For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.
How this chapter applies
Eclat core: standard-deviation calculation and comparison are retained as an Eclat IP bridge where the school includes statistics in upper-secondary Mathematics.
School-sensitive extension: the whole chapter is outside K341 core, although its content overlaps K310 Topic S1 and may support later statistics pathways.
2027 national comparison: K341 has no statistics topic; K310 Topic S1 includes standard deviation for grouped and ungrouped data and comparison using mean and standard deviation.
Check your school: check whether the chapter belongs to E-Math, an IP extension, or later pathway preparation in the current school.
Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 20) Statistics - Standard Deviation cover? A: Compute mean, variance, and standard deviation for raw and grouped data in IP AMaths.
Standard deviation quantifies spread. Handle both raw datasets and grouped frequencies.
Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.
The core idea is simple: Standard deviation measures how spread out a dataset is.
Use it as a working check: Find the mean first, measure each value's distance from the mean, square those distances, average them, then take the square root.
Then go one layer deeper: Example: for grouped data, replace each class interval with its midpoint before multiplying by frequency. The midpoint is an estimate, so state grouped-data answers to a sensible accuracy.
1 Formulae
Mean: xˉ=n∑x.
Variance (population): σ2=n∑(x−xˉ)2.
For grouped data with midpoints mi and frequencies fi:
xˉ=∑fi∑fimi
Working-table checkpoint
Before calculating standard deviation, build the table that matches the data format. This prevents you from using the raw-value formula on a frequency table or treating a class interval as one exact value.
Data format
Columns to create
Mean uses
Variance uses
Raw list
x, (x−xˉ)2
∑x÷n
∑(x−xˉ)2÷n
Frequency table
x, f, fx, f(x−xˉ)2
Grouped data
class interval, midpoint m, f, fm, f(m−xˉ)2
Calculator shortcut
x, f, x2, fx2 or m2
Worked check: if the table says value 6 occurs 4 times, its contribution to the mean numerator is 4×6=24, not just 6. After finding the mean, its contribution to the variance numerator is 4(6−xˉ)2.
Misconception check: frequency is a multiplier. It tells you how many times a value or midpoint is counted, so it must appear in both the mean numerator and the variance numerator.
2 Spread interpretation checkpoint
Before calculating, identify whether the question changes every value in the same way or changes the spread itself. This helps you avoid recomputing when the spread can be reasoned out.
Data change
What happens to the mean?
What happens to standard deviation?
Why
Add k to every value.
Increases by k.
Unchanged.
Every value and the mean move together, so distances from the mean stay the same.
Subtract k from every value.
Decreases by k.
Unchanged.
The whole dataset shifts left by the same amount.
Multiply every value by k.
Multiplies by k.
Multiplies by ∣k∣.
Distances from the mean are stretched by the scale factor.
Replace grouped data by midpoints.
Becomes an estimate.
Becomes an estimate.
Each class is represented by its midpoint, not by the exact original values.
Common trap: standard deviation measures spread around the mean, not the size of the mean itself. Adding the same constant to every value changes the mean but not the spread.
Coded-data reversal checkpoint
If a question defines a coded variable, undo the coding in the right order. Translation changes the mean only; scaling changes both the mean and the standard deviation.
Coding statement
Mean relationship
Standard deviation relationship
Common trap
y=x−a
xˉ=yˉ+a
σx=σy
Adding a to the standard deviation as well.
y=bx−a
xˉ=a+byˉ
y=bx+a
yˉ=bxˉ+a
Worked check: if y=10x−50, yˉ=1.8, and σy=0.6, then
xˉ=50+10(1.8)=68,σx=10(0.6)=6.
Misconception check: the standard deviation ignores the +50 shift because all values and the mean move together. It responds only to the stretch factor 10.
3 Worked example - Raw data
Scores: 4,6,8,10,12.
Mean: xˉ=54+6+8+10+12=8.
Compute squared deviations: (4−8)2=16, (6−8)2=4
Variance: σ2=516+4+0+4+16=8
Standard deviation: σ=8=22
4 Worked example - Grouped data
Class intervals 0≤x<5, 5≤x<10, 10≤x<15 with frequencies 3,5,2 respectively.
5 Worked example - Frequency table with repeated values
A dataset records test attempts with the following frequencies: value 2 occurs four times, 5 occurs three times, and 9 occurs three times. Compute the mean and standard deviation.
Total frequency n=4+3+3=10.
Mean: xˉ=104(2)+3(5)+3(9)=108+15+27=5.
Squared deviations:
For x=2: (2−5)2=9; contribution 4×9=36
Variance: σ2=1036+0+48=1084=8.4
Standard deviation: σ=8.4≈2.90 (3 s.f.).
Answer: mean 5, standard deviation ≈2.90.
6 Practice Quiz
Review how shifts and scaling affect spread, and rehearse grouped-data calculations so you can respond quickly to statistics prompts.
7 Try this
Given grouped data with class width 4 and frequencies 2,7,9,4, construct the midpoints, compute xˉ, and evaluate σ accurately.
.
σ2=∑fi∑fi(mi−xˉ)2.
∑fx÷∑f
∑f(x−xˉ)2÷∑f
∑fm÷∑f
∑f(m−xˉ)2÷∑f
,
fm2
Same mean column as above
∑f∑fx2−xˉ2, using m for grouped data
σx=∣b∣σy
Forgetting to multiply the spread by the scale factor.
