A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
A student scores 78 on an exam modeled with a mean of 70 and a standard deviation of 8. What fraction of scores are lower? You do not need to inspect every score. If the normal model is reasonable, one standardization formula and a probability table answer the question.
The important phrase is if the normal model is reasonable. A normal distribution is a specific mathematical model, not a promise that every dataset forms a perfect bell. It can describe some continuous measurements well, and it is also the reference distribution behind many later inference methods. This lesson teaches both how to use the model and how to recognize when its assumptions are doubtful.
This work extends the cumulative-probability language from L-PR-4 and the mean, standard deviation, Empirical Rule, and z-scores from L-DS-5.
By the end of this lesson, you will be able to:
Describe the normal distribution and use the Empirical Rule to estimate central proportions.
Use the z-table to find left-tail, right-tail, and between-values probabilities.
Standardize and solve both probability and inverse-normal problems.
Judge whether a normal model is appropriate from a distribution’s shape and context.
Section 2: Prerequisites
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
What you need coming in — and why it matters today:
Mean and standard deviation (L-DS-5): locates a distribution’s centre and measures its spread. Those two parameters determine the location and scale of a normal model.
Empirical Rule (L-DS-5): For approximately normal data, about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean. Today the z-table refines those estimates.
Z-scores (L-DS-5): expresses a value’s signed distance from the mean in standard-deviation units. Today that conversion connects every normal model to one common table.
Quick check — can you recall these?
If a distribution is approximately normal with , about 68% of values fall within which range of ?
What changes in this lesson: The Empirical Rule remains an estimate, while the z-table provides more precise normal-model probabilities. A large dataset does not automatically make its values normally distributed, and a z-score is not itself a probability. First decide that a normal model is reasonable; then standardize and read the appropriate area.
Section 3: Core Concepts
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
How this section is organized: Five concepts build in order. Read C1–C2 for the shape and table structure, then C3 for the conversion formula that links any normal distribution to the table. C4 reverses the process (finding values from probabilities), and C5 teaches when not to use the normal model.
C1: Shape and Empirical Rule — what the normal curve looks like and the 68–95–99.7 pattern
C2: The standard normal Z and the z-table — how to read three types of probabilities
C3: Standardizing any normal variable — the bridge formula
C4: Inverse normal — finding a value from a probability
C5: Recognizing when the normal model fits
C0 — Areas as Proportions of Populations
The normal distribution is often introduced as a mathematical curve. But it is more intuitive to think of it as a description of a population.
Suppose a large population of heights is modeled by a normal distribution with mean cm and standard deviation cm. Imagine 100,000 values generated from that model.
Range
Approx. count
Proportion
Between 156 cm and 170 cm (within 1σ)
About 68,270 values
68.3%
Between 149 cm and 177 cm (within 2σ)
About 95,450 values
95.5%
Between 142 cm and 184 cm (within 3σ)
About 99,730 values
99.7%
Shorter than 149 cm (more than 2σ below)
About 2,275 values
2.3%
Taller than 184 cm (more than 3σ above)
About 135 values
0.13%
This is the key insight: probabilities are areas, and areas are proportions of populations. When the table gives , it means a randomly selected modeled value has about a 2.3% chance of falling below 149 cm — equivalently, about 2.3% of this modeled population lies below 149 cm.
The z-table you will use in this lesson is simply a lookup table for these proportions. Every area under the standard normal curve corresponds to a proportion of a standard normal population — and, after standardization, to a proportion of any normal population.
C1 — Properties of the Normal Distribution
Connect to L-PR-4: a continuous random variable like height or test score cannot use a PMF — probabilities correspond to areas under a curve, not point masses. The normal distribution provides the specific curve.
Normal Distribution
A random variable follows a normal distribution, written with , when its probability density function is
This specific density is symmetric and bell-shaped. A graph that merely looks symmetric and bell-shaped is not, by appearance alone, proof of a normal distribution.
Key properties:
Symmetric about — the left and right halves are mirror images.
Mean = Median = Mode = (all three coincide at the center).
Tails are asymptotic — they approach but never touch the horizontal axis.
Total area under the curve = 1 (as required for any probability distribution).
Fully described by two parameters: (location) and (variance); is the standard deviation.
Example: For a population modeled by (mean 162 cm, cm), about 95% of modeled values lie between 150 cm and 174 cm because that interval is .
The second parameter in is the variance, not the standard deviation . Writing when you mean (with ) is a common error. Always check: is the number after the comma or ? Convention throughout this course is .
μ =σ =
Figure 1: The Empirical Rule for a normal distribution. Set μ and σ, then pick a band: about 68%
of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ — leaving only about
0.3% in the two tails beyond ±3σ. The endpoints μ ± kσ recompute live in data units.
C2 — Standard Normal Distribution and the Z-Table
Rather than computing a different area formula for every possible and , we use a single reference distribution.
We write to mean “Z follows the standard normal distribution” — the special case where and .
Standard Normal Distribution
has mean 0 and standard deviation 1.
The z-table gives the left-tail cumulative area for each listed value of .
From this single entry, we derive three probability types:
Left-tail: — read directly from the table.
Right-tail: — complement rule.
Between two values: — subtract the smaller left-tail from the larger.
The displayed course table covers nonnegative z-scores from 0.00 through 3.09. Rows give the ones and tenths digits of ; columns give the hundredths digit. For a negative score, use symmetry:
Standard Normal Distribution (Z) Table
Area to the left of z
Standard Normal Distribution (Z) Table
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.5000
0.5040
0.5080
0.5120
0.5160
0.5199
0.5239
0.5279
0.5319
0.5359
0.1
0.5398
0.5438
0.5478
0.5517
0.5557
0.5596
0.5636
0.5675
0.5714
0.5753
0.2
0.5793
0.5832
0.5871
0.5910
0.5948
0.5987
0.6026
0.6064
0.6103
0.6141
0.3
0.6179
0.6217
0.6255
0.6293
0.6331
0.6368
0.6406
0.6443
0.6480
0.6517
0.4
0.6554
0.6591
0.6628
0.6664
0.6700
0.6736
0.6772
0.6808
0.6844
0.6879
0.5
0.6915
0.6950
0.6985
0.7019
0.7054
0.7088
0.7123
0.7157
0.7190
0.7224
0.6
0.7257
0.7291
0.7324
0.7357
0.7389
0.7422
0.7454
0.7486
0.7517
0.7549
0.7
0.7580
0.7611
0.7642
0.7673
0.7704
0.7734
0.7764
0.7794
0.7823
0.7852
0.8
0.7881
0.7910
0.7939
0.7967
0.7995
0.8023
0.8051
0.8078
0.8106
0.8133
0.9
0.8159
0.8186
0.8212
0.8238
0.8264
0.8289
0.8315
0.8340
0.8365
0.8389
1.0
0.8413
0.8438
0.8461
0.8485
0.8508
0.8531
0.8554
0.8577
0.8599
0.8621
1.1
0.8643
0.8665
0.8686
0.8708
0.8729
0.8749
0.8770
0.8790
0.8810
0.8830
1.2
0.8849
0.8869
0.8888
0.8907
0.8925
0.8944
0.8962
0.8980
0.8997
0.9015
1.3
0.9032
0.9049
0.9066
0.9082
0.9099
0.9115
0.9131
0.9147
0.9162
0.9177
1.4
0.9192
0.9207
0.9222
0.9236
0.9251
0.9265
0.9279
0.9292
0.9306
0.9319
1.5
0.9332
0.9345
0.9357
0.9370
0.9382
0.9394
0.9406
0.9418
0.9429
0.9441
1.6
0.9452
0.9463
0.9474
0.9484
0.9495
0.9505
0.9515
0.9525
0.9535
0.9545
1.7
0.9554
0.9564
0.9573
0.9582
0.9591
0.9599
0.9608
0.9616
0.9625
0.9633
1.8
0.9641
0.9649
0.9656
0.9664
0.9671
0.9678
0.9686
0.9693
0.9699
0.9706
1.9
0.9713
0.9719
0.9726
0.9732
0.9738
0.9744
0.9750
0.9756
0.9761
0.9767
2.0
0.9772
0.9778
0.9783
0.9788
0.9793
0.9798
0.9803
0.9808
0.9812
0.9817
2.1
0.9821
0.9826
0.9830
0.9834
0.9838
0.9842
0.9846
0.9850
0.9854
0.9857
2.2
0.9861
0.9864
0.9868
0.9871
0.9875
0.9878
0.9881
0.9884
0.9887
0.9890
2.3
0.9893
0.9896
0.9898
0.9901
0.9904
0.9906
0.9909
0.9911
0.9913
0.9916
2.4
0.9918
0.9920
0.9922
0.9925
0.9927
0.9929
0.9931
0.9932
0.9934
0.9936
2.5
0.9938
0.9940
0.9941
0.9943
0.9945
0.9946
0.9948
0.9949
0.9951
0.9952
2.6
0.9953
0.9955
0.9956
0.9957
0.9959
0.9960
0.9961
0.9962
0.9963
0.9964
2.7
0.9965
0.9966
0.9967
0.9968
0.9969
0.9970
0.9971
0.9972
0.9973
0.9974
2.8
0.9974
0.9975
0.9976
0.9977
0.9977
0.9978
0.9979
0.9979
0.9980
0.9981
2.9
0.9981
0.9982
0.9982
0.9983
0.9984
0.9984
0.9985
0.9985
0.9986
0.9986
3.0
0.9987
0.9987
0.9987
0.9988
0.9988
0.9989
0.9989
0.9989
0.9990
0.9990
Three worked examples using the table above:
Type 1 — Left-tail:. Find row 1.2, column 0.03: .
Type 2 — Right-tail:.
Type 3 — Between: First use symmetry: . Then .
The z-table gives — a left-tail area. If you look up and get 0.9332, that means approximately 93.32% of the distribution is below 1.50. The table-based right-tail probability is . Using 0.9332 directly as a right-tail probability is the single most common z-table error.
When computing , you must subtract the two left-tail areas, not add them. By symmetry, . Therefore, . Adding the two areas () would incorrectly include overlapping regions. The intuition: you want the area in the middle strip, which is the big left area minus the small left area that you do not want.
Try it interactively. Use the visualization below to shade normal curve areas. Enter any z-score to see the corresponding probability, or switch to a general normal distribution to practice standardization.
Figure 2: Interactive normal curve. Switch between Standard N(0,1) and General N(μ, σ²) using the tabs. In General mode, enter μ, σ, and x — the z-score and probability update live, showing
how standardization works. Use the Shade buttons to explore left-tail, right-tail, and between-values
probabilities. Right-tail and between modes show the calculation, not just the answer.
C3 — Standardizing a Normal Variable
Any can be converted to the standard normal scale using the standardization formula.
Standardization (Z-Score Transformation)
For , standardize the random variable:
For a particular observed value , its corresponding standard score is:
Interpretation: tells you how many standard deviations is above (positive) or below (negative) the mean.
Mini-example: IQ scores follow (mean 100, ). Find .
Standardize: .
Look up .
About 88.5% of people score below 118.
You must standardize before using the z-table. The z-table works only for . Looking up directly in a table designed for produces a nonsense answer. Every non-standard normal problem requires the conversion step first.
μ =σ =x =
Figure 3: The standardization transformation. Set μ and σ, then drag x. The orange marker on the
X curve and the blue marker on the Z curve sit at the same horizontal position — the same
fraction of the bell — confirming that z = (x − μ) / σ is an axis relabelling that leaves the
shaded probability unchanged.
C4 — Finding Values Given a Probability (Inverse Normal)
Sometimes the question is reversed: given a probability, find the corresponding value. This is the percentile or quantile problem.
Inverse Normal Lookup
Given a cumulative probability , find such that :
Look up in the body of the z-table to find (the z-score corresponding to the th quantile).
Unstandardize:
We use (read “z-star”) to distinguish this critical value from an ordinary observed z-score.
Mini-example: Exam scores follow (). Find the 90th percentile.
Step 1: Find such that . Scanning the z-table body, the closest entry is 0.8997 at . So .
Step 2: Unstandardize: .
The 90th percentile score is approximately 80.24.
In the inverse problem, search the table body (the probability values) to find , then use the row/column labels to read . Students sometimes look up the probability in the margins (the z-values) instead of the body — this produces a random number unrelated to the problem. Remember: body → → unstandardize.
Figure 4: Inverse normal lookup. Set a target cumulative probability p (the percentile).
The left tail shades to area p and the boundary slides to z*, the critical
value whose left-tail area equals p — this is the body → z* reading the z-table requires.
Then set μ and σ to watch the unstandardize step
x = μ + z*·σ compute live. Choose p → find z* → unstandardize to x.
C5 — Recognizing Non-Normal Shapes
The normal model is powerful, but it is not universal. Applying it blindly to non-normal data produces incorrect probabilities.
When to Use (and Not Use) the Normal Model
Normal model is appropriate when:
The variable is continuous, or sufficiently fine-grained for a continuous approximation
A histogram is roughly symmetric, unimodal, and bell-shaped
A normal-probability plot is approximately a straight line, without strong systematic curvature or isolated departures
There is no strong skew, second mode, boundary pile-up, or unusually heavy tail
Central proportions are reasonably consistent with the Empirical Rule
Normal model is NOT appropriate when:
The observed distribution is strongly right-skewed or left-skewed
Values pile up at a meaningful lower or upper boundary
The distribution has heavy tails, large outliers, or multiple modes
The data are sparse counts or proportions near 0 or 1 and the conditions for a normal approximation are not met
Figure 5: When is the normal model a good fit? A symmetric, unimodal bell (left) is well
described by the normal distribution. A right-skewed shape (like incomes), a bimodal shape (two
distinct groups), and a few separated discrete counts are not — the normal model would misdescribe
their shape.
Looking ahead: a normal distribution can approximate binomial counts when and , together with a continuity correction. L-INF-1 introduces conditions under which the sampling distribution of a mean becomes approximately normal as the sample size grows; the NIST central-limit discussion gives the broader principle and its conditions.
Do not decide from the variable’s name or sample size alone. Judge the distribution actually described or observed. A large sample can reveal a skewed shape more clearly; it does not turn the original values into a normal distribution. Likewise, count data are not automatically normal, though some count distributions can have useful normal approximations under stated conditions.
Section 4: Worked Examples
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Example 1 — Fully Worked: Three Probability Types for Z
Find (a) , (b) , and (c) .
(a) Left tail — read the table directly.
I notice that “less than” asks for the left-tail area stored by the table.
(b) Right tail — use the complement.
The displayed table lists positive z-scores, so first read the entry for and use symmetry.
Sanity check: Because is below the mean, more than half the curve lies to its right.
(c) Between two values — subtract left-tail areas.
The desired strip is the larger left-tail area with the smaller one removed.
Example 2 — Prediction Checkpoint: Standardize and Find Probability
IQ scores are modeled as (). Find .
Before revealing the solution, record the z-score you expect, whether the answer should be above or below 0.50, and the table operation you will use.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Step 1 — Standardize.
Because is above the mean, the right-tail probability should be below 0.50.
Step 2 — Read the left-tail table entry.
Step 3 — Take the complement.
About 11.5% of scores in this model are above 118. Compare your recorded sign, operation, and answer with these three decisions.
Example 3 — Minimally Scaffolded: Inverse Normal (90th Percentile)
Exam scores follow (). Find the 90th percentile.
Before revealing the solution, record which table entry you will search for, the sign you expect for , and how you will return to the exam-score scale.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Step 1 — Translate the percentile.
Step 2 — Find the standard-normal quantile.
The closest table entry is 0.8997 at , so .
Step 3 — Unstandardize.
The 90th-percentile score is approximately 80.24. About 90% of modeled scores are below this value.
Example 4 — Precision Comparison: Empirical Rule Application
Analysis to examine:
A population is modeled by , with measurements in centimetres. A student is asked for the modeled fraction between 158 cm and 182 cm.
Student A says: “The interval is , so the Empirical Rule gives about 95%.”
Student B says: “The interval is , so a four-decimal z-table calculation gives 95.45%.”
Decide which parts of each analysis are correct. Report the Empirical-Rule estimate and the four-decimal z-table result, and explain why they need not display the same rounded percentage.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Step 1 — Identify the standardized bounds.
The interval is , so Student A’s Empirical-Rule estimate of about 95% is appropriate.
Step 2 — Use the displayed table precision.
The four-decimal table therefore gives about 95.44%, not 95.45%. A higher-precision calculation gives approximately 95.45%; the one-hundredth-of-a-percentage-point difference comes from rounding the table entries. Student B has the right method but mixes two different precision levels.
Section 5: Guided Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Note: The z-table in Section 3 is scrollable and reusable. Keep it open in a separate browser tab or scroll back to Section 3 as needed while working through these problems.
Problem 1 — Standard Normal Probabilities (C2)
Problem 2 — Standardize and Find Probability (C3)
Problem 3 — Inverse Normal (C4)
Problem 4 — Classify the Distribution (C5)
Section 6: Independent Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Reminder: The z-table in Section 3 is scrollable and reusable. Open Section 3 in a separate tab or scroll back as needed.
Problem 1 — Standard Normal Probability (C2)
Problem 2 — Standardize a Raw Value (C3)
Problem 3 — Recover a Percentile Value (C4)
Problem 4 — Judge the Model (C5)
Problem 5 — Estimate a Central Band (C1)
Problem 6 — Compare Two Quality-Control Settings (C2 + C3)
Error-Analysis Review — Locate the First Broken Step
Spaced Review — Z-Scores and the Empirical Rule
Spaced Review — Binomial Parameters
Section 7: Mastery Check
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Question 1 — Feynman Test
In your own words, explain why using only the symmetry of the normal distribution. Do not use the z-table or specific numerical areas.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Compare with a model answer
The standard normal curve is a mirror image across zero. The point is the reflection of , so the area farther into the left tail beyond matches the reflected area farther into the right tail beyond .
Question 2 — Novel Application
Question 3 — Diagnose a Flawed Analysis
Self-Assessment
Still confusedReady for the Boss Fight
Section 8: Boss Fight
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Choose the context that interests you. In either path, decide which normal-distribution ideas each task needs, show your reasoning, and complete all four tasks before comparing with the solution.
⚙️ Path A: The Engineer
Evaluate a bolt-production process, redesign one threshold, and estimate when an equipment upgrade pays back.
📊 Path B: The Analyst
Evaluate a percentile-based grading policy and determine how changing the spread changes its raw-score cutoff.
⚙️ Path A: The Engineer
📊 Path B: The Analyst
Section 9: Challenge Problems
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Ready for more? These go beyond the lesson objectives.
Challenge 1 — Mirror Tails
Starting from the normal distribution’s symmetry, determine how the left-tail probability below relates to the left-tail probability below , for any .
Then verify numerically for : look up both and in the z-table and confirm the relationship holds.
Show Solution
Proof:
The standard normal is symmetric about 0: for all .
By this symmetry, the area to the left of equals the area to the right of :
Now apply the complement rule — the total area is 1:
(For a continuous distribution, , so .)
Combining: .
Numerical verification for :
From the z-table: .
.
Also from the z-table: . ✓
The values match. This also confirms the familiar fact that the critical values each cut off 2.5% in their respective tails — the basis for the 95% confidence interval (covered in L-INF-2).
Challenge 2 — Recovering a Distribution
Section 10: Solutions Reference
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
The solutions page includes worked solutions for all Guided Practice problems, Independent Practice problems, Mastery Check questions, Boss Fight tasks, and Challenge Problems.
Quick-Reference Formula Card
Formula
Purpose
When to use
Standardize to
Before any z-table lookup for a non-standard normal
Right-tail from left-tail
When the table gives left-tail but you need right-tail
Between two values
Subtract smaller left-area from larger
Unstandardize (inverse normal)
After finding from a given cumulative probability
Z-Table Reading Guide
The z-table in Section 3 gives for listed nonnegative values of — always a left-tail area.
Row = ones and tenths digit of (e.g., row 1.4 for ).
Column = hundredths digit (e.g., column 0.05 for ).
For inverse problems: search the table body for the target probability, then read the row and column labels to get .
Symmetry shortcut: — you can always convert negative z-values using the positive half of the table.
Common Errors to Avoid
P1 (Right-tail confusion): The table gives . For , use the complement: .
P2 (Forgetting to standardize): Always compute before looking up the table. Never look up raw values.
P3 (Adding instead of subtracting): For , subtract — never add — the two left-tail areas.