B. Probability Distributions · Continuous Distributions and the Normal Distribution

The Empirical Rule: 68, 95, 99.7

B2.6About 65 minutes

Learning goals

I can state the empirical rule (68–95–99.7) for normal distributions.
I can split a normal curve into the 34%, 13.5%, 2.35% and 0.15% pieces and add them to answer questions.
I can use the rule to estimate percents and counts, and to work backwards to μ and σ.
I can use the rule to judge whether a data set is approximately normal.

Words to know

empirical rule
For any normal distribution, about 68% of values lie within 1σ of the mean, about 95% within 2σ, and about 99.7% within 3σ. Also called the 68–95–99.7 rule.

within kσ of the mean
Between μ − kσ and μ + kσ.

unusual value
A common guideline: a value more than 2 standard deviations from the mean (it happens only about 5% of the time in a normal distribution).

outlier (normal model)
A value more than 3σ from the mean, which happens only about 0.3% of the time if the model is right.

A quick way to estimate normal probabilities

Every normal curve has the same shape; only the centre and the scale change. So the same fractions of the area always lie within 1, 2 and 3 standard deviations of the mean. Memorize three numbers and you can answer many questions in your head:

about 68% of values are within 1σ of μ;
about 95% are within 2σ;
about 99.7% are within 3σ.

The curriculum states the first two as "approximately 68%" and "approximately 95%". More exact values are 68.27%, 95.45% and 99.73%; you will find these from the z-table later in the unit.

The empirical rule. Using symmetry, 68% splits into 34% on each side; the next band adds 13.5% on each side (95% − 68% = 27%, halved); the third adds 2.35% each side; and only 0.15% is beyond 3σ on each side.

The pieces

Using symmetry you can break the curve into 8 pieces:

between μ and μ + σ: 34% (half of 68%);
between μ + σ and μ + 2σ: 13.5% (half of 95% − 68% = 27%);
between μ + 2σ and μ + 3σ: 2.35% (half of 99.7% − 95% = 4.7%);
beyond μ + 3σ: 0.15% (half of 0.3%).

The left side mirrors the right. Useful combinations: above μ + σ is 13.5 + 2.35 + 0.15 = 16%; above μ + 2σ is 2.5%.

Worked example

Example 1: Maple syrup bottles

A machine fills bottles with maple syrup. The volumes are normally distributed with μ = 500 mL and σ = 4 mL. (a) What percent of bottles hold between 492 and 508 mL? (b) What percent hold more than 504 mL? (c) What percent hold between 496 and 512 mL? (d) In a run of 2000 bottles, about how many hold less than 492 mL?

Mark the scale: 488, 492, 496, 500, 504, 508, 512.
(a) 492 and 508 are μ ± 2σ, so about 95%.
(b) 504 is μ + σ. Above it: 13.5 + 2.35 + 0.15 = 16%.
(c) 496 is μ − σ and 512 is μ + 3σ. Add 34 + 34 + 13.5 + 2.35 = 83.85%.
(d) Below 492 (μ − 2σ) is 2.35 + 0.15 = 2.5%. 2.5% of 2000 = 0.025 × 2000 = 50.

(a) about 95% (b) about 16% (c) about 83.85% (d) about 50 bottles.

Maple syrup volumes N(500, 4²). The shaded region from 496 mL (μ − σ) to 512 mL (μ + 3σ) holds about 34% + 34% + 13.5% + 2.35% = 83.85% of the bottles.

Worked example

Example 2: Working backwards to μ and σ

Marks on a large provincial test are approximately normal. About 16% of students scored above 80 and about 2.5% scored below 50. Estimate μ and σ.

16% above means 80 is one SD above the mean: μ + σ = 80.
2.5% below means 50 is two SDs below: μ − 2σ = 50.
Subtract: 3σ = 30, so σ = 10.
Then μ = 80 − 10 = 70.

μ ≈ 70 and σ ≈ 10.

Worked example

Example 3: Is the seedling sample approximately normal?

The 40 seedlings from the lesson on histograms have x̄ ≈ 31.86 cm and s ≈ 7.30 cm. Counting from the sorted list: 28 heights lie within 1 SD of the mean (24.57 to 39.16), 36 within 2 SD (17.27 to 46.46) and all 40 within 3 SD. Compare with the empirical rule.

Within 1 SD: 28/40 = 70%. Rule: about 68%. Close.
Within 2 SD: 36/40 = 90%. Rule: about 95%. A little low: the three shortest seedlings (16.3, 16.8 and 17.0 cm) and the tallest (47.5 cm) are just outside 2 SD.
Within 3 SD: 40/40 = 100%. Rule: about 99.7%. Consistent.
The histogram also had one main peak and was roughly symmetric.

The percentages are reasonably close to 68–95–99.7 for a sample of only 40, so a normal model is reasonable. With small samples, expect differences of a few percent.

Unusual values

Because only about 5% of values are more than 2σ from the mean, statisticians often call such a value unusual. Values more than 3σ away happen only about 0.3% of the time; if you see one, check whether it is an error or whether the normal model is wrong. You will use this kind of thinking again in the unit One-Variable Data Analysis.

Just for fun

Statistician's weather report: "Tomorrow will be within one standard deviation of normal, with a 68% chance of nothing surprising."

Hands-on activity

Test the rule on your 5-second data

You need: the 40 stopwatch times from the last lesson (or 40 new ones), a calculator.

Find x̄ and s of your 40 times.
Work out x̄ − s, x̄ + s, x̄ − 2s, x̄ + 2s, x̄ − 3s and x̄ + 3s.
Count how many times are within 1, 2 and 3 standard deviations of the mean, and change each count to a percent.
Compare with 68%, 95% and 99.7%.

What to look for: with 40 values, results like 60% to 75% within 1 SD and 90% to 100% within 2 SD are typical for roughly normal data. Much lower values, or a lopsided count (far more above the mean than below), suggest your times are not normal, perhaps because you tended to stop late.

Your turn

Heights of a group of Grade 12 students are N(170, 8²) cm. What percent are between 162 and 178 cm?

For N(170, 8²), what percent are taller than 186 cm?

For N(170, 8²), what percent are between 154 and 178 cm?

For N(170, 8²), what percent are shorter than 162 cm?

In a group of 600 students modelled by N(170, 8²), about how many are between 170 and 194 cm?

A normal distribution has 2.5% of its values above 130 and 16% below 90. Find μ and σ.

Is a value 3.4 standard deviations above the mean unusual? Explain.

More practice: warm-up

About what percent of a normal distribution is within 2σ of the mean?

About what percent is between μ and μ + σ?

About what percent is above μ + 2σ?

About what percent is beyond 3σ on both sides together?

For N(50, 5²), what values are 1 standard deviation from the mean?

More practice: on your own

Battery lives are N(400, 25²) hours. What percent last between 350 and 425 hours?

For the batteries, what percent last less than 325 hours or more than 475 hours?

A teacher says a mark is "unusually high" if it is more than 2 SD above the mean. The marks are N(68, 9²). Which marks are unusually high? About what percent of students get one?

The lengths of icicles in a sample of 50 have x̄ = 22 cm and s = 6 cm. 30 are within 1 SD, 44 within 2 SD and 50 within 3 SD. Is a normal model reasonable?

Explain why the band from μ + σ to μ + 2σ holds 13.5%.

Challenge

A cereal company fills boxes with mean μ g and σ = 5 g. It wants only about 2.5% of boxes to contain less than the 750 g printed on the box. What should μ be?

Snowfall totals for a city's winters are approximately normal. In about 84% of winters the total is below 230 cm, and in about 97.5% it is above 110 cm. Find μ and σ.

Use the empirical rule to estimate the 84th, 97.5th and 16th percentiles of N(500, 4²).

Check your understanding

Quiz

About what percent of values in a normal distribution lie within 1 standard deviation of the mean?

X ~ N(60, 5²). About what percent of values are above 70?

X ~ N(60, 5²). About what percent of values are between 55 and 70?

A normal distribution has about 16% of values above 45 and 50% above 40. What is σ?

In a sample of 200, which result best suggests the data are NOT normal?

For N(500, 4²), about how many of 1000 bottles hold more than 504 mL?

For parents and teachers

Big idea (B2.6): the empirical rule: in any normal distribution, about 68% of values are within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD. Splitting the curve into 34%, 13.5%, 2.35% and 0.15% pieces answers many questions quickly.

How to help: Have your teen draw the "34, 13.5, 2.35, 0.15" curve from memory. Then give everyday questions: "If boys' heights in a grade are N(175, 7²), what percent are over 189 cm?" (2.5%).

Common difficulties:

Using 68% when the question asks about only one side of the mean (that is 34%).
Forgetting to mark the scale (μ ± σ, μ ± 2σ, ...) before shading.
Applying the rule to values that are not whole numbers of SDs from the mean. For those you need z-scores and the z-table (the next lessons).
Treating the rule as exact. It is approximate.