H2 Maths Normal Distribution Formula Sheet | Standardisation

Q: What does H2 Maths Notes (JC 1-2): 6.3) Normal Distribution cover?
A: Standardisation, symmetry, inverse normal, and linear transformations for normal models in the 2026 H2 Maths syllabus.

Download: Get the H2 Maths Normal Distribution formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
Bookmark the GC normal menu (normalcdf, invNorm) and keep a sketch pad handy-exam scripts still expect hand-drawn bell curves with shaded regions. The 2026 syllabus excludes normal approximation to binomial distribution, so treat binomial and normal questions as separate models.

• A normal question is a bell-curve area question: Sketch and shade the region.
• Standardisation turns the original variable into a z-score: Subtract the mean, then divide by the standard deviation.
• Inverse normal works backwards from an area to a value: Decide whether the given percentage is a left-tail or right-tail area.

Concrete example: If the top 5 percent is needed, use the 95th percentile from the left, then convert it back to the original score scale.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 6.3 is assessed in Paper 2 Section B (Probability and Statistics, 60 marks); normal approximation to binomial distribution is excluded.

Formulas at a glance

Every result the 9758 syllabus expects you to apply, on one screen. MF27 does not include standard normal tables or normal critical values, so you need to know how to standardise, use your graphing calculator, and combine variances correctly. Worked examples appear in the sections below.

Distribution & standardisation

Inverse normal

Linear transformation $Y = aX + b$

Combining independent $X, Y$

Core Concepts

• A normal model is written $X \sim \mathcal{N}(\mu, \sigma^2)$ with mean $\mu$ and variance $\sigma^2$.
• Standardisation converts

Standardisation Workflow

1. Sketch the bell curve, mark $\mu$ and shade the requested region.
2. Convert bounds using $z = \frac{x - \mu}{\sigma}$.
3. Evaluate the corresponding standard normal probability with the table or GC.
4. State the final probability to three significant figures unless otherwise required.

Example -- Tail probability

Industrial bearings have $X \sim \mathcal{N}(45.0, 1.2^2)$ mm. Bearings longer than $46.5$ mm are scrapped. Find the scrap rate.

1. Compute $z = \frac{46.5 - 45.0}{1.2} = 1.25$.
2. $P(X \geq 46.5) = P(Z \geq 1.25) = 1 - \Phi(1.25) \approx 0.106$

Inverse Normal Problems

• Typical prompt: find $x$ such that $P(X \leq x) = p$.
• Use $x = \mu + z_p \sigma$

Example -- Percentile target

Chemistry practical scores obey $X \sim \mathcal{N}(68.5, 4.2^2)$. What is the minimum score for the top 5%?

1. Need $p = 0.95$, so $z_{0.95} = 1.645$ (from tables or invNorm(0.95, 0, 1)).
2. $x = 68.5 + 1.645 \times 4.2 \approx 75.4$

Tail-area checkpoint

Before using normalcdf or invNorm, translate the wording into the area your calculator needs.

Misconception check: invNorm usually asks for the area to the left of the cut-off. For a top-tail question, convert the right-tail percentage into its left-tail complement before finding the value.

Central-interval checkpoint

When a question gives a central percentage such as "middle 90 percent" or $P(|Z| < a) = 0.90$, split the leftover probability equally between the two tails before using inverse normal.

Worked check: for the middle $90\%$ of a standard normal distribution, the two tails take $10\%$ altogether, so each tail is $5\%$. The upper cut-off is therefore $z_{0.95} \approx 1.645$, and the interval is approximately $-1.645 < Z < 1.645$

Misconception check: do not enter $0.90$ directly into invNorm when the question asks for a symmetric central interval. $0.90$ gives the 90th percentile, not the two-sided middle 90%.

Unknown parameter checkpoint

Some normal-distribution questions give a probability and ask for $\mu$ or $\sigma$. Do not guess the parameter from the picture. Convert the probability into a standard-normal cut-off first, then write the z-score equation with the unknown still inside it.

Worked check: if $X \sim \mathcal{N}(\mu, 6^2)$ and $P(X > 72) = 0.10$, the boundary $72$

Misconception check: a right-tail probability is not automatically the value to enter into invNorm. Convert it to the left-tail area first, then solve the z-score equation.

Linear Transformations and Sums

If $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$ and $Y \sim \mathcal{N}(\mu_Y, \sigma_Y^2)$

Variance-combination checkpoint

Before combining normal variables, decide whether the question describes one variable being rescaled or several independent observations being added.

One observation is rescaled:
  Y = aX + b          -> multiply variance by a^2

Independent observations are added:
  T = X_1 + X_2       -> add their variances

Same symbol repeated without independence:
  X + X = 2X          -> treat it as a rescale, not a sum of copies

Misconception check: independence is what allows variances to add. A repeated symbol such as $X + X$ is not independent evidence; it is just $2X$.

Example -- Total mass

Let the content mass $X \sim \mathcal{N}(52.0, 3.2^2)$ g and packaging mass $Y \sim \mathcal{N}(10.0, 1.5^2)$

1. Total $T = X + Y \sim \mathcal{N}(62.0, 3.2^2 + 1.5^2) = \mathcal{N}(62.0, 12.49)$

Calculator Workflows

• Casio (fx-CG / fx-9860): NORMALCD(a, b, μ, σ) returns $P(a \leq X \leq b)$; set a = -1E99 for $-\infty$.
• TI (84/89): normalcdf(lower, upper, μ, σ) and invNorm(area, μ, σ) cover forward and inverse queries.
• Document the command in working: e.g. normalcdf(64, 10^99, 62, √12.49) for the total-mass example.

Exam Watch Points

• Always draw the diagram first; SEAB mark schemes allocate method marks for the sketch.
• Quote the standardisation line explicitly, e.g. “Let $Z = \frac{X - 45.0}{1.2} \sim \mathcal{N}(0, 1)$.”
• If asked to solve for $\mu$

Practice Quiz

Put normal standardisation, percentile inverses, and linear-combination questions into timed practice.

Quick Revision Checklist

• Convert raw bounds to $z$-scores confidently and use symmetry shortcuts.
• Use inverse normal commands to retrieve critical values for percentiles and control limits.
• Combine independent normal variables by adding means and variances before standardising.
• Explain modelling decisions: why a normal model fits the context, and which assumptions were made.

Want weekly guided practice on Normal Distribution? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Standardising when the GC can handle the original parameters directly: Many students convert to $Z$-scores unnecessarily. GC commands like normalcdf(lower, upper, μ, σ) accept the original distribution directly - no standardisation needed for probability questions unless the question explicitly asks for a $z$-score.
• Adding variances instead of scaling them when multiplying a variable: For $Y = 2X$, $\operatorname{Var}(Y) = 4\operatorname{Var}(X)$

Frequently asked questions

Is the normal distribution in Paper 1 or Paper 2?
Topic 6.3 is part of Probability & Statistics and is assessed in Paper 2 Section B (60 marks). Paper 1 is Pure Mathematics only.

Is normal approximation to the binomial examinable in 2026?
No. The 2026 H2 Maths (9758) syllabus explicitly excludes normal approximation to the binomial distribution, so continuity correction is not needed. Treat the two distributions as entirely separate.

Do I need to show the standardisation step if I use the GC directly?
You still need to write the distribution model (e.g. "$X \sim \mathcal{N}(45, 1.44)$") and the probability statement (e.g. "$P(X > 46.5)$") before quoting the GC output. The GC result alone is not sufficient for method marks.

Is there a formula sheet for H2 Maths normal distribution?
Yes - the "Formulas at a glance" section near the top collects standardisation, the symmetry relations, inverse normal, and the mean and variance rules for linear transformations and sums of independent normal variables. MF27 does not provide a standard normal table or normal critical values, so learn the setup and use your graphing calculator as permitted.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

• Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
• Sequences & series: Sequences & Series
• Vectors: Vectors
• Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals · Differential Equations
• Statistics: Probability · Discrete Random Variables · Normal Distribution (this page) · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 6 Probability and statistics sub-topic 6.3 Normal distribution (standardisation, $\Phi$, inverse normal, linear transformations; excludes normal approximation to binomial distribution): SEAB 9758 syllabus PDF

Next steps: stay on the H2 Maths notes hub and advance to Topic 6.4 - Sampling distributions for CLT-based inference.