At a glance. One lesson of about 60 minutes. Students use the frequency form and the summary form, estimate σ from a grouped table, prove that the forms agree, and work backwards from a given σ or variance.
Calculator: GDC optional. Totals can be built by hand for small tables. The optional section shows value and frequency lists and which standard deviation to read.
What students will learn
Know
- σ = √(Σf(x − μ)² ÷ n) with n = Σf.
- σ = √(Σfx² ÷ n − μ²).
- From grouped data, mid-interval values give an estimate.
Understand
- Σ(x − μ)² = Σx² − nμ², so the two forms are one measure.
- The variance is never negative, and is 0 only when every value equals the mean.
- The summary form is the practical choice when the mean is a long decimal.
Do
- Calculate σ from a frequency table by both forms.
- Estimate σ from grouped data.
- Rearrange the summary form to find a missing total or frequency.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: both forms, grouped data, and working backwards (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if, deducing from the summary form that the variance cannot be negative.
- Criterion C. Communicating: setting out columns x, f, fx, fx² with totals, and following the proof line by line (worked example 3).
- Criterion D. Applying mathematics in real-life contexts: the library tally, choosing the form that suits the data.
The one thing to do with the figure
Make a prediction with the form switch. On table 2, with the deviation form showing, ask: “When I flip to the summary form, will the variance change?” Let a student explain before pressing. The point is that the readout does not change: only the working does.
Then ask why the summary form is gentler on table 2, whose mean is 1.4: the squares of whole numbers are whole, and only one decimal enters at the end.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the library tally. Collect predictions on whether two forms can disagree. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples (including the proof); pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the units of the two terms. |
| 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 | 3; x² − 10x + 25; 4.5. |
|---|---|
| Hook | No: the two forms are the same quantity, so they always give the same variance. |
| Worked examples | σ = 0.9 by deviations; variance 0.81 by the summary form; the proof gives Σx² − nμ². |
| Faded example | 17.1. |
| 1. Setting out the summary form | B. A subtracts μ, not μ²; C and D divide or root in the wrong place. |
| 2. Standard deviation from a frequency table | 0.7. √0.49. |
| 3. Summary form from totals | 2. √(20 − 16). |
| 4. Standard deviation of grouped data | 7. √49. |
| 5. Working back to a sum of squares | 1160. 20 × (9 + 49). |
| 6. Using the expanded form | 324. 30 + 6 × 49. |
| 7. Missing frequency from a variance | 2. n = 10. |
| 8. Effect of doubling every frequency | D. A doubles the values instead; B doubles only the sum of squares; C doubles only n. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Subtracted μ instead of μ², or divided or rooted in the wrong place. | Write the mean of the squares and the square of the mean on separate lines. |
| 0.5, 0.8 or 1.3 in question 2 | Gave the variance, ignored frequencies, or divided by the number of rows. | Make the column f and total it to find n. |
| 4 or 0.47 in question 3 | Gave the variance, or rooted before subtracting. | Root is the last step. |
| 49 or 8.2 in question 4 | Gave the variance, or left out frequencies. | Add the columns f, fx and fx². |
| 58 or 320 in question 5 | Stopped at the mean of the squares, or added μ. | Rearrange: Σfx² ÷ n = σ² + μ². |
| 72 or 79 in question 6 | Added unlike totals, or used μ² once. | Show the expansion with nμ². |
| 10 or 6 in question 7 | Gave n, or removed only one 4. | Write n = 4 + f + 4. |
| A, B or C in question 8 | Treated more data as more spread, or doubled only the top or only the bottom of the variance. | Write Σf(x − μ)² ÷ n and double both parts. |
Questions worth asking
- Why is the summary form a risk with a calculator that rounds the mean? The two terms are nearly equal when the spread is small, and subtracting them can lose accuracy.
- Why can the mean of the squares never be less than the square of the mean? Their difference is the variance, which is a mean of squares and so never negative.
- What does a grouped σ estimate depend on? The mid-interval values; the true spread inside each class is unknown.
- If one more value equal to the mean is added to a table, what happens to σ? It falls a little: Σf(x − μ)² is unchanged, because the new deviation is 0, but n grows by 1, so the variance is smaller.
Differentiation
A way in for anyone stuck
- Give a table template with columns x, f, fx and fx², and a totals row.
- Provide the mean in advance so students can focus on the squares.
- Rehearse expanding (x − μ)² with a numerical example before the proof.
Stretch
- Show that if every x in a table is multiplied by 3, with the frequencies unchanged, the standard deviation is multiplied by 3.
- Show that the variance of the whole numbers 1 to n is (n² − 1) ÷ 12.
Students learning in English as an additional language
“Frequency form” and “summary form” are two ways of writing the same formula. Say “sigma f x squared” for Σfx² and “mu squared” for μ². “Estimate” means a value that is probably close, not exact.
| Word | Plain meaning | Example |
|---|---|---|
| frequency form | the standard deviation with each squared deviation counted f times | σ² = Σf(x − μ)² ÷ n |
| summary form | a second way to write the variance | it uses Σfx and Σfx², not deviations |
| mean of the squares | Σfx² divided by n | (1 + 9) ÷ 2 = 5 |
| square of the mean | the mean multiplied by itself | the mean 1.5 gives 2.25 |
| grouped table | a table of classes with a frequency each | 0 ≤ x < 10: 4 values |
| total frequency | the sum of all the frequencies, n | 3 + 5 + 2 = 10 |
Sentence starters for explaining aloud or in writing:
- I use the summary form because …
- Σfx² is … and μ² is …
- The variance is the mean of the squares minus …
Two things not to say
“Use whichever form gives a different answer to check.” Both forms give the same variance; a difference is an arithmetic slip.
“fx² means (fx) all squared.” Square x, then multiply by f.
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.
- EvaluateFind the variance of 1, 1, 3, 5 using Σx² ÷ n − μ².
2.75. The mean is 2.5 and Σx² = 36, so Σx² ÷ n = 9 and 9 − 6.25 = 2.75. - Totalsn = 5, Σfx = 15 and Σfx² = 60. Find σ.
1.73. The mean is 3, 60 ÷ 5 = 12, 12 − 9 = 3, and √3 = 1.73.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextParcels at a post office in Cebu: classes 0 to under 2 kg (f = 6), 2 to under 4 (f = 10), 4 to under 6 (f = 4). Estimate σ to one decimal place.
1.4. Mid-interval values 1, 3, 5; Σfx = 56, μ = 2.8; Σfx² = 6 + 90 + 100 = 196; 9.8 − 7.84 = 1.96; σ = 1.4. - Earlier work (Lesson 3)Find σ for 6, 8, 10, 12 (population).
2.24. The mean is 9, the squares are 9, 1, 1, 9, the variance is 5 and √5 = 2.24.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA table has n = 8, mean 5 and Σfx² = 190. Is that possible? Say why.
No. Σfx² ÷ n = 23.75 is less than μ² = 25, which would make the variance negative. - ProofShow that Σf(x − μ)² = Σfx² − nμ² for a frequency table.
Expand: Σfx² − 2μΣfx + μ²Σf = Σfx² − 2nμ² + nμ² = Σfx² − nμ².
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.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.