PAPER · REASONING · PRACTICE

Variance: calculation, stability and transformations

Use the textbook’s n-denominator definition, compare data with the same centre and reason about shifts and scales.

40 minutes · Year 8, lower volume · text pp. 83–84 · textbook lesson 42

Textbook reference: Year 8, lower volume · Chapter 20: Analysing data · lessons 42–42

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

3 min: Predict which is more spread out: A={4,5,6} or B={1,5,9}, both with mean 5. 8 min: On paper compute every deviation and squared deviation. Divide by n=3: variances $\frac23$ and $\frac{32}{3}$ . Add 3 to every value of A, then separately double every original value of A. The existing outlier engine does not calculate variance. 5 min: Explain unchanged variance after adding 3 and variance $\frac83$ after doubling, four times the original. State squared units and retain the course convention of dividing by n. 2 min: Individually state how multiplying every value by −2 changes the variance, with a squared-deviation reason.

Organise the key idea

This descriptive variance is $s^{2}=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^{2}$ . Compare variability only in a stated context and unit.

Open the original chapter

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Find the descriptive variance of $\{2,4,4,6\}$ , using the number of observations as the denominator. Show the mean first.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Mean=(2+4+4+6)/4=4.
Squared deviations are 4,0,0,4; their sum is 8.
Variance=8/4=2.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

Find the descriptive variance of $\{1,3,5\}$ , using the number of observations as the denominator. Show the mean first.

Mean = $\frac{1+3+5}{3}=\square$ .
Deviations: $1-\square,\quad3-\square,\quad5-\square$ .
Squared total = $(-2)^{2}+0^{2}+2^{2}=\square$ ; variance = $\frac{\square}{3}$ .

The three values sum to 9, so their mean is 3.
Subtract 3 from each value: −2,0,2. Their squares are 4,0,4.
Add the squares to get 8, then divide by 3 observations, following the n-denominator definition.

Mean=3; deviations are −2,0,2.
The square total is 8, divided by 3 observations.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Find the mean of $\{2,4\}$ . Show the relevant ordering or calculation.

Identify the observations and any frequencies or weights.
Form a weighted total and divide by the total weight.
Check the original data; do not discard a value merely because it is extreme.

Add every observation, then divide by the number of observations.
The result is $3$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

The reference mean is 5. Find the sum of squared deviations for 3,5,7.

Subtract the mean before squaring.
A negative deviation has a positive square.
Sum 4,0,4.

The deviations are −2,0,2 and their squares are 4,0,4.
Add the squares: 8.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Find the descriptive variance of $\{1,1,3,3\}$ , using the number of observations as the denominator. Show the mean first.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $1$ ; its unit is the square of the data unit.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Find the descriptive variance of $\{2,2,2\}$ , using the number of observations as the denominator. Show the mean first.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $0$ ; its unit is the square of the data unit.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Group A is 4,5,6 and group B is 3,5,7. Check both means, compare both n-denominator variances, and decide which set is more stable on the same scale.

Identify the observations and any frequencies or weights.
Form a weighted total and divide by the total weight.
Check the original data; do not discard a value merely because it is extreme.

The two means agree, so compare spread rather than centre.
Square totals are 2 and 8, and both sets contain 3 values.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

Find the descriptive variance of $\{0,2,4\}$ , using the number of observations as the denominator. Show the mean first.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $8/3$ ; its unit is the square of the data unit.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Find the descriptive variance of $\{1,3\}$ , using the number of observations as the denominator. Show the mean first.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $1$ ; its unit is the square of the data unit.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Find the descriptive variance of $\{5,5,5,5\}$ , using the number of observations as the denominator. Show the mean first.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $0$ ; its unit is the square of the data unit.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

The data 2,4,4,6 have variance 2. Add 3 to every value. Find the new variance and explain the unchanged deviations.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

The mean also increases by 3, from 4 to 7.
Every deviation remains −2,0,0,2; the variance remains 2.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

The data 2,4,4,6 have variance 2. Multiply every value by 2. Find the new variance and explain the scale factor.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Each deviation doubles, so each squared deviation is multiplied by 4.
The variance becomes 4×2=8.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

Find the descriptive variance of $\{-2,0,2\}$ , using the number of observations as the denominator. Show the mean first.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $8/3$ ; its unit is the square of the data unit.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Values 1 and 3 each occur twice. Find the mean and n-denominator variance, counting all four observations.

Value	Frequency / weight
1	2
3	2

Use the stated conditions and the labelled scales.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

The data are 1,1,3,3; the mean is 2.
The squared-deviation total is 4 over 4 observations.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

The data 1,1,5,5 have mean 3. Find their variance and explain why knowing the mean alone does not determine spread.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

Every deviation is −2 or 2, so every squared deviation is 4.
Variance=16/4=4; data 3,3,3,3 have the same mean but zero variance.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

For this chapter’s descriptive variance of 2,4, a learner divides the squared-deviation total 2 by n−1 and reports 2. Correct the calculation under the stated n-denominator definition.

Identify the observations and any frequencies or weights.
Use the mean, square each deviation, sum, and divide by the number of observations.
Check the original data; do not discard a value merely because it is extreme.

The mean is 3 and the square total is 2.
Here n=2, so the descriptive variance is 2/2=1. A different statistical estimator uses another definition.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Preparation and assessment notes

Prerequisites

Mean, signed differences, squares and fractions.

Look for in students’ work

Use the two prerequisite checks to choose support. Project the worked and guided tasks, then collect the eight paper core questions. Follow-ups and extensions are optional; accept other complete valid methods. This page does not save learners’ paper answers.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original chapter-core practice after the cited lower-volume textbook lessons. References identify skills; textbook exercises are not reproduced.

大湾区初中数学三维课堂（八年级下册）正文.pdf

Year 8, lower volume · text pp. 83–84 · textbook lesson 42

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Variance: calculation, stability and transformations

Year 8, lower volume · text pp. 83–84 · textbook lesson 42 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Find the descriptive variance of $\{2,4,4,6\}$ , using the number of observations as the denominator. Show the mean first.

Guided attempt2

Find the descriptive variance of $\{1,3,5\}$ , using the number of observations as the denominator. Show the mean first.

Mean = $\frac{1+3+5}{3}=\square$ .
Deviations: $1-\square,\quad3-\square,\quad5-\square$ .
Squared total = $(-2)^{2}+0^{2}+2^{2}=\square$ ; variance = $\frac{\square}{3}$ .

Prerequisite3

Find the mean of $\{2,4\}$ . Show the relevant ordering or calculation.

Prerequisite4

The reference mean is 5. Find the sum of squared deviations for 3,5,7.

Variance: calculation, stability and transformations

Year 8, lower volume · text pp. 83–84 · textbook lesson 42 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Find the descriptive variance of $\{1,1,3,3\}$ , using the number of observations as the denominator. Show the mean first.

Independent practice6

Find the descriptive variance of $\{2,2,2\}$ , using the number of observations as the denominator. Show the mean first.

Independent practice7

Group A is 4,5,6 and group B is 3,5,7. Check both means, compare both n-denominator variances, and decide which set is more stable on the same scale.

Independent practice8

Find the descriptive variance of $\{0,2,4\}$ , using the number of observations as the denominator. Show the mean first.

Exit question9

Find the descriptive variance of $\{1,3\}$ , using the number of observations as the denominator. Show the mean first.

Exit question10

Find the descriptive variance of $\{5,5,5,5\}$ , using the number of observations as the denominator. Show the mean first.

Variance: calculation, stability and transformations — Further practice

Year 8, lower volume · text pp. 83–84 · textbook lesson 42

Further practice11

The data 2,4,4,6 have variance 2. Add 3 to every value. Find the new variance and explain the unchanged deviations.

Further practice12

The data 2,4,4,6 have variance 2. Multiply every value by 2. Find the new variance and explain the scale factor.

Further practice13

Find the descriptive variance of $\{-2,0,2\}$ , using the number of observations as the denominator. Show the mean first.

Further practice14

Values 1 and 3 each occur twice. Find the mean and n-denominator variance, counting all four observations.

Value	Frequency / weight
1	2
3	2

Use the stated conditions and the labelled scales.

Extension15

The data 1,1,5,5 have mean 3. Find their variance and explain why knowing the mean alone does not determine spread.

Extension16

For this chapter’s descriptive variance of 2,4, a learner divides the squared-deviation total 2 by n−1 and reports 2. Correct the calculation under the stated n-denominator definition.

Variance: calculation, stability and transformations — Teacher answers

Year 8, lower volume · text pp. 83–84 · textbook lesson 42

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Mean, signed differences, squares and fractions.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

3′ Predict which is more spread out: A={4,5,6} or B={1,5,9}, both with mean 5.
8′ On paper compute every deviation and squared deviation. Divide by n=3: variances $\frac23$ and $\frac{32}{3}$ . Add 3 to every value of A, then separately double every original value of A. The existing outlier engine does not calculate variance.
5′ Explain unchanged variance after adding 3 and variance $\frac83$ after doubling, four times the original. State squared units and retain the course convention of dividing by n.
2′ Individually state how multiplying every value by −2 changes the variance, with a squared-deviation reason.

Worked example1

Find the descriptive variance of $\{2,4,4,6\}$ , using the number of observations as the denominator. Show the mean first.

Reference answer

$s^{2}=2$

Mean=(2+4+4+6)/4=4.
Squared deviations are 4,0,0,4; their sum is 8.
Variance=8/4=2.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

Find the descriptive variance of $\{1,3,5\}$ , using the number of observations as the denominator. Show the mean first.

Mean = $\frac{1+3+5}{3}=\square$ .
Deviations: $1-\square,\quad3-\square,\quad5-\square$ .
Squared total = $(-2)^{2}+0^{2}+2^{2}=\square$ ; variance = $\frac{\square}{3}$ .

Reference answer

$s^{2}=\frac{8}{3}$

Mean=3; deviations are −2,0,2.
The square total is 8, divided by 3 observations.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Find the mean of $\{2,4\}$ . Show the relevant ordering or calculation.

Reference answer

$3$

Add every observation, then divide by the number of observations.
The result is $3$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

The reference mean is 5. Find the sum of squared deviations for 3,5,7.

Reference answer

8

The deviations are −2,0,2 and their squares are 4,0,4.
Add the squares: 8.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Find the descriptive variance of $\{1,1,3,3\}$ , using the number of observations as the denominator. Show the mean first.

Reference answer

$s^{2}=1$

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $1$ ; its unit is the square of the data unit.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Find the descriptive variance of $\{2,2,2\}$ , using the number of observations as the denominator. Show the mean first.

Reference answer

$s^{2}=0$

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $0$ ; its unit is the square of the data unit.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Group A is 4,5,6 and group B is 3,5,7. Check both means, compare both n-denominator variances, and decide which set is more stable on the same scale.

Reference answer

Both means are 5. A has variance 2/3, B has 8/3, so A is less variable.

The two means agree, so compare spread rather than centre.
Square totals are 2 and 8, and both sets contain 3 values.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

Find the descriptive variance of $\{0,2,4\}$ , using the number of observations as the denominator. Show the mean first.

Reference answer

$s^{2}=\frac{8}{3}$

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $8/3$ ; its unit is the square of the data unit.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Find the descriptive variance of $\{1,3\}$ , using the number of observations as the denominator. Show the mean first.

Reference answer

$s^{2}=1$

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $1$ ; its unit is the square of the data unit.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Find the descriptive variance of $\{5,5,5,5\}$ , using the number of observations as the denominator. Show the mean first.

Reference answer

$s^{2}=0$

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $0$ ; its unit is the square of the data unit.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

The data 2,4,4,6 have variance 2. Add 3 to every value. Find the new variance and explain the unchanged deviations.

Reference answer

$s^{2}=2$

The mean also increases by 3, from 4 to 7.
Every deviation remains −2,0,0,2; the variance remains 2.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

The data 2,4,4,6 have variance 2. Multiply every value by 2. Find the new variance and explain the scale factor.

Reference answer

$s^{2}=8$

Each deviation doubles, so each squared deviation is multiplied by 4.
The variance becomes 4×2=8.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

Find the descriptive variance of $\{-2,0,2\}$ , using the number of observations as the denominator. Show the mean first.

Reference answer

$s^{2}=\frac{8}{3}$

Compute the mean from every observation.
Square each deviation from the mean, add, and divide by n.
The variance is $8/3$ ; its unit is the square of the data unit.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Values 1 and 3 each occur twice. Find the mean and n-denominator variance, counting all four observations.

Value	Frequency / weight
1	2
3	2

Use the stated conditions and the labelled scales.

Reference answer

Mean 2 and variance 1.

The data are 1,1,3,3; the mean is 2.
The squared-deviation total is 4 over 4 observations.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

The data 1,1,5,5 have mean 3. Find their variance and explain why knowing the mean alone does not determine spread.

Reference answer

$s^{2}=4$

Every deviation is −2 or 2, so every squared deviation is 4.
Variance=16/4=4; data 3,3,3,3 have the same mean but zero variance.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

For this chapter’s descriptive variance of 2,4, a learner divides the squared-deviation total 2 by n−1 and reports 2. Correct the calculation under the stated n-denominator definition.

Reference answer

$s^{2}=1$

The mean is 3 and the square total is 2.
Here n=2, so the descriptive variance is 2/2=1. A different statistical estimator uses another definition.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.