Saturday, 29 August 2026·Indigo — tuition·Mathematics · Statistics & Data Analysis
Everything from this session, rebuilt as a study page: the two standard-deviation
formulas with the worked numbers, how to read all five values off a box plot, the rules for
cumulative frequency curves, and the exact sentence structure the exam wants when you compare
two sets of data.
What this lesson covered
Five things, in the order they were taught:
Mean, median and mode — and which one actually represents a set of data when there is
an extreme value in it.
Standard deviation — two equivalent formulas, by hand, then the fast calculator route.
Box-and-whisker plots — the five values you must be able to read off in seconds.
Cumulative frequency curves — finding n, reading quartiles, and the "more than" flip.
Comparing two data sets — one measure of average, one measure of spread, written as
two sentences.
The whole topic is built on one split: one number for the centre
(mean / median / mode) and one number for the spread (range / interquartile range /
standard deviation). Almost every question asks you to produce one of each and then say what
they mean.
The four blocks of the topic. Every exam question is somewhere on this map.
1 · Mean, median, mode — which one represents the data?
Measure
How you get it
Use it when
It lets you down when
Mean (x̄)
Add every value, divide by n
Data is fairly even, no wild values
One extreme value drags it away from where the data actually sits
Median
Sort, take the middle position. With an even n, average the two middle values
There is an outlier, or the data is skewed
You need every value to count (it ignores how far away the extremes are)
Mode
The value that occurs most often
Data is categorical, or you want the most common outcome
There is no repeat, or several values tie
The worked set from the lesson
Eight values were used all the way through:
5 5 5 6 7 7 8 17
7.5
Mean — 60 ÷ 8
6.5
Median — average of 4th and 5th (6 and 7)
5
Mode — occurs three times
3.74
Standard deviation
The question that was put to the class: mean or median — which better represents this set?
The median. The 17 is an outlier: it sits far from every other value and pulls the mean up
to 7.5, above seven of the eight numbers. The median, at 6.5, sits where the data actually is.
Trap. With an even number of values the median is the average of the two middle ones,
not "the fourth one". With eight values it lives between the 4th and 5th positions.
2 · Standard deviation — the same answer by two roads
Standard deviation answers one question: on average, how far is a value from the mean?
Two sets can share a mean and be nothing alike.
The mean cannot tell these two sets apart. The standard deviation can.
Version 1 — deviations from the mean
σ = √( Σ(x − x̄)² ÷ n )
Find how far each value is from the mean, square it (so negatives and positives stop cancelling),
average the squares, then square-root back to the original units.
Value x
Frequency
x − x̄
(x − x̄)²
f × (x − x̄)²
5
3
−2.5
6.25
18.75
6
1
−1.5
2.25
2.25
7
2
−0.5
0.25
0.50
8
1
0.5
0.25
0.25
17
1
9.5
90.25
90.25
Total
8
112
σ = √(112 ÷ 8) = √14 = 3.7416… ≈ 3.74
Trap. A value that occurs three times contributes three times. Multiply each squared
deviation by its frequency — this is where marks quietly disappear.
Version 2 — mean of the squares minus the square of the mean
σ = √( Σx²/n − x̄² ) "average of the squares − average, squared"
Faster once you already know the mean: square each value, average those, subtract the mean squared.
The calculator route — what the lesson said to practise
The exam expects the calculator, not the table. On a ClassWiz-type calculator
(the Casio FX-97SG X is the model approved for use in Singapore schools):
SETUP → Statistics → Frequency: ON. Without this you cannot enter "5 occurs 3 times"
in one row and you will mis-key long lists.
MENU → Statistics → 1-Variable.
Type the values in the x column and the counts in the FREQ column.
AC, then OPTN → 1-Variable Calc.
Read x̄ = 7.5 and σx = 3.7416…
σx or sx? The calculator gives you both. σx divides by n — that is the one these
questions want. sx divides by n − 1 (sample standard deviation) and will be slightly larger.
Picking the wrong one costs the mark even though your method was right.
3 · Box-and-whisker plots
A box plot is a picture of five numbers. The lesson's point: you should be able to read all five
in a few seconds, because the marks are for what you say about them, not for finding them.
The five readings. Quartile positions here are drawn for shape; the lesson's own
minimum and maximum were 12 and 25, giving a range of 13.
The trap that was called out twice. The median line does not have to sit halfway
across the box. Its position is decided by the data, not by the drawing. A median pushed towards
one end tells you the data is skewed — that is information, not a badly drawn diagram.
Why the middle 50% at all?
Because the extremes are the least trustworthy part of a data set — one unusual value can sit far
out and say nothing about the group. The interquartile range throws away the outer quarters and
measures the part where the data actually lives.
4 · Cumulative frequency curves
25% of 124 is 31, 50% is 62, 75% is 93. Read across to the curve, then down.
You are asked for
What you do
How many data values in total (n)
Read where the curve ends on the vertical axis — not the largest number on the
horizontal axis
Median
Go in at 50% of n, across to the curve, down to the value
Lower quartile Q₁
25% of n
Upper quartile Q₃
75% of n
Interquartile range
Q₃ − Q₁ — write down both readings first, then subtract
"How many scored more than 55?"
Read the value at 55 (that is the "≤ 55"
count), then n − that reading
Every reading off this curve is a "less than or equal to" reading. This is the single
biggest source of lost marks in the topic. If the question says more than, you must
subtract from the total. The value exactly on the boundary is included in the "≤" side.
A marks tip that came straight from the lesson. When you find an interquartile range,
write down the two values you read — for example "Q₃ = 62, Q₁ = 49, IQR = 13" — rather than only
the final 13. Graph readings vary slightly between students, so schools accept a range of answers,
but only if they can see which readings you took. A bare "13" with no workings gives the marker
nothing to accept.
Reading between the gridlines
If your line falls between two gridlines, take the halfway value. Between 52 and 54, read 53.
A reading that is a little off is normal; a reading that is wildly off — answering 50 minutes when
the whole axis only goes to 10 — means you have misread the axis, and that is worth a second look
before you write it down.
The reversed curve, and the trick with two curves
Most cumulative frequency curves rise: they count "≤ this value". Some questions plot the
opposite — "number of customers who waited more than t minutes" — and that curve
falls. Everything flips: reading across now gives you the "more than" count directly, and
it is the "less than" count you have to get by subtracting.
Where the two curves cross, the count above equals the count below — so that value
is the median.
5 · Comparing two sets of data — the answer structure
This is the part that carries the marks, and it has a fixed shape. One sentence about the
average, one sentence about the spread, each naming the measure you used and what it means in
the context of the question.
Sentence
Measure you quote
What it tells the reader
1 — average performance
Median (or mean)
"On average, School A performed better, because its median mark of 40 is higher than
School B's 38."
2 — consistency / spread
Interquartile range (or standard deviation)
"The two schools were equally consistent, because their interquartile ranges are the
same." Or: "School A was less consistent, because its interquartile range is larger, so its
marks are more spread out."
Larger spread → more spread out → less consistent. Say all three; the marks are
for the interpretation, not for the number.
Never compare a median with a standard deviation. Compare like with like: median against
median, IQR against IQR. And always quote the actual figures — a comparison with no numbers in it
does not earn the mark.
What happens if everyone's marks change
A favourite follow-up: "5 marks are added to every student's score. What happens to the median?
To the interquartile range?"
Statistic
Add c to every value
Multiply every value by k
Mean, median, mode, Q₁, Q₃
increase by c
multiply by k
Range
unchanged
multiply by |k|
Interquartile range
unchanged
multiply by |k|
Standard deviation
unchanged
multiply by |k|
Variance (σ²)
unchanged
multiply by k²
Why. Adding the same number to everyone slides the whole data set along the number line.
Every value moves, so every measure of position moves with it — but nothing gets closer to
or further from anything else, so every measure of spread stays exactly the same.
The traps, in one list
Reading n off the horizontal axis instead of the end of the curve.
Forgetting that every cumulative frequency reading is "≤", so "more than" needs n − reading.
Giving only the final IQR without the two readings it came from.
Forgetting frequency when computing standard deviation by hand.
Using sx instead of σx on the calculator.
Assuming the median sits in the middle of the box.
Quoting the mean when an outlier is present and the median is the honest answer.
Comparing two sets with only one sentence — the spread sentence is a separate mark.
Treating a falling "more than" curve like a normal rising one.
Practice
Work these before looking. They mirror the questions the lesson went through.
1 · For 4, 4, 6, 9, 12, 13 — find the mean, median and standard deviation.
Mean = 48 ÷ 6 = 8. Median = average of 3rd and 4th = (6 + 9) ÷ 2 = 7.5.
Σx² = 16 + 16 + 36 + 81 + 144 + 169 = 462. σ = √(462/6 − 8²) = √(77 − 64) = √13 ≈ 3.61.
2 · A box plot shows min 12, Q₁ 16, median 18, Q₃ 22, max 25. Give the range and the IQR,
and say what the median's position tells you.
Range = 25 − 12 = 13. IQR = 22 − 16 = 6.
The median sits 2 above Q₁ but 4 below Q₃, so the lower half of the middle 50% is tighter than the
upper half — the data is skewed towards the higher values.
3 · A cumulative frequency curve for 124 students ends at 124. Reading at 55 marks gives
86. How many scored more than 55?
The 86 is the number who scored 55 or fewer. More than 55 = 124 − 86 = 38.
4 · Two classes have median 38 and 40, and both have an interquartile range of 11.
Write the comparison.
On average the second class performed better, as its median mark of 40 is higher than the first
class's 38. The two classes were equally consistent, as both have an interquartile range of 11, so
their marks are equally spread out.
5 · Every student is given 5 bonus marks. State the effect on the median, the range and
the standard deviation.
The median increases by 5. The range is unchanged. The standard deviation is unchanged — adding
the same amount to everyone shifts the data without spreading it.
6 · A curve is drawn for "number of customers waiting more than t minutes", and a second
curve for "waiting t minutes or less" is drawn on the same axes. What does the crossing point give?
The median waiting time. At that point the number waiting longer equals the number waiting less,
which is the definition of the middle value.
Worth checking with the teacher
Whether your school wants the phrase "on average" in the comparison sentence — it was
mentioned that schools differ on the exact wording they accept.
Whether comparisons should quote the median and IQR, or the mean and standard
deviation, when the question does not say.
The worked example on the last page of the handout was interrupted; the reversed-curve
question is the one to redo carefully.
Samuel Study · Mathematics · Statistics and Data Analysis
Lesson of Saturday, 29 August 2026 · Indigo
Built from the session; formulas, worked figures and practice answers independently recomputed.