Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Lesson 3: a set has mean 40 and σ 6. What value is 2 standard deviations above the mean?
Answer
52. 40 + 2 × 6 = 52. - From Year 8: what is 15% of 400?
Answer
60. 0.15 × 400 = 60. - From Year 9: what is (100 − 80) ÷ 2?
Answer
10. Half of 20.
Words used in this lesson
- normal distribution
- a symmetric bell-shaped distribution set by its mean and standard deviation. For example: heights of adults in a large city.
- 68-95-99.7 rule
- the approximate shares of a bell-shaped set within 1, 2 and 3 standard deviations of the mean. For example: the share between μ − 2σ and μ + 2σ.
- tail
- the thin end of a distribution beyond a stated value. For example: values above μ + 2σ.
- symmetric
- the left half is a mirror image of the right half. For example: the same share above and below the mean.
- z-score
- the number of standard deviations a value is from the mean. For example: z = (x − μ) ÷ σ.
- expected count
- the share multiplied by the total, a typical number. For example: 0.3% of 1 000 is 3.
The bakery scales in Dakar
A bakery in Dakar weighs a sample of its loaves every morning. The target mass is 500 g. Most loaves are close to 500 g; a few are a little light or heavy, and the masses pile up in the middle and thin out on both sides.
Predict first: if the standard deviation of the masses is 10 g, what share of the loaves do you expect within 10 g of 500? Within 20 g? Within 30 g? Then use the figure to check, and to see what changes when the standard deviation changes.
Within 0 standard deviations of the mean, from 500 g to 500 g: about 0% of the loaves.
Show me the steps
- Set the number of standard deviations to 1 and keep the size at 10 g. The interval is 490 g to 510 g and it holds about 68% of the loaves.
- Set it to 2. The interval is 480 g to 520 g, and it holds about 95% of the loaves.
- Set it to 3. The interval is 470 g to 530 g and it holds about 99.7% of the loaves, nearly all of them.
- Go back to 1 and make the size of one standard deviation 20 g. The interval is now 480 g to 520 g, twice as wide, but it still holds about 68%: the share depends on the number of standard deviations, not on the grams.
A normal distribution is symmetric and bell-shaped, centred on its mean μ, with a spread set by its standard deviation σ. For any normal distribution, about 68% of the values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.
A value x is z = (x − μ) ÷ σ standard deviations from the mean. The rule is stated in z, so it holds in any units, and it can be combined with symmetry: both halves of the curve are mirror images.
Real data are only approximately normal, so shares are “about” and counts are expected counts.
One standard deviation from the mean is not unusual. About a third of the values of a bell-shaped set are further than that from the mean, split between the two tails. A value more than 2 standard deviations away is about 1 in 20, and only beyond 3 is it very rare.
Worked examples
1. Bus waits are bell-shaped with mean 12 minutes and σ = 3. Which waits include about 95% of the buses?
- 95% is within 2 standard deviations: 2 × 3 = 6 minutes either side.
- From 12 − 6 to 12 + 6, which is 6 to 18 minutes.
2. What percentage of waits lie between 1 and 2 standard deviations above the mean?
- Within 2 standard deviations (both sides) is 95%, within 1 is 68%. The ring between them, on both sides, is 95 − 68 = 27%.
- By symmetry, half of that is above the mean: 27 ÷ 2 = 13.5%.
3. Of 600 buses, how many are expected to wait more than 3 standard deviations above the mean?
- Outside 3 standard deviations, in both tails, is 100 − 99.7 = 0.3%. One tail is 0.15%.
- 0.15% of 600 = 0.0015 × 600 = 0.9, so about 1 bus.
Your go. The first step is given. Lengths of bolts are bell-shaped with mean 50 mm and standard deviation 4 mm. What percentage are between 46 mm and 62 mm?
- 46 is 1 standard deviation below the mean, and 62 is 3 standard deviations above it.
- Now find the share between the mean and each bound, and add them.
Check the answer
The share between the mean and 1 below is half of 68% = 34%. The share between the mean and 3 above is half of 99.7% = 49.85%. Together: 34 + 49.85 = 83.85%, about 84%.
Check before you go on
1. Marks in a large test are bell-shaped with mean 62 and standard deviation 8. Which interval holds about 95% of the marks?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A bell-shaped set has mean 140 and standard deviation 12. What value is 2 standard deviations above the mean?
3. A set has mean 140 and standard deviation 12. About what percentage of values are above 152?
Level 2 · Building
4. The lengths of 400 bolts are bell-shaped with mean 40.0 mm and standard deviation 0.2 mm. About how many bolts are between 39.6 mm and 40.4 mm?
5. Heights are bell-shaped with mean 172 cm and standard deviation 7 cm. What percentage of heights are between 172 cm and 186 cm? Give the answer to one decimal place.
Level 3 · Stretch
6. In test A (mean 55, standard deviation 10) a pupil scores 78. In test B the mean is 70 and the standard deviation is 4. What score in test B is equally far above the mean, in standard deviations? Give the answer to one decimal place.
7. Journey times are bell-shaped with mean 20 minutes and standard deviation 5 minutes. Over 200 journeys, about how many take longer than 30 minutes?
Something to think about
Two bakeries both make loaves with a mean of 500 g, but the standard deviation is 5 g at one and 25 g at the other. A customer rejects any loaf more than 2 standard deviations from 500 g. Which bakery's customer is more likely to find a rejected loaf, and what is different about the rejected loaves at the two bakeries?
One way to think about it
About 1 loaf in 20 at both bakeries, because the share beyond 2 standard deviations is the same for any bell shape. What differs is the rejection zone in grams: more than 10 g from 500 g at the first bakery, and more than 50 g at the second.
Exit ticket
8. Two bell-shaped sets have the same mean. Set X has standard deviation 3 and set Y has standard deviation 9. Which statement is true?