H2 Maths Normal Distribution Formula Sheet | Standardisation
H2 Maths Normal Distribution Formula Sheet | Standardisation
Study guide/
H2 Maths normal distribution formula sheet: standardisation, inverse normal, linear combinations of normal variables, and sums and means of n observations - every key result on...
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.
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. The standard normal distribution function Φ is tabulated in MF27, but standardising and combining variances correctly is on you. Worked examples appear in the sections below.
Distribution & standardisation
Quantity
Formula
Reviewed by
Marcus Pang·Managing Director (Maths)
Model
X∼N(μ,σ2)
Standardisation
Z=σX−μ∼N(0,1)
Symmetry
P(Z≤−a)=P(Z≥a)
Two-sided
P(∣Z∣≤a)=2P(Z≤a)−1
Inverse normal
Goal
Formula
Value from percentile
x=μ+zp,σ, where Φ(zp)=p
Linear transformation Y=aX+b
Quantity
Formula
Mean
E(Y)=aE(X)+b
Variance
Var(Y)=a2Var(X)
Combining independent X,Y
Quantity
Formula
Mean
E(aX+bY)=aE(X)+bE(Y)
Variance
Var(aX+bY)=a2Var(X)+b2Var(Y)
Sum
X+Y∼N(μX+μY,σX2+σY2)
Core Concepts
A normal model is written X∼N(μ,σ2) with mean μ and variance σ2.
Standardisation converts X to Z=σX−μ∼N(0,1); tables and graphing calculator (GC) functions assume this form.
Symmetry gives P(Z≤−a)=P(Z≥a) and P(∣Z∣≤a)=2P(Z≤a)−1
For Y=aX+b: E(Y)=aE(X)+b and Var(Y)=a2Var(X)
For independent X and Y: E(aX+bY)=aE(X)+bE(Y)
The 2026 syllabus excludes normal approximation to binomial distribution (so continuity correction is not required).
Standardisation Workflow
Sketch the bell curve, mark μ and shade the requested region.
Convert bounds using z=σx−μ.
Evaluate the corresponding standard normal probability with the table or GC.
State the final probability to three significant figures unless otherwise required.
Example -- Tail probability
Industrial bearings have X∼N(45.0,1.22) mm. Bearings longer than 46.5 mm are scrapped. Find the scrap rate.
Compute z=1.246.5−45.0=1.25.
P(X≥46.5)=P(Z≥1.25)=1−Φ(1.25)≈0.106.
About 10.6% exceed the upper tolerance.
Inverse Normal Problems
Typical prompt: find x such that P(X≤x)=p.
Use x=μ+zpσ, where zp satisfies Φ(zp)=p.
Example -- Percentile target
Chemistry practical scores obey X∼N(68.5,4.22). What is the minimum score for the top 5%?
Need p=0.95, so z0.95=1.645 (from tables or invNorm(0.95, 0, 1)).
x=68.5+1.645×4.2≈75.4.
Scores ≥75.4 lie in the top 5%.
Tail-area checkpoint
Before using normalcdf or invNorm, translate the wording into the area your calculator needs.
Question wording
Area to enter or shade
First sketch check
Trap to avoid
"Less than" or "below" a value
Left-tail area up to the value
Shade from far left to the boundary
Entering the right-tail complement
"More than" or "above" a value
Right-tail area beyond the value
Shade from the boundary to far right
Using invNorm with the right-tail area directly
"Between two values"
Middle area between both bounds
Mark both boundaries before calculating
Standardising only one boundary
"Top p percent"
Left-tail area of 100 minus p percent
Boundary should sit to the right of the mean if p is small
Entering p percent instead of its complement
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.
Central area
Total tail area
Each tail
Left-tail area for upper cut-off
0.90
0.10
0.05
0.95
0.95
0.05
0.025
0.975
0.98
0.02
0.01
0.99
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 z0.95≈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 μ or σ. 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.
Unknown
First move
Equation to write
Common trap
Mean μ
Convert the stated area into a z-value.
z=σx−μ, so μ=x−zσ.
Moving μ to the wrong side after standardising.
Standard deviation σ
Convert the stated area and check whether the boundary is above or below the mean.
z=σx−μ, so σ=zx−μ
A boundary value x
Use inverse normal, then return to the original scale.
x=μ+zσ.
Leaving the answer as a z-score instead of the original measurement.
Worked check: if X∼N(μ,62) and P(X>72)=0.10, the boundary 72 is the 90th percentile from the left. Hence z0.90≈1.282, so 672−μ=1.282 and μ≈64.3.
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∼N(μX,σX2) and Y∼N(μY,σY2) are independent, then X+Y∼N(μX+μY,σX2+σY2).
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
Expression
What it represents
Variance rule
Common wrong move
2X+5
One measurement doubled, then shifted.
Var(2X+5)=4Var(X)
Multiplying the variance by 2 instead of 22.
X1+X2, independent and identically distributed
Two separate observations from the same model.
Var(X1+X2)=2Var(X)
X+X
The same random variable written twice.
Var(X+X)=Var(2X)=4Var(X)
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∼N(52.0,3.22) g and packaging mass Y∼N(10.0,1.52) g be independent. Find P(X+Y>64).
Total T=X+Y∼N(62.0,3.22+1.52)=N(62.0,12.49).
σT=12.49=3.534.
Standardise: z=3.53464−62=0.566.
P(T>64)=1−Φ(0.566)≈0.286.
Calculator Workflows
Casio (fx-CG / fx-9860): NORMALCD(a, b, μ, σ) returns P(a≤X≤b); set a = -1E99 for −∞.
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=1.2X−45.0∼N(0,1).”
If asked to solve for μ or σ, rewrite the statement as Φ(z)=p first, then use invNorm (or table values) carefully.
Normal approximation to binomial distribution is excluded for 2026, so do not introduce continuity correction in exam solutions.
Round intermediate z-values to at least 3 decimal places; round the final probability at the end.
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, Var(Y)=4Var(X), not 2Var(X). Forgetting to square the constant is a very common error in linear transformation questions.
Using Var(X+X)=2Var(X) instead of Var(2X)=4Var(X)
Forgetting to sketch the bell curve: SEAB mark schemes award a method mark for a correct shaded diagram even when the final probability is wrong. Always draw the curve, mark μ, and shade the region.
Rounding intermediate z-values to only 2 decimal places: Rounding z to 2 d.p. before looking up Φ(z) introduces avoidable error. Keep at least 3 decimal places for z until the final answer is stated.
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∼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. The standard normal table is provided in MF27, but standardising and combining variances correctly is on you.
Other H2 Maths formula sheets
Revising more than one topic? Grab the matching one-page formula sheet:
SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 6 Probability and statistics sub-topic 6.3 Normal distribution (standardisation, Φ, inverse normal, linear transformations; excludes normal approximation to binomial distribution): SEAB 9758 syllabus PDF
Accepting a negative standard deviation because the tail was read backwards.
Treating the two observations as the same random variable.
Adding variances as if the two terms were independent.
: X+X and 2X are algebraically equal for a single observation, but when X1 and X2 are independent copies, Var(X1+X2)=2Var(X)=4Var(X). Read the question carefully to know which scenario applies.
