average = total.” Then label what part of that total is already known. This reduces a verbal puzzle to a part-whole relationship.
For combined groups, find each group’s total before combining. The group averages cannot usually be averaged directly unless the groups contain the same number of observations.
Ask what claim the data support
“The highest bar is Category A” may be supported. “People prefer A because it is better” is a different causal claim. Teach the learner to separate a numerical description from an explanation that requires more evidence.
This habit also improves word-problem reading. Mathematics is not only producing a number; it is deciding what statement that number is allowed to support.
Check a representation against the original data
Sum frequencies when each observation belongs once. Compare a graph value with its table entry. Multiply an average by the number of observations to recover the total. These return paths make compressed representations auditable.
If the representation cannot reconstruct the original totals or conditions, find the first point where information was lost or misread.
Continue through the Primary Mathematics series
For percentages used to compare groups of different sizes, use Ratio, Rate and Percentage. For quantities and units on graph axes, use Measurement, Units, Perimeter, Area and Volume. For positions on coordinate grids, use Geometry, Angles, Symmetry and Coordinates.
The broader concept owner remains Statistics and Data. Return to the BTT Primary Mathematics Learning Hub for the full worked-guide collection.
Original learning guide. Curriculum reference checked 6 September 2026. All tables and graph scenarios are original text-based teaching examples rather than official examination items.