Q: What does IP Maths Notes (Lower Sec, Year 1-2): 09) Data Handling & Statistics cover? A: Construct tables, compute mean/median/mode, interpret box-and-whisker plots, and justify graph choices for IP assignments.
The core idea is simple: Clean tables make statistics easier to calculate and explain.
Use it as a working check: Choose the graph from the data type, then use mean, median, mode, range, quartiles, and correlation language to describe what the data shows.
Then go one layer deeper: Use the examples to practise sorting data, building frequency tables, estimating from grouped data, and writing one clear sentence that links the statistic back to context.
Statistics converts raw data into decisions. Learn how to summarise, display, and interpret data sets with precision.
These notes align with MOE Lower Secondary Mathematics syllabus used in IP pathways (aligned to O-Level Mathematics 4052 foundations).
Status: MOE Lower Secondary Mathematics syllabus (latest release) checked 2025-11-30 - scope unchanged; remains the reference for these lower-sec notes.
Add frequencies progressively to track the number of observations below a threshold. Useful for median estimates in grouped data.
Cumulative-frequency position checkpoint
For a cumulative-frequency table, first find the position you need, then locate the first cumulative total that reaches or passes that position.
Target
Position to find
How to locate it
Common trap
Median
2n
Find the first cumulative frequency at least this large.
Choosing the class with the largest frequency instead of the running total position.
Lower quartile
4n
Find the first cumulative frequency at least this large.
Dividing by 4, then forgetting to use cumulative frequency.
Upper quartile
43n
Find the first cumulative frequency at least this large.
Reading the third class automatically as Q3.
Percentile
Required percentage of n
Convert the percentage to a position, then locate it in the cumulative column.
Treating the percentage as a frequency count.
Worked check: if n=40, the median position is 20, the lower-quartile position is 10, and the upper-quartile position is 30. If the cumulative totals are 6, 17, 31, and 40, then both the median and upper quartile lie in the third class because the running total first reaches positions 20 and 30 there.
Misconception check: cumulative frequency answers "how many values have appeared so far?" It is not the same as the frequency in that class.
2 Measures of central tendency
Mean: xˉ=n∑x.
Median: Middle value after sorting; if even number of data, average the two middle values.
Mode: Most frequent value.
Worked example: Data set: 12, 15, 18, 19, 20, 20, 21, 25.
xˉ=812+15+18+19+20+20+21+25=8150=18.75.
Median = average of 4th and 5th values → 219+20=19.5.
Mode = 20.
Range = 25 - 12 = 13.
Which average should you quote?
Before calculating, decide what the question is trying to compare.
Situation
Statistic to prioritise
Why it fits
Common trap
Data has one unusually high or low value
Median
It is less affected by an outlier than the mean.
Quoting only the mean when one extreme value pulls it away from most results.
Data is categorical, such as favourite CCA or transport type
Mode
It names the most common category.
Trying to calculate a mean for labels.
Data is numerical and roughly balanced
Mean
It uses every value in the data set.
Forgetting that the mean can be a decimal even when all data values are whole numbers.
Question asks how spread out the middle half is
Interquartile range
It compares the central 50% of the data.
Using range when one extreme value makes the spread look larger than most results.
Misconception check: "average" is not automatically the mean. State which measure you used and why it suits the data.
3 Grouped data estimates
Estimate mean using midpoints xi and frequencies fi:
xˉ≈∑fi∑fixi.
Median class: locate where cumulative frequency crosses 2n.
Grouped-mean setup checkpoint
For grouped data, you do not know every exact value. Use each class midpoint as an estimate, then weight it by frequency.
Step
What to write
Common trap
Find midpoint
Add the lower and upper class limits, then divide by 2.
Using the lower endpoint as if every value in the class is there.
Multiply by frequency
Calculate fx for each class.
Adding midpoints without weighting by how many values are in each class.
Add the weighted totals
Find ∑fx and ∑f.
Dividing by the number of classes instead of the total frequency.
Estimate the mean
Use xˉ≈∑f∑fx.
Reporting the answer as exact when grouped data only gives an estimate.
Worked check: for classes 0-10, 10-20, and 20-30 with frequencies 6, 11, and 5, the midpoints are 5, 15, and 25. Then ∑fx=6(5)+11(15)+5(25)=320 and ∑f=22, so the estimated mean is 22320≈14.5.
Misconception check: the midpoint is a representative value for the class, not a claim that every observation in that class equals the midpoint.
4 Data representations
Bar chart: discrete categories.
Histogram: continuous data (no gaps between bars).
Pie chart: proportionate sectors (rare for IP exams, but know how to compute angles: angle=nfrequency×360∘).
Box-and-whisker plot: shows median, quartiles, and spread.
Scatter plot: pairs of values; look for correlation.
Graph choice checkpoint
Before drawing, identify whether the data is categorical, continuous, paired, or already summarised.
Data or question cue
Best display
First check before drawing
Common trap
Categories such as CCA, colour, or transport type
Bar chart
Keep bars separated and label each category clearly.
Joining bars as if the categories are continuous.
Continuous measurements grouped into intervals
Histogram
Use touching bars and check that class intervals are consistent.
Leaving gaps between bars or treating interval labels as separate categories.
Parts of one whole
Pie chart
Convert each frequency to a sector angle out of 360 degrees.
Using a pie chart when the totals come from different groups.
Comparing centre and spread
Box-and-whisker plot
Mark minimum, lower quartile, median, upper quartile, and maximum in order.
Comparing only the maximum when the question asks about typical value or consistency.
Paired numerical values, such as height and arm span
Scatter plot
Plot each pair as one point and describe the trend.
Connecting the points like a line graph before judging correlation.
Worked check: if a question compares students' heights grouped as 150-154 cm, 155-159 cm, and 160-164 cm, use a histogram because height is continuous. If it compares favourite sports, use a bar chart because the choices are categories.
Scatter-plot correlation checkpoint
For scatter plots, describe the relationship before making a conclusion. A good sentence names the direction, strength, variables, and any unusual point.
Feature to check
What to look for
Sentence move
Common trap
Direction
Points generally rise, fall, or show no clear pattern.
"There is positive correlation between revision time and score."
Saying "higher correlation" without naming what increases.
Strength
Points lie close to or far from an imagined trend line.
"The correlation is strong because the points are close to a straight-line pattern."
Calling every upward trend strong.
Outlier
One point is far from the main cluster.
"One result is unusual and should be checked before drawing a firm conclusion."
Ignoring the outlier because most points fit the trend.
Causation
Whether the graph alone proves one variable causes the other.
"The graph suggests an association, but it does not by itself prove cause."
Writing that one variable definitely causes the other from a scatter plot alone.
Worked check: if points for revision time and test score slope upward but one student revised for many hours and still scored low, write: "There is a generally positive correlation between revision time and score, but one outlier means the conclusion should be cautious." This is stronger than just writing "positive correlation".
Misconception check: correlation describes a pattern between two variables. It does not automatically prove why the pattern happens.
Worked example - Box plot interpretation
Five-number summary: minimum 28, lower quartile 34, median 41, upper quartile 47, maximum 55.
Interquartile range (IQR) = 47 - 34 = 13 (measure of middle spread).
50% of data lies between 34 and 47.
If comparing classes, comment on centres (median) and spreads (IQR).
Box-plot comparison checkpoint
When comparing two box plots, write about centre and spread separately. A higher maximum alone is not enough to say one group usually did better.
Feature to compare
What to calculate or read
Sentence frame
Trap to avoid
Typical value
Compare the medians
Class A has a higher median, so a typical Class A score is higher.
Comparing only the highest score.
Consistency
Compare the IQRs
Class B has a smaller IQR, so the middle 50% of scores are more consistent.
Saying smaller spread always means better performance.
Overall spread
Compare the ranges
Class A has a larger range, so its scores are more spread out overall.
Ignoring outliers or extremes.
Overlap
Check whether the boxes overlap strongly
The middle 50% of both groups overlap, so the difference in typical performance may be small.
Treating a small median difference as a large gap without checking spread.
Worked check: Class A has median 72 and IQR 8. Class B has median 68 and IQR 18. Class A has the higher typical score and is more consistent, because its median is higher and its middle 50% is less spread out.
Misconception check: "better" needs context. In test scores, higher centre is usually better; in timings or error counts, lower centre may be better.
5 Writing statistical commentary
Answer three questions:
What is the key statistic?
So what does it mean in context?
Now what: what action or conclusion follows?
Example sentence: “Class A's median score is 41, 3 marks higher than Class B's, suggesting stronger overall performance despite similar spreads.”
Commentary sentence checkpoint
When a question asks you to "comment" or "compare", do not list numbers only. Turn the statistic into a sentence that names the group, the measure, and the context.
Weak answer
Stronger answer move
Why it earns more credit
"Class A is 72."
Name the statistic: "Class A's median score is 72."
The marker knows which measure you are using.
"Class A is higher."
Compare against another value: "Class A's median is 4 marks higher than Class B's."
The comparison is quantified.
"Class B is more spread out."
Name the spread measure: "Class B has a larger IQR, so the middle 50% of scores is less consistent."
The conclusion is tied to the correct feature of the box plot.
"There is positive correlation."
Add context: "As revision time increases, test score tends to increase."
The trend is linked to the variables in the question.
Worked check: if Class A has median 72 and IQR 8, while Class B has median 68 and IQR 18, write: "Class A has the higher typical score because its median is 4 marks higher, and it is more consistent because its IQR is 10 marks smaller."
Misconception check: more numbers do not automatically make better commentary. Pick the measure that answers the question, then explain what it means in context.
Practice Quiz
Test your mastery of averages, grouped-data estimations, data displays, and commentary language with the quiz below.
Try it yourself
A data set has values 4, 7, 7, 8, 9, 10, 10, 13. Compute mean, median, mode, and range.
Construct a frequency table and histogram for the following heights (cm): 150, 152, 154, 154, 155, 156, 157, 158, 158, 159, 160.
Given grouped data classes 0-10, 10-20, 20-30 with frequencies 6, 11, 5, estimate the mean.
Create a box plot for the data set {12, 18, 19, 23, 24, 25, 27, 31, 34} and interpret the median and IQR.
