At a glance. One lesson of about 60 minutes. Students state intervals for given shares, find shares between bounds using symmetry, calculate expected counts, and compare results using the number of standard deviations from the mean.
Calculator: no calculator. The rule supplies every share needed, and counts come from percentages of n. No calculator is needed.
What students will learn
Know
- About 68%, 95% and 99.7% lie within 1, 2 and 3 standard deviations of the mean.
- z = (x − μ) ÷ σ.
- The curve is symmetric about the mean.
Understand
- The standard deviation is the natural unit of distance on a bell-shaped set.
- The share within k standard deviations does not depend on the units or the size of σ.
- Real data are only approximately normal, so counts are expected counts.
Do
- Give an interval holding a stated share.
- Find shares between bounds using the rule and symmetry.
- Compare values from different distributions by z.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the rule and symmetry to find intervals, shares and counts (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if, explaining why the share beyond 2 standard deviations is the same at both bakeries while the rejection zone in grams is not.
- Criterion C. Communicating: sketching and shading a bell curve with the mean and the bounds marked in the units of the data.
- Criterion D. Applying mathematics in real-life contexts: bakery scales, bolts and journey times, deciding what counts as an unusual value.
The one thing to do with the figure
Predict, then shade. Before moving the slider, ask for the share within 1 standard deviation and write the class's guesses. Then reveal. After 2 and 3, change the size of one standard deviation and ask: “Will the share change?” Students should say no, then see the interval double in width.
The point to land: the shares belong to the number of standard deviations, and the grams belong to the size of one.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the bakery. Collect predictions of the three shares. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to the figure at 1, 2 and 3. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 52; 60; 10. |
|---|---|
| Hook | About 68%, 95% and 99.7%. |
| Worked examples | 6 to 18 minutes; 13.5%; about 1 bus. |
| Faded example | 83.85%, about 84%. |
| 1. Interval holding about 95% | C. A is 1 standard deviation, B is 3, D is 4. |
| 2. Value a number of standard deviations above the mean | 164. 140 + 2 × 12. |
| 3. Percentage above 1 standard deviation | 16. (100 − 68) ÷ 2. |
| 4. Expected count within 2 standard deviations | 380. 95% of 400. |
| 5. Share between the mean and 2 standard deviations | 47.5. 95 ÷ 2. |
| 6. Equally exceptional result on another scale | 79.2. 70 + 2.3 × 4. |
| 7. Expected count in one tail | 5. 2.5% of 200. |
| 8. What a larger standard deviation means | B. A says the mean differs; C and D are not implied by the spread. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Used 1, 3 or 4 standard deviations instead of 2. | Make the intervals in z first: 1, 2, 3. |
| 24 or 152 in question 2 | Gave the distance only, or one standard deviation. | Write mean + zσ as a line. |
| 32, 84 or 68 in question 3 | Gave both tails, the lower part, or the central share. | Sketch the curve, shade the part asked for. |
| 272, 399 or 20 in question 4 | Used the wrong number of standard deviations, or the share outside. | Divide the distance by σ first. |
| 95, 34 or 49.85 in question 5 | Gave both sides, one standard deviation, or three. | How many σ is 186 above the mean? |
| 93 or 78 in question 6 | Compared raw distances, or the same score. | Convert to z in the first test. |
| 10 or 195 in question 7 | Gave both tails, or the complement. | One tail is half of the outside share. |
| A, C or D in question 8 | Confused spread with mean, share, or count. | Which statement is about position, and which about spread? |
Questions worth asking
- Why are the shares the same for any normal distribution? Every normal curve is the same shape, stretched and shifted; z removes the units.
- Is a value 2.5 standard deviations from the mean unusual? Yes: only about 1 value in 160 is that far above the mean (about 1 in 80 counting both tails), so it is rare but not impossible. The 68-95-99.7 rule does not give this share; it lies between the 2σ and 3σ values.
- How would you check whether real data are approximately normal? Look at a histogram for symmetry and a bell shape, and test the 68-95-99.7 shares.
- What might make a measured set not bell-shaped? A hard limit, skew, or a mixture of two groups.
Differentiation
A way in for anyone stuck
- Give a pre-drawn bell with the 1, 2 and 3 standard deviation marks and the shares printed on the bands (34, 13.5, 2.35 and so on).
- Use a number line in the units of the data under the curve.
- Practise z with whole-number results before decimals.
Stretch
- Find the percentage between 1 and 2 standard deviations below the mean, then between 2 and 3.
- Decide which of two results, 70 in a test with mean 58 and standard deviation 5 or 84 in a test with mean 66 and standard deviation 6, is more exceptional.
Students learning in English as an additional language
“Bell-shaped” describes a curve that is highest in the middle and falls away in both directions, like a bell. “Tail” is the thin part at either end. “Symmetric” means the left half is a mirror image of the right half. Say “z score” for the number of standard deviations from the mean.
| Word | Plain meaning | Example |
|---|---|---|
| normal distribution | a symmetric bell-shaped distribution set by its mean and standard deviation | 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 | the share between μ − 2σ and μ + 2σ |
| tail | the thin end of a distribution beyond a stated value | values above μ + 2σ |
| symmetric | the left half is a mirror image of the right half | the same share above and below the mean |
| z-score | the number of standard deviations a value is from the mean | z = (x − μ) ÷ σ |
| expected count | the share multiplied by the total, a typical number | 0.3% of 1 000 is 3 |
Sentence starters for explaining aloud or in writing:
- The value is … standard deviations above the mean because …
- By symmetry the share above is …
- About …% of the values lie between … and ….
Two things not to say
“A value more than one standard deviation from the mean is exceptional.” About a third of values are further than that.
“The standard deviation is the largest distance from the mean.” It is a typical distance.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- IntervalMean 80 and σ 5, bell-shaped. Which interval holds about 68% of the values?
75 to 85. - ShareAbout what percentage of a bell-shaped set is below the mean?
50%.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextRainfall totals for a month in a region of India are bell-shaped with mean 150 mm and σ 20 mm. In what range do about 99.7% of months fall?
90 mm to 210 mm. - Earlier work (Lesson 5)A bell-shaped set with mean 20 and σ 3 is multiplied by 2 and then 1 is added. Give the new mean and σ.
41 and 6.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA pupil says: "My score is 1.5 standard deviations above the mean, so I am in the top 1.5%." Is she right?
No: 1.5 standard deviations above the mean has about 7% above it, so she is in about the top 7%. - GeneralizeFor a bell-shaped set, show that the share between 1 and 2 standard deviations above the mean is 13.5% using only the 68% and 95% shares.
(95 − 68) ÷ 2 = 13.5, because the ring between 1 and 2 standard deviations on both sides is 27%, and the curve is symmetric.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is required. The shares to use are 68%, 95% and 99.7%.