The shape of the lesson
By the end of the lesson, learners can
- Squared deviations: Each value's deviation is \(x - \bar{x}\).
- Frequency tables and grouped data: With frequencies, every sum gets an \(f\): \(\bar{x} = \dfrac{\sum fx}{\sum f}\) and variance \(= \dfrac{\sum fx^2}{\sum f} - \bar{x}^2\).
- Coded totals: To keep numbers small, data are often given as \(\sum (x - a)\) and \(\sum (x - a)^2\).
- Two data sets together: To combine two data sets, never average the means or the standard deviations.
- Mean: \(\bar{x} = \dfrac{\sum x}{n}\) (or \(\dfrac{\sum fx}{\sum f}\)).
- Variance: \(\dfrac{\sum x^2}{n} - \bar{x}^2\); standard deviation is its square root.
- Coded: mean \(= a + \dfrac{\sum (x - a)}{n}\); variance is the same as for the coded values.
- Reverse: \(\sum x = n\bar{x}\) and \(\sum x^2 = n(\sigma^2 + \bar{x}^2)\).
- Combine: add the totals, then use the formulas once.
One picture every idea returns to
\(\text{Variance} = \dfrac{\sum (x - \bar{x})^2}{n} = \dfrac{\sum x^2}{n} - \bar{x}^2\) and standard deviation \(= \sqrt{\text{variance}}\). Coding by subtracting \(a\) shifts the mean by \(a\) but leaves the standard deviation unchanged.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, The typical distance from the mean.. Standard deviation measures how far values typically sit from the mean. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Squared deviations | Each value's deviation is \(x - \bar{x}\). Model the worked example From raw data, both ways on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Frequency tables and grouped data | With frequencies, every sum gets an \(f\): \(\bar{x} = \dfrac{\sum fx}{\sum f}\) and variance \(= \dfrac{\sum fx^2}{\sum f} - \bar{x}^2\). Model the worked example Discrete frequency table on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: Coded totals | To keep numbers small, data are often given as \(\sum (x - a)\) and \(\sum (x - a)^2\). Model the worked example Mean and standard deviation from coded totals on the board, one line at a time. | Pairs |
| 23 to 30 min | Teach and check: Two data sets together | To combine two data sets, never average the means or the standard deviations. Model the worked example Combined mean and standard deviation on the board, one line at a time. Your turn: Ten values have mean 6 and standard deviation 2. Find \(\sum x\) and \(\sum x^2\). | Rally Coach |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 23 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Two data sets together, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Writing \(\dfrac{\sum x^2}{n} - \bar{x}\) or \(\dfrac{(\sum x)^2}{n}\). | It is the MEAN that is squared and subtracted: \(\dfrac{\sum x^2}{n} - \bar{x}^2\). |
| Using \((\sum fx)^2\) or \(\sum (fx)^2\) in place of \(\sum fx^2\). | Square the value \(x\) first, then multiply by its frequency. |
| Thinking \(\sum x^2 = \sum (x - a)^2 + na^2\). | The cross term \(-2a\sum x\) matters; expand the bracket properly or work via the variance. |
| The combined standard deviation (3.95) is bigger than both separate ones (2 and 3), because the two means are far apart. | Averaging the standard deviations misses this completely. |
Variance and standard deviation, including coded data: worksheet
6 questions, 23 marks. Written in the style of the papers; no past paper question is reproduced.
Find the mean and standard deviation of 3, 5, 6, 9, 12.
For 15 values, \(\sum x = 270\) and \(\sum x^2 = 5100\). Find the mean and standard deviation.
The scores \(x\) of 20 players are: \(x = 1\): 2, 2: 5, 3: 8, 4: 4, 5: 1. Calculate the mean and standard deviation.
The masses \(x\) grams of 30 packets satisfy \(\sum (x - 100) = -60\) and \(\sum (x - 100)^2 = 1470\).
(a) Find the mean and standard deviation of the masses. (b) Find \(\sum x\) and \(\sum x^2\).
20 girls have mean mark 52 and standard deviation 6. 30 boys have mean mark 47 and standard deviation 8. Find the mean and standard deviation of all 50 marks.
A factory measures the lengths \(x\) cm of rods from two machines. For 10 rods from machine A, \(\sum x = 120\) and \(\sum x^2 = 1530\). When 15 rods from machine B are included, the mean of all 25 rods is 14 cm and the standard deviation is 3 cm.
Find the mean and standard deviation of the lengths of the 15 rods from machine B.
Answer key and marking notes
- \(\bar{x} = 35 \div 5 = 7\); \(\sum x^2 = 9 + 25 + 36 + 81 + 144 = 295\).
- Variance \(= 59 - 49 = 10\), standard deviation \(= \sqrt{10} = 3.16\) (3 s.f.).
- \(\bar{x} = 270 \div 15 = 18\).
- Variance \(= 5100 \div 15 - 18^2 = 340 - 324 = 16\), standard deviation 4.
- \(\sum fx = 57\), mean \(= 2.85\).
- \(\sum fx^2 = 183\), variance \(= 9.15 - 8.1225 = 1.0275\).
- Standard deviation \(= 1.01\) (3 s.f.).
- (a) Mean \(= 100 + (-60 \div 30) = 98\) g.
- Variance \(= \dfrac{1470}{30} - (-2)^2 = 49 - 4 = 45\); standard deviation \(= \sqrt{45} = 6.71\) g (3 s.f.).
- (b) \(\sum x = -60 + 3000 = 2940\).
- \(\sum x^2 = 30(45 + 98^2) = 30 \times 9649 = 289\,470\).
- Girls: \(\sum x = 1040\), \(\sum x^2 = 20(36 + 2704) = 54\,800\).
- Boys: \(\sum x = 1410\), \(\sum x^2 = 30(64 + 2209) = 68\,190\).
- All: \(\sum x = 2450\), mean \(= 49\); \(\sum x^2 = 122\,990\).
- Variance \(= 2459.8 - 49^2 = 58.8\), standard deviation \(= 7.67\) (3 s.f.).
- All 25 rods: \(\sum x = 25 \times 14 = 350\), \(\sum x^2 = 25(3^2 + 14^2) = 5125\).
- Machine B: \(\sum x = 350 - 120 = 230\), \(\sum x^2 = 5125 - 1530 = 3595\).
- Mean \(= 230 \div 15 = 15.3\) cm (3 s.f.).
- Variance \(= \dfrac{3595}{15} - \left(\dfrac{230}{15}\right)^2 = 239.67 - 235.11 = 4.56\), standard deviation \(= 2.13\) cm (3 s.f.).
8 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Find the standard deviation of 5, 8, 9, 12, 16.
- A3.74
- B14
- C4.18
- D10.7
For 16 values, \(\sum x = 192\) and \(\sum x^2 = 2560\). Find the standard deviation.
- A4
- B16
- C12.2
- D12.6
A data set \(x\) has mean 8 and standard deviation 2. Each value is transformed by \(y = 3x - 5\). Find the mean and standard deviation of \(y\).
- AMean 19, standard deviation 6
- BMean 19, standard deviation 1
- CMean 24, standard deviation 6
- DMean 19, standard deviation 36
The numbers \(x\) of errors on 20 pages are: \(x = 0\): 4 pages, \(x = 1\): 7, \(x = 2\): 6, \(x = 3\): 3. Find the standard deviation of the number of errors.
- A0.970
- B0.94
- C3.43
- D1.22
For 20 values, \(\sum (x - 30) = -40\) and \(\sum (x - 30)^2 = 500\). Find the mean and standard deviation of \(x\).
- AMean 28, standard deviation 4.58
- BMean 32, standard deviation 4.58
- CMean 28, standard deviation 5
- DMean 28, standard deviation 21
For 10 values, \(\sum (x - 5) = 20\) and \(\sum (x - 5)^2 = 100\). Find \(\sum x^2\).
- A550
- B350
- C1050
- D800
Set A has 12 values with mean 20 and standard deviation 3. Set B has 8 values with mean 25 and standard deviation 4. Find the standard deviation of all 20 values together.
- A4.22
- B3.44
- C3.5
- D17.8
Ten values have mean 12 and standard deviation 3. One value, 18, is removed. Find the standard deviation of the remaining 9 values.
- A2.36
- B3
- C6.45
- D2.32
Quiz key
\(\bar{x} = 50 \div 5 = 10\). Deviations \(-5, -2, -1, 2, 6\); squares add to \(25 + 4 + 1 + 4 + 36 = 70\). Variance \(= 70 \div 5 = 14\), so the standard deviation is \(\sqrt{14} = 3.74\) (3 s.f.). The tempting 14 is the variance, with no square root. 4.18 divides by \(n - 1 = 4\), and 10.7 is \(\sqrt{\sum x^2 / n}\), forgetting to subtract \(\bar{x}^2\).
\(\bar{x} = 192 \div 16 = 12\). Variance \(= \dfrac{2560}{16} - 12^2 = 160 - 144 = 16\), so the standard deviation is 4. 16 is the variance. The tempting 12.2 is \(\sqrt{160 - 12}\), subtracting the mean instead of its square, and 12.6 is \(\sqrt{160}\), forgetting the mean altogether.
\(\bar{y} = 3 \times 8 - 5 = 19\). Multiplying by 3 multiplies every deviation by 3, and subtracting 5 changes no deviation, so the standard deviation of \(y\) is \(3 \times 2 = 6\). The tempting "1" applies the whole rule, \(3 \times 2 - 5\), to the standard deviation. 24 forgets to subtract 5 from the mean, and 36 is the variance of \(y\).
\(\sum fx = 0 + 7 + 12 + 9 = 28\), so \(\bar{x} = 1.4\). \(\sum fx^2 = 0 + 7 + 24 + 27 = 58\). Variance \(= \dfrac{58}{20} - 1.4^2 = 2.9 - 1.96 = 0.94\), standard deviation \(= \sqrt{0.94} = 0.970\) (3 s.f.). 0.94 is the variance. The tempting 3.43 uses \(\sum (fx)^2 = 274\) in place of \(\sum fx^2\), and 1.22 subtracts the mean 1.4 instead of \(1.4^2\).
Mean of \(x - 30\) is \(-40 \div 20 = -2\), so \(\bar{x} = 30 - 2 = 28\). Variance \(= \dfrac{500}{20} - (-2)^2 = 25 - 4 = 21\), so the standard deviation is \(\sqrt{21} = 4.58\) (3 s.f.). Mean 32 adds the 2 instead of subtracting it. The tempting 5 is \(\sqrt{25}\), forgetting to subtract the square of the coded mean, and 21 is the variance.
\(\sum x = 20 + 10 \times 5 = 70\). Then \(100 = \sum x^2 - 10(70) + 250\), so \(\sum x^2 = 100 + 700 - 250 = 550\). Check: variance \(= 55 - 7^2 = 6\), the same as the coded variance \(10 - 2^2 = 6\). The tempting 350 is \(100 + 10 \times 25\), which forgets the cross term \(-10\sum x\). 1050 adds the 250 instead of subtracting it, and 800 leaves the 250 out.
A: \(\sum x = 240\), \(\sum x^2 = 12(9 + 400) = 4908\). B: \(\sum x = 200\), \(\sum x^2 = 8(16 + 625) = 5128\). Combined: \(\sum x = 440\), mean 22, \(\sum x^2 = 10\,036\). Variance \(= 501.8 - 484 = 17.8\), standard deviation \(= 4.22\) (3 s.f.). The tempting 3.44 combines only the two variances, \(\sqrt{(12 \times 9 + 8 \times 16) \div 20}\), and ignores the gap between the means. 3.5 averages the standard deviations, and 17.8 is the variance.
\(\sum x = 120\), \(\sum x^2 = 10(9 + 144) = 1530\). Without 18: \(\sum x = 102\), \(\sum x^2 = 1530 - 324 = 1206\), \(n = 9\). Mean \(= 11.33\), variance \(= \dfrac{1206}{9} - \left(\dfrac{102}{9}\right)^2 = 134 - 128.44 = 5.56\), standard deviation \(= 2.36\) (3 s.f.). The tempting 3 assumes removing a value leaves the spread alone, but 18 was far from the mean. 6.45 forgets to take \(18^2\) out of \(\sum x^2\), and 2.32 removes the squared deviation \(6^2\) but still divides by 10.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.