IP AMaths Notes (Upper Sec, Year 3-4): 20) Statistics - Standard Deviation

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

Compute mean, variance, and standard deviation for raw and grouped data in IP AMaths.

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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.
Exam-track route: use the separate O-Level and SEC G3 Additional Mathematics notes for K341 topic ownership.

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.

New to the Integrated Programme? Start with What is IP? | Browse all free IP 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: $\bar{x} = \dfrac{\sum x}{n}$

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Marcus Pang·Managing Director (Maths)

Data format	Columns to create	Mean uses	Variance uses
Raw list	$x$ , $(x-\bar{x})^2$	$\sum x \div n$	$\sum (x-\bar{x})^2 \div n$
Frequency table	$x$ , $f$ , $fx$ , $f(x-\bar{x})^2$
Grouped data	class interval, midpoint $m$ , $f$ , $fm$ , $f(m-\bar{x})^2$
Calculator shortcut	$x$ , $f$ , $x^2$ , $fx^2$ or $m^2$

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 $\lvert k\rvert$ .	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.

Coding statement	Mean relationship	Standard deviation relationship	Common trap
$y = x - a$	$\bar{x} = \bar{y} + a$	$\sigma_x = \sigma_y$	Adding $a$ to the standard deviation as well.
$y = \dfrac{x - a}{b}$	$\bar{x} = a + b\bar{y}$
$y = bx + a$	$\bar{y} = b\bar{x} + a$

IP AMaths Notes (Upper Sec, Year 3-4): 20) Statistics - Standard Deviation

How this chapter applies

1 Formulae

Sources

Working-table checkpoint

2 Spread interpretation checkpoint

Coded-data reversal checkpoint

3 Worked example - Raw data

4 Worked example - Grouped data

5 Worked example - Frequency table with repeated values

6 Practice Quiz

7 Try this

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