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Teacher Guide: Normal Probability

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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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Probability Distributions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Compute P(X<a)P(X < a), P(X>a)P(X > a) and P(a<X<b)P(a < X < b) for a normal variable in real units
  • Standardize values and read left areas from the z-table
  • Solve inverse problems: find the value for a given percentile or cut-off with x=μ+zσx = \mu + z\sigma
  • Interpret probabilities as proportions and counts of a population
Prerequisites
  • • Introduction to Normal Distribution: mean, standard deviation and the bell curve
  • • Z-Scores: z=x−μσz = \frac{x - \mu}{\sigma}
  • • Standard Normal Distribution: reading the z-table
  • • Percentages and decimals
Discussion Starters
  • 1. Before calculating, how can a sketch tell you whether the answer is above or below 0.5?
  • 2. Why can one table serve every normal distribution, whatever the mean and standard deviation?
  • 3. A battery maker promises that 90% of batteries last at least 18 hours. How could they have found that number?
  • 4. If the top 10% of scores earn a certificate, why is the cut-off not simply 10% above the mean?
  • 5. What does it mean if a measured value has a z-score of 4?
Common Misconceptions

The table value is the answer to every question

Remediation: Require a sketch for every problem. Ask: is the shaded region the left side? If not, what must be done with the table value?

The area between two values is found from the difference of the z-scores

Remediation: Use the visual: drag both bounds and compare the shaded area with P(Z<z2−z1)P(Z < z_2 - z_1). Show that the region from -1 to 2 is not the same as the region left of 3.

A percentile and a z-score are the same number

Remediation: Contrast the 90th percentile (an area of 0.90) with its z-score (1.28) and its raw value (80.24 points). Keep three columns: area, z, x.

Differentiation Ideas

For Struggling Students:

  • • Start with left areas only and whole-number z-scores
  • • Provide a template: sketch, z, table, adjust
  • • Give the table values in the problem so students focus on standardizing and adjusting

For On-Level Students:

  • • Mix left, right and between questions in different real-world units
  • • Convert probabilities into counts for a given population
  • • Solve inverse problems for top and bottom percentages

For Advanced Students:

  • • Find the mean a machine must be set to so that only 2.5% of products fall below a limit
  • • Find the standard deviation from a mean and one known percentile
  • • Compare the 68-95-99.7 rule with exact table values
Standards Alignment
  • HSS-ID.A.4 (CCSS.MATH.CONTENT.HSS.ID.A.4)

    Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages; use calculators, spreadsheets, and tables to estimate areas under the normal curve

  • MP4 (CCSS.MATH.PRACTICE.MP4)

    Model with mathematics

Lesson Resources
  • visualDraggable Normal Curve

    Students drag the bounds in real units and watch the z-scores and shaded area change

  • activityPercentile Hunt

    In inverse mode students find the z-score for a target area, then convert it to a cut-off score

  • worksheetFour Regions, One Table

    Mixed problems with left, right, between and inverse questions in different units

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Normal probability answers questions like "What fraction of bottles hold less than 494 mL?" when a quantity XX follows a normal distribution with mean μ\mu and standard deviation σ\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. 1.Sketch and shade the region: left of aa, right of aa, or between aa and bb.
  2. 2.Standardize: turn each value into a z-score
z=x−μσz = \frac{x - \mu}{\sigma}
  1. 3.Read the z-table: it gives the area to the left of zz, that is P(Z<z)P(Z < z) (see the lesson on the standard normal distribution).
  2. 4.Adjust for the region:
  • Left: P(X<a)=P(Z<za)P(X < a) = P(Z < z_a)
  • Right: P(X>a)=1−P(Z<za)P(X > a) = 1 - P(Z < z_a)
  • Between: P(a<X<b)=P(Z<zb)−P(Z<za)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 zz whose left area matches in the table, then go back to real units:
x=μ+zσx = \mu + z\sigma
A probability of 0.93320.9332 also means 93.32% of the population, so every answer can be read as a proportion of people, bottles or batteries.

Worked Examples

Women's heights in a country are normally distributed with μ=165\mu = 165 cm and σ=6\sigma = 6 cm. What proportion of women are shorter than 174 cm?
1

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
2

Standardize

z=174−1656=96z = \frac{174 - 165}{6} = \frac{9}{6} → z=1.5z = 1.5
3

Read the z-table

Row 1.5, column 0.00 → P(Z<1.5)=0.9332P(Z < 1.5) = 0.9332
4

Interpret

A left area needs no adjustment → P(X<174)=0.9332P(X < 174) = 0.9332

Common Mistakes

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

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

Correct: Subtract from 1: P(X>14)=1−0.8944=0.1056P(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)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=0.81850.9772 - 0.1587 = 0.8185.

Standardizing in the wrong order or dividing by the wrong number: z=μ−xσz = \frac{\mu - x}{\sigma} or z=x−μσ2z = \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=x−μσz = \frac{x - \mu}{\sigma}: values above the mean get a positive zz, values below get a negative zz.

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≈1.28z \approx 1.28 (positive, above the mean). Bottom 10% gives z≈−1.28z \approx -1.28.

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 μ=330\mu = 330 mL and σ=2\sigma = 2 mL. Any can below 327 mL is flagged as underfilled.

1Try It Yourself

Cans are filled with μ=330\mu = 330 mL and σ=2\sigma = 2 mL, normally distributed. The table gives P(Z<−1.5)=0.0668P(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 μ=8\mu = 8 kg and σ=0.5\sigma = 0.5 kg. A baby below the 10th percentile gets an extra check-up.

2Try It Yourself

Baby weights at 6 months: μ=8\mu = 8 kg, σ=0.5\sigma = 0.5 kg. The table shows that z≈−1.28z \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=μ+zσx = \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 μ=240\mu = 240 minutes and σ=30\sigma = 30 minutes. Organizers want to know the share of runners finishing between 225 and 285 minutes.

3Try It Yourself

Finishing times: μ=240\mu = 240 min, σ=30\sigma = 30 min. Table values: P(Z<1.5)=0.9332P(Z < 1.5) = 0.9332 and P(Z<−0.5)=0.3085P(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=x−μσz = \frac{x - \mu}{\sigma}, read the z-table, then adjust
  • 2The z-table gives the LEFT area: P(X<a)=P(Z<za)P(X < a) = P(Z < z_a)
  • 3Right tail: P(X>a)=1−P(Z<za)P(X > a) = 1 - P(Z < z_a)
  • 4Between two values: P(a<X<b)=P(Z<zb)−P(Z<za)P(a < X < b) = P(Z < z_b) - P(Z < z_a)
  • 5Inverse problems: find zz for the given area inside the table, then x=μ+zσx = \mu + z\sigma (top 10% means z≈1.28z \approx 1.28)
  • 6A probability is also a proportion of the population: 0.06680.0668 of 2000 bottles is about 134 bottles

Frequently Asked Questions

Is P(X<174)P(X < 174) different from P(X≤174)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.

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=1.645z = 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=4z = 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.

Glossary

Standardizing
Turning a value xx into a z-score with z=x−μσz = \frac{x - \mu}{\sigma}, so that the standard normal table can be used
Z-table
Table of the standard normal distribution: for each zz it gives the area to the left, P(Z<z)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)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=μ+zσx = \mu + z\sigma that separates a given area, for example the top 10%

Formula Card

Standardize

z=x−μσz = \frac{x - \mu}{\sigma}

From real units to a z-score

Left area

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

Read straight from the z-table

Right area

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

Complement of the table value

Area between

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

Difference of two left areas

Cut-off value

x=μ+zσx = \mu + z\sigma

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

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