B. Probability Distributions · Continuous Distributions and the Normal Distribution

Working Backwards: Percentiles, Cut-offs and Unknown Parameters

B2.8About 70 minutes

Learning goals

I can use the z-table in reverse to find the z-score for a given area.
I can find percentiles and cut-off values of a normal distribution.
I can find an unknown mean or standard deviation from a given probability.
I can find the percentile rank of a given value.

Words to know

percentile
The kth percentile is the value with k% of the distribution below it. The median is the 50th percentile.

percentile rank
The percent of the distribution below a given value.

quartiles
Q1 is the 25th percentile and Q3 is the 75th percentile; the interquartile range is IQR = Q3 − Q1.

inverse normal
Finding the value x (or z) that has a given area to its left. Spreadsheet: =NORM.INV(area, μ, σ) or =NORM.S.INV(area).

The question turned around

So far you were given a value and asked for a probability. Often the question runs the other way:

"How tall must a seedling be to be in the top 5%?"
"Between which volumes do the middle 80% of bottles fall?"
"Where should a machine's mean be set so that only 2% of boxes are underweight?"

Now you know an area and need a value. You use the same table, backwards.

Using the table in reverse

Sketch the curve and shade the given area. Turn it into an area to the left (the kind the table gives).
Search inside the table for the area closest to it. Read off z from the row and column.
Convert to x with x = μ + zσ.

If the area is below 0.5, z is negative: look up 1 minus the area and attach a minus sign. If the area is exactly halfway between two table entries (0.95 lies halfway between 0.9495 and 0.9505), use the z halfway between them (1.645).

z	P(Z < z)	P(Z < −z) = 1 − P(Z < z)
0.52	0.6985	0.3015
0.67	0.7486	0.2514
0.68	0.7517	0.2483
0.84	0.7995	0.2005
0.88	0.8106	0.1894
1.00	0.8413	0.1587
1.04	0.8508	0.1492
1.14	0.8729	0.1271
1.28	0.8997	0.1003
1.29	0.9015	0.0985
1.64	0.9495	0.0505
1.65	0.9505	0.0495
1.96	0.9750	0.0250
2.00	0.9772	0.0228
2.05	0.9798	0.0202
2.06	0.9803	0.0197
2.33	0.9901	0.0099
2.68	0.9963	0.0037

z-table values used in this lesson (taken from the full standard normal table in the lesson Using the z-Table to Find Normal Probabilities). The middle column is the area to the left of z; the last column is the area to the left of −z.

To find the 90th percentile of Z, look for 0.90 inside the table. The closest entry is 0.8997, in row 1.2, column .08, so z ≈ 1.28.

Worked example

Example 1: Percentiles of the standard normal

Find (a) the 90th percentile of Z and (b) the first quartile (25th percentile) of Z.

(a) Look for 0.9000. Nearby entries: P(Z < 1.28) = 0.8997 and P(Z < 1.29) = 0.9015. 0.8997 is closer.
So the 90th percentile is z ≈ 1.28.
(b) 0.25 is less than 0.5, so z is negative. Look for 1 − 0.25 = 0.75: P(Z < 0.67) = 0.7486 and P(Z < 0.68) = 0.7517. 0.7486 is closer, so z = 0.67 for the 75th percentile.
By symmetry, the 25th percentile is z ≈ −0.67.

(a) z ≈ 1.28 (b) z ≈ −0.67

Worked example

Example 2: The top 5% of seedlings

Heights of maple seedlings are N(32, 10.2²) cm. A nursery labels the tallest 5% as "premium". What is the minimum height for a premium seedling?

Top 5% means 95% are below: we need the 95th percentile.
In the table, P(Z < 1.64) = 0.9495 and P(Z < 1.65) = 0.9505. 0.95 is exactly halfway, so use z = 1.645.
x = 32 + 1.645(10.2) = 32 + 16.779 = 48.779.

A premium seedling must be at least about 48.8 cm tall.

Worked example

Example 3: The middle 80%

Syrup volumes are N(500, 4²) mL. Between which two volumes do the middle 80% of bottles lie?

The middle 80% leaves 10% in each tail. So we need the 10th and 90th percentiles.
90th percentile of Z: z ≈ 1.28. 10th percentile: z ≈ −1.28.
x = 500 ± 1.28(4) = 500 ± 5.12.

Between about 494.88 mL and 505.12 mL.

The middle 80% of syrup volumes lies between 494.88 mL and 505.12 mL, which are 1.28 standard deviations either side of the mean. Each tail holds 10%.

Worked example

Example 4: Setting the mean of a machine

A cereal box says 750 g. The filling machine has σ = 6 g. Where should the mean be set so that only 2% of boxes contain less than 750 g?

The area to the left of 750 must be 0.02. Look for 1 − 0.02 = 0.98 in the table.
P(Z < 2.05) = 0.9798 and P(Z < 2.06) = 0.9803; 0.9798 is closer, so z = 2.05 and 750 is at z = −2.05.
x = μ + zσ gives 750 = μ + (−2.05)(6) = μ − 12.3.
μ = 750 + 12.3 = 762.3.

Set the mean to about 762.3 g.

Worked example

Example 5: Finding σ

On a large exam the mean is 70, and a mark of 85 is at the 90th percentile. Assuming normal marks, estimate σ.

The 90th percentile has z ≈ 1.28.
85 = 70 + 1.28σ, so 1.28σ = 15.
σ = 15/1.28 ≈ 11.72.

σ ≈ 11.7

Worked example

Example 6: Percentile rank of a value

What is the percentile rank of a seedling 41 cm tall in N(32, 10.2²)?

z = (41 − 32)/10.2 ≈ 0.88.
P(Z < 0.88) = 0.8106.

About 81.1% of seedlings are shorter, so 41 cm is about the 81st percentile.

Just for fun

A student told their parents they scored in the 99th percentile. The parents asked, "Out of how many percentiles?"

Hands-on activity

Where do you stand?

You need: a measuring tape, a calculator, the z-table above, and a helper.

Measure your arm span (fingertip to fingertip with arms stretched out) to the nearest centimetre.
Suppose arm spans of Canadian teens your age are modelled by a normal distribution with μ = 170 cm and σ = 9 cm (an example model for practice, not official data).
Find your z-score and your percentile rank.
Find the arm spans that mark the 10th and 90th percentiles of the model.
Measure two or three family members and find their percentile ranks too (they are not teens, so talk about whether the model fits them).

What to look for: the 10th and 90th percentiles are 170 ± 1.28(9), about 158.5 cm and 181.5 cm. Most people land between them. Arm span is usually close to a person's height, so you can check your measurement against your height.

Your turn

Find the z-score of the 75th percentile.

Find the z-score with 15% of the area to its left.

Syrup volumes are N(500, 4²). Find the 99th percentile.

Find the z-score that has 30% of the area to its right.

Heights are N(170, 8²) cm. Find Q1, Q3 and the IQR.

Marks are N(68, 9²). What mark is needed to be in the top 10%?

A commute has mean 42 min. On 95% of days it takes less than 55 min. Estimate σ.

More practice: warm-up

What percentile is the median?

What z-score is the 50th percentile of Z?

Which z has P(Z < z) = 0.8413?

Which z has P(Z < z) = 0.0228?

Write the formula for changing z back into x.

More practice: on your own

Birth masses are N(3.4, 0.5²) kg. Between which masses do the middle 95% of babies lie?

Seedlings are N(32, 10.2²). A nursery discards the shortest 2%. What is the cut-off height?

What is the percentile rank of a commute of 50 min if commutes are N(42, 7²)?

Milk cartons should hold 1 L. A machine has σ = 8 mL. Where should the mean be set so only 1% hold less than 990 mL?

The same machine is set to μ = 1000 mL. How small must σ be so that only 1% hold less than 990 mL?

Challenge

A normal distribution has 10% of its values below 40 and 20% above 70. Find μ and σ.

Show that for any normal distribution the IQR is about 1.34σ.

A common rule calls a value an outlier if it is more than 1.5 IQR above Q3 or below Q1. For a normal distribution, where is the upper fence in z-terms, and what proportion of values are outliers?

Check your understanding

Quiz

Which z-score is the 90th percentile of the standard normal distribution?

Heights are N(170, 8²) cm. What height is at the 95th percentile?

Which z-scores bound the middle 95% of a normal distribution?

A machine has σ = 5 g. Only 2% of packages should be under 500 g. What should μ be?

A value is at the 30th percentile. What does that mean?

The mean is 60 and a value of 72 is at the 84th percentile (z ≈ 1). What is σ approximately?

For parents and teachers

Big idea (B2.8): you can use the z-table in reverse: from an area (a percentile, a top or bottom percent, a middle percent) find z, then x = μ + zσ. With an unknown μ or σ, set up the same equation and solve.

How to help: Ask your teen to turn each question into "what area is to the left?" before looking anything up. Check answers with a spreadsheet: =NORM.INV(0.95, 32, 10.2) gives the 95th percentile of the seedling heights.

Common difficulties:

Looking up the area in the z-column instead of inside the table.
Using z = 1.96 for "top 5%" (that is 1.645; 1.96 is for 2.5% in each tail).
Forgetting the negative sign for percentiles below the 50th.
Sign errors in μ − zσ when z is negative.