IB Maths AA — Normal Distribution and z-Scores: Tutor Prep Guide | Standardising, tail areas, inverse look-up

2026-11-09

Students can quote z equals (x minus mu) over sigma, then freeze when a raw score appears. One mean of 62 and one sigma of 8 carry both directions of the table.

Objectives — Students can: (1) standardise a raw score and read the sign aloud; (2) separate upper tail, lower tail and interval probabilities; (3) reverse a given probability into a z value and back into a raw score.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Subtract then divide	(70 – 62) / 8	1
Complement	1 minus 0.8413	0.0668
Symmetry	How phi of minus 1 relates to phi of 1	They sum to 1

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	One score, two directions	Ask whether 70 is high	Say why standardising is needed
8-20 min	Standardise	z equals (x minus mu) over sigma	Get z equals 1
20-34 min	Table direction	Upper, lower, interval	Get 0.8413 and 0.0668
34-46 min	Interval	phi of 1 minus phi of minus 1	Get 0.6827
46-56 min	Inverse look-up	Find z for 0.9	Get 72.25
56-60 min	Exit check	Collect three items	Work alone

03 Teaching script (read it out)

Scene 1: Standardise: subtract the mean first, divide by sigma second

With mu 62 and sigma 8, x equals 70 gives z equals (70 – 62) / 8 equals 1.
Make them say it: z equals 1 means “one standard deviation above the mean”, not “eight marks higher”.

Ask them to repeat: what unit z carries (none, it counts standard deviations).

Scene 2: Table reading: the upper tail always needs 1 minus

phi of 1 is 0.8413, so the probability above 70 is 1 minus 0.8413, which is 0.1587.
For x equals 74, z equals 1.5, phi of 1.5 is 0.9332, so the upper tail is 0.0668.
The usual mistake is quoting 0.9332 as the chance of being above 74; that figure is the chance of being below it.

Ask them to repeat: above means subtract from 1, below means read the table directly.

Scene 3: Reverse: find z, then multiply by sigma before adding mu

For the top ten per cent boundary, solve phi of z equals 0.9, giving z equals 1.2816.
Then x equals 62 plus 1.2816 times 8, which is 72.25. The order matters: scale first, shift second.

Have them compute: the same boundary when sigma becomes 10 (62 plus 12.8155, so 74.82).

04 Question types and worked steps

Type	Example	Steps	Answer
Standardise	x 66, mu 62, sigma 8	z equals (x minus mu) over sigma	z 0.5, area 0.6915
Upper tail	Probability above 74	1 minus phi of 1.5	0.0668
Interval	Probability between 54 and 70	phi of 1 minus phi of minus 1	0.6827
Inverse	Top ten per cent boundary	Look up 0.9, then rescale	72.25

Sanity check (teach this): A probability always sits between 0 and 1; a result above 1 or below 0 means the tail was read in the wrong direction. A negative z means below the mean, never a negative probability.

05 Independent practice (3 questions)

Question	Answer	What to watch
z and lower tail for x equals 66	z 0.5, 0.6915	Did they subtract 62 first
Probability above 74	0.0668	Did they subtract from 1
Top ten per cent boundary	72.25	Was sigma applied after z

06 Common mistakes and what to say

Mistake	Student writes	Say this
Reversing the order	Writes (62 – 70) / 8	“Subtract the mean first, then divide by sigma. The order is fixed.”
Forgetting the complement	Answers 0.9332	“The table reads below. The question asks above, so subtract from one.”
Using 0.1 in the inverse	Gets z minus 1.28, then 51.75	“Top ten per cent means ninety per cent below. Look up 0.9.”
Skipping the rescale	Answers 63.28	“Going back is mu plus z times sigma, not mu plus z.”

07 Homework (paper, section and question numbers)

IB Maths chapter index — Use the three items on this page

08 Exit check (last 5 minutes)

Question	Answer	Decision
z for x equals 66	z 0.5	All three right, move to hypothesis testing
Probability above 74	0.0668	One wrong, reteach the upper tail
Top ten per cent boundary	72.25	Two wrong, return to act three

09 Tutor FAQ

Is a normal table provided?

A standard table is available, though many questions expect the calculator function.

Must students memorise 68-95-99.7?

Yes, it is the quickest sanity check for a one-sigma interval.

Does the calculator model matter?

normalcdf and invNorm are fine, but method marks still need the written steps.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.