My z-score has three decimals, like 1.645. What do I do?

Round to two decimals for the table, or take a value between the two neighbouring entries. For $z = 1.645$ the areas 0.9495 and 0.9505 are equally close, so 0.95 is a good answer.

What if the z-score is beyond the table, like $z = 4$?

Tables usually stop at about 3.49. Beyond that, the left area is practically 1 (for large positive z) or practically 0 (for large negative z). Such values are extremely rare.

Why does the percentile give the area to the left?

The 90th percentile is the value with 90% of the population below it. "Below" is the left side of the curve, which is exactly what the z-table measures.

Teacher Resources

Normal Probability

Q: Is $P(X < 174)$ different from $P(X \le 174)$?

No. For a continuous variable a single value has probability 0, so $<$ and $\le$ give the same area. This is different from the binomial distribution, where it matters.

Find probabilities and percentiles for normal data in real units: standardize, read the z-table, and work backwards to cut-off values.

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Definition

Normal probability answers questions like "What fraction of bottles hold less than 494 mL?" when a quantity

X

follows a normal distribution with mean

\mu

and standard deviation

\sigma

, measured in real units (cm, points, hours, mL).

The probability is the area under the bell curve over the values you ask about. Every normal curve becomes the standard normal curve after one step, so you only ever need one z-table.

The four-step method

1.Sketch and shade the region: left of $a$ , right of $a$ , or between $a$ and $b$ .
2.Standardize: turn each value into a z-score

z = \frac{x - \mu}{\sigma}

3.Read the z-table: it gives the area to the left of $z$ , that is $P(Z < z)$ (see the lesson on the standard normal distribution).
4.Adjust for the region:

Left: $P(X < a) = P(Z < z_a)$
Right: $P(X > a) = 1 - P(Z < z_a)$
Between: $P(a < X < b) = P(Z < z_b) - P(Z < z_a)$

Working backwards: percentiles and cut-offs

Sometimes the area is given and the value is missing ("Which score marks the top 10%?"). Find the

z

whose left area matches in the table, then go back to real units:

x = \mu + z\sigma

A probability of

0.9332

also means 93.32% of the population, so every answer can be read as a proportion of people, bottles or batteries.

Try it now

Test scores are normally distributed with

\mu = 60

and

\sigma = 10

. Which expression gives the z-score of a score of 75?

Worked Examples

Women's heights in a country are normally distributed with

\mu = 165

cm and

\sigma = 6

cm. What proportion of women are shorter than 174 cm?

Sketch and shade

Shade the area to the left of 174 cm, which lies to the right of the mean 165 cm → The answer should be more than 0.5

Standardize

z = \frac{174 - 165}{6} = \frac{9}{6}

→

z = 1.5

Read the z-table

Row 1.5, column 0.00 →

P(Z < 1.5) = 0.9332

Interpret

A left area needs no adjustment →

P(X < 174) = 0.9332

Answer: $P(X < 174) = 0.9332$ : about 93.32% of women are shorter than 174 cm.

Common Mistakes

Using the table value directly for a "greater than" question, for example $P(X > 14) = 0.8944$

Why it's wrong: The z-table always gives the area to the LEFT of $z$ . A right tail is everything else.

Correct: Subtract from 1: $P(X > 14) = 1 - 0.8944 = 0.1056$ . A quick sketch shows the right tail is small, so 0.8944 cannot be right.

Subtracting the z-scores first: $P(-1 < Z < 2) = P(Z < 2 - (-1)) = P(Z < 3)$

Why it's wrong: Areas under a curve do not work like that. You need the area up to each bound, then the difference between those areas.

Correct: Look up both left areas and subtract them: $0.9772 - 0.1587 = 0.8185$ .

Standardizing in the wrong order or dividing by the wrong number: $z = \frac{\mu - x}{\sigma}$ or $z = \frac{x - \mu}{\sigma^2}$

Why it's wrong: Reversing the subtraction flips the sign, so you land on the wrong side of the mean. Dividing by the variance gives the wrong distance.

Correct: Always $z = \frac{x - \mu}{\sigma}$ : values above the mean get a positive $z$ , values below get a negative $z$ .

In inverse problems, using the percentage as the z-score (z = 0.90) or using the wrong tail (z = -1.28 for the top 10%)

Why it's wrong: The percentage is an AREA, which you look for inside the table. The z-score is read from the row and column edges.

Correct: Top 10% means 0.90 to the left, which gives $z \approx 1.28$ (positive, above the mean). Bottom 10% gives $z \approx -1.28$ .

Interactive Visual

Women's heights: μ = 165 cm, σ = 6 cm. Drag the bound or switch the shading

μ = 165 cm, σ = 6 cm · Drag a handle or type x or z.

z = (x − μ)/σ = (174 − 165)/6 = 1.5

P(X ≤ 174) = P(Z ≤ 1.5) ≈ 0.9332 = 93.32%

About 93.32% of all values fall in the shaded region.

Exam scores: μ = 70, σ = 8. Area between 62 and 86

μ = 70 points, σ = 8 points · Drag a handle or type x or z.

z = (x − μ)/σ = (62 − 70)/8 = -1

z = (x − μ)/σ = (86 − 70)/8 = 2

P(62 ≤ X ≤ 86) = P(-1 ≤ Z ≤ 2) ≈ 0.8186 = 81.86%

About 81.86% of all values fall in the shaded region.

Cut-off: which score marks the top 10%?

μ = 70 points, σ = 8 points · Drag a handle or type x or z.

Move the handle until the shaded area is 0.1000 (10%).

zP(Z ≥ 0) ≈ 0.5000

x = μ + zσ = 70 + 0 · 8 ≈ 70 points

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Practice Problems

18 problems

Problem 1 of 18

Easy

Test scores are normally distributed with

\mu = 60

and

\sigma = 10

. Which expression gives the z-score of a score of 75?

Practice Problems: Normal Probability

1. Test scores are normally distributed with \mu = 60 and \sigma = 10. Which expression gives the z-score of a score of 75?
2. Heights of adult men are normally distributed with \mu = 170 cm and \sigma = 8 cm. What is the z-score of a man who is 182 cm tall?
3. The z-table gives P(Z < 1) = 0.8413. What is P(Z > 1)?
4. Apples from an orchard weigh \mu = 150 g with \sigma = 20 g, normally distributed. Use P(Z < 0.5) = 0.6915. What proportion of apples weigh less than 160 g? Give a decimal.
5. Exam scores are normally distributed with \mu = 72 and \sigma = 6. The table gives P(Z < 1.5) = 0.9332. Find P(X < 81).
6. Pizza delivery times are normally distributed with \mu = 30 min and \sigma = 5 min. The table gives P(Z < 2) = 0.9772. Find the probability that a delivery takes more than 40 minutes.
7. Bottles are filled with \mu = 500 mL and \sigma = 5 mL, normally distributed. Using P(Z < 2) = 0.9772 and P(Z < -1) = 0.1587, what is P(495 < X < 510)?
8. A laptop battery lasts \mu = 8 hours with \sigma = 0.8 hours, normally distributed. Use P(Z < 1.25) = 0.8944. What is the probability that a battery lasts more than 9 hours? Give a decimal.
9. Women's heights are normally distributed with \mu = 165 cm and \sigma = 6 cm. The table gives P(Z < 1) = 0.8413 and P(Z < -1.5) = 0.0668. Find P(156 < X < 171).
10. IQ scores are normally distributed with \mu = 100 and \sigma = 15. Find the 90th percentile.
11. For women's heights (\mu = 165 cm, \sigma = 6 cm) we found P(X < 174) = 0.9332. What does this mean?
12. Exam scores are normally distributed with \mu = 70 and \sigma = 8. The table gives P(Z < 0.5) = 0.6915. What proportion of students score between 66 and 74? Give a decimal.
13. A factory makes 1000 light bulbs. Their lifetimes are normally distributed with \mu = 1200 hours and \sigma = 100 hours. The table gives P(Z < 2) = 0.9772. About how many bulbs last more than 1400 hours?
14. A machine fills bottles with \mu = 500 mL and \sigma = 4 mL. The lowest 5% of bottles are rejected. Find the cut-off volume below which a bottle is rejected.
15. Men's heights are normally distributed with \mu = 175 cm and \sigma = 7 cm. Using z \approx 1.28 for the 90th percentile, what height marks the tallest 10%?
16. A phone battery lasts \mu = 20 hours with \sigma = 2 hours, normally distributed. The table gives P(Z < -0.67) = 0.2514. At least how many hours do 75% of batteries last? Give the answer to two decimal places.
17. A warehouse ships 4000 packages a day. Their weights are normally distributed with \mu = 250 g and \sigma = 10 g. The table gives P(Z < 1.5) = 0.9332 and P(Z < -1) = 0.1587. About how many packages weigh between 240 g and 265 g?
18. A cereal company fills boxes with \sigma = 6 g. The label says 500 g, and only 2.5% of boxes may contain less. What mean \mu should the machine be set to?

Why It Matters

Normal probabilities turn a mean and a standard deviation into concrete decisions:

Manufacturing: what share of bottles is underfilled, and where to set the machine so almost none are
Health: doctors compare a child's height or weight to growth charts, which are percentiles of a normal curve
Education: exam boards set the score for the top 10% or decide how many students pass
Products: a company can promise "90% of batteries last at least 18 hours" only after this calculation
Sports and science: deciding whether a result is unusual means asking how much area lies beyond it

The same four steps answer all of these, whatever the units.

Real World Applications

Quality Control: Underfilled Cans

Drink factories check what share of cans fall below the amount printed on the label, and adjust the filling machines.

Example:

A line fills cans with $\mu = 330$ mL and $\sigma = 2$ mL. Any can below 327 mL is flagged as underfilled.

1Try It Yourself

Cans are filled with $\mu = 330$ mL and $\sigma = 2$ mL, normally distributed. The table gives $P(Z < -1.5) = 0.0668$ .

What proportion of cans contain less than 327 mL? Give a decimal.

Step 1: Write the mathematical expression

Standardize 327 mL, then read the left area:

Growth Charts at the Doctor

Pediatricians compare babies with growth charts. Each curve on the chart is a percentile of a normal distribution.

Example:

Weights of 6-month-old babies in a study are about normal with $\mu = 8$ kg and $\sigma = 0.5$ kg. A baby below the 10th percentile gets an extra check-up.

2Try It Yourself

Baby weights at 6 months: $\mu = 8$ kg, $\sigma = 0.5$ kg. The table shows that $z \approx -1.28$ has 10% of the area to its left.

Below what weight (in kg) is a baby in the lowest 10%?

Step 1: Write the mathematical expression

Use $x = \mu + z\sigma$ :

Planning a City Marathon

Race organizers estimate how many runners will be on the course at certain times, to plan water stations and staff.

Example:

Finishing times are about normal with $\mu = 240$ minutes and $\sigma = 30$ minutes. Organizers want to know the share of runners finishing between 225 and 285 minutes.

3Try It Yourself

Finishing times: $\mu = 240$ min, $\sigma = 30$ min. Table values: $P(Z < 1.5) = 0.9332$ and $P(Z < -0.5) = 0.3085$ .

What proportion of runners finish between 225 and 285 minutes?

Step 1: Write the mathematical expression

Subtract the two left areas:

Key Takeaways

1To find a normal probability, sketch the region, standardize with $z = \frac{x - \mu}{\sigma}$ , read the z-table, then adjust
2The z-table gives the LEFT area: $P(X < a) = P(Z < z_a)$
3Right tail: $P(X > a) = 1 - P(Z < z_a)$
4Between two values: $P(a < X < b) = P(Z < z_b) - P(Z < z_a)$
5Inverse problems: find $z$ for the given area inside the table, then $x = \mu + z\sigma$ (top 10% means $z \approx 1.28$ )
6A probability is also a proportion of the population: $0.0668$ of 2000 bottles is about 134 bottles

Frequently Asked Questions

No. For a continuous variable a single value has probability 0, so

<

and

\le

give the same area. This is different from the binomial distribution, where it matters.

Glossary

Standardizing: Turning a value $x$ into a z-score with $z = \frac{x - \mu}{\sigma}$ , so that the standard normal table can be used
Z-table: Table of the standard normal distribution: for each $z$ it gives the area to the left, $P(Z < z)$
Left tail / right tail: The area below a value (left) or above a value (right) under the normal curve
Complement rule: $P(X > a) = 1 - P(X < a)$ , because the whole area under the curve is 1
Percentile: The value with a given percentage of the population below it, for example the 90th percentile
Cut-off value: The value $x = \mu + z\sigma$ that separates a given area, for example the top 10%

Formula Card

Standardize

z = \frac{x - \mu}{\sigma}

From real units to a z-score

Left area

P(X < a) = P(Z < z_a)

Read straight from the z-table

Right area

P(X > a) = 1 - P(Z < z_a)

Complement of the table value

Area between

P(a < X < b) = P(Z < z_b) - P(Z < z_a)

Difference of two left areas

Cut-off value

x = \mu + z\sigma

From a z-score back to real units (percentiles)

Today's challengeOne problem, three tries

Normal Probability

Definition

The four-step method

Working backwards: percentiles and cut-offs

Worked Examples

Common Mistakes

Interactive Visual

Women's heights: μ = 165 cm, σ = 6 cm. Drag the bound or switch the shading

Exam scores: μ = 70, σ = 8. Area between 62 and 86

Cut-off: which score marks the top 10%?

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Quality Control: Underfilled Cans

Growth Charts at the Doctor

Planning a City Marathon

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Standardize

Left area

Right area

Area between

Cut-off value

More in This Topic

Introduction to Probability Distributions

Binomial Probability

Discrete vs Continuous Distributions

Expected Value