At a glance. One lesson of about 60 minutes. Students build the standard deviation from deviations, explain why they are squared, work from a sum of squares in either direction, and compare data sets with equal means.
Calculator: GDC optional. Everything can be done by hand with small sets. The optional section shows the one-variable statistics screen and warns which of its two standard deviations to read.
What students will learn
Know
- A deviation is x − μ and the deviations add to zero.
- σ = √(Σ(x − μ)² ÷ n), and the variance is σ².
- σ is in the units of the data.
Understand
- The standard deviation is a typical distance from the mean, built so that deviations cannot cancel.
- Two sets with the same mean can have very different spread.
- The range uses two values; σ uses all of them.
Do
- Calculate σ by hand for a small set.
- Work backwards from σ to the sum of squares.
- Compare the spread of data sets.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: deviations, variance and σ by hand, and working backwards (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if, describing every data set with σ = 0 and the effect of adding a value equal to the mean.
- Criterion C. Communicating: the table x, x − μ, (x − μ)², and units kept with the answer.
- Criterion D. Applying mathematics in real-life contexts: the festival hook, where the mean temperature alone does not support a decision.
The one thing to do with the figure
Ask for the mean square before revealing it. On set 2, ask: “The deviations are −10, −5, 0, 5, 10. Without the readout, estimate the standard deviation.” Most will say about 5 to 7: the typical distance. Then reveal 7.07.
Then compare set 3 with set 4. Set 3's range is twice set 4's, 16 against 8, but its σ is only about 1.4 times as large, 5.06 against 3.58, because three of its values sit on the mean. The range sees only the two end values; σ sees all five.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the two towns. Collect ideas for one number that measures variation. |
| 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 C, return to the warning card: the deviations cancel, so each is squared first. |
| 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 | 7; 25; 22. |
|---|---|
| Hook | The standard deviation: the root of the mean of the squared distances from the mean. |
| Worked examples | σ = 1.9; the deviations sum to Σx − nμ = 0; σ of Q is twice σ of P. |
| Faded example | 2.7. |
| 1. Standard deviation from deviations | B. A adds the signed deviations; C averages the distances without squaring; D stops at the variance. |
| 2. Variance from deviations | 8. 40 ÷ 5. |
| 3. Standard deviation from a sum of squares | 3. √(72 ÷ 8). |
| 4. Standard deviation of a small set | 2.8. √8. |
| 5. Adding a value equal to the mean | 3.2. √(50 ÷ 5). |
| 6. Sum of squares from the standard deviation | 90. 10 × 3². |
| 7. Finding an unknown value then the spread | 4. x = 16, then √16. |
| 8. Comparing spread | C. Mean squares 0, 2, 20 and 14.4; D has the widest range but is not the most spread. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Added the signed deviations, averaged the distances without squaring, or stopped at the variance. | Say the steps in order: square, mean, root. |
| 40, 0 or 10 in question 2 | Did not divide, added signed deviations, or divided by n − 1. | Table: x, x − μ, (x − μ)², with a total and a division. |
| 9 or 8.49 in question 3 | Stopped at the variance, or took the root of the sum. | Say the three steps aloud: mean of the squares, then root. |
| 8, 3.3 or 5.7 in question 4 | Gave the variance, divided by 3, or skipped dividing. | Check that n is the number of values. |
| 3.5 or 10 in question 5 | Used the old standard deviation, or gave the variance. | Recompute Σ and n separately. |
| 30 or 9 in question 6 | Multiplied σ by n, or gave σ². | Rearrange σ² = Σ ÷ n. |
| 16 or 8.9 in question 7 | Gave the variance, or the root of the sum. | Find x first, then list deviations. |
| A, B or D in question 8 | Chose the set with the widest range, or the smallest spread. | Compute each mean square; the range is not the spread. |
Questions worth asking
- Why does the standard deviation use the mean and not the median? The deviations from the mean have the smallest sum of squares; the mean is the centre this measure is built around.
- Why take the root at the end? The squares are in squared units; the root returns to the units of the data.
- If one value is changed to a far greater one, which changes more, the range or σ? Both change; σ also depends on how many other values are away from the mean.
- Could σ be larger than the range? No: no value can be further from the mean than the range.
Differentiation
A way in for anyone stuck
- Give a three-column template: x, x − μ, (x − μ)².
- Use whole-number means so that deviations are quick to find.
- Write the formula in words first: root of the mean of the squared distances from the mean.
Stretch
- Find a data set of four whole numbers in which σ is exactly half of the range, and explain why σ can never be more than half the range.
- Find a data set with five values, mean 10 and σ exactly 4 using whole numbers.
Students learning in English as an additional language
“Deviation” is how far a value is from the mean. “Variance” is the mean of the squared deviations. “Standard deviation” is the square root of the variance. Say “mu” for μ and “sigma” for σ.
| Word | Plain meaning | Example |
|---|---|---|
| deviation | the signed distance of a value from the mean | x − μ = −1.5 |
| variance | the mean of the squared deviations | σ² = 2.25 |
| standard deviation | the square root of the variance, a typical distance from the mean | σ = 1.5 |
| spread | how widely values are scattered | the same mean, a different spread |
| population | the whole set of values being described | every pupil in the year group |
| root mean square | the square root of the mean of the squares | √(Σd² ÷ n) |
Sentence starters for explaining aloud or in writing:
- The mean is … and the deviations are …
- This set is more spread out because …
- The standard deviation is … so a typical value is … from the mean.
Two things not to say
“Standard deviation is just the range.” The range uses only the highest and lowest values.
“Divide the deviations by n and you have the spread.” Signed deviations add to zero, so the squares are needed.
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.
- CalculateFind the variance of 1, 3, 5.
2.67 to two decimal places. Deviations −2, 0, 2; the squares add to 8; 8 ÷ 3 = 2.67. - FactWhat is Σ(x − μ) for any data set?
0.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextBus delays in a town in Kenya have σ = 4 minutes, and 30 buses are timed. What is Σ(x − μ)²?
480. 30 × 16. - Earlier work (Lesson 1)Estimate the mean from classes 0 to under 10 (f = 5) and 10 to under 20 (f = 15).
12.5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSet A is 2, 4, 6. Set B is 12, 14, 16. Without computing, how do their standard deviations compare, and why?
Equal: B is A with 10 added to every value, so every deviation from the mean is unchanged. - GeneralizeA data set has n values, all equal to c except one value c + d. Find σ in terms of n and d.
σ = d ÷ n × √(n − 1). The mean is c + d ÷ n; the deviations are d − d ÷ n once and −d ÷ n (n − 1) times, giving mean square d²(n − 1) ÷ n².
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 the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.