PAPER · REASONING · PRACTICE

Data centres: weights, ordering and choosing a summary — Teacher reference

Calculate weighted means, medians and all tied modes; distinguish grouped estimates from raw-data answers.

40 minutes · Year 8, lower volume · text pp. 75–82 · textbook lessons 38–41

Textbook reference: Year 8, lower volume · Chapter 20: Analysing data · lessons 38–41

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.

Student classroom page 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 whether moving just one extreme value changes the mean and median equally. 8 min: In the existing outlier inquiry keep 4,5,6,7 fixed and record fifth values 0,8,40. Sort each set: means 4.4,6,12.4; medians 5,6,6. Record every value, not only the preferred summary. 5 min: Choose a summary for a stated purpose and explain its sensitivity. Include the extreme value when describing the data; do not remove it merely to improve a summary. 2 min: Individually explain which of these summaries changes most and why, without claiming one is always best.

Organise the key idea

A summary depends on the question and distribution. A frequency is a weight; a grouped midpoint replaces unknown individual observations only for an estimate.

Open the original chapter Open the interactive inquiry

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

Use the given values and weights to find the weighted mean. State the total weight.

Value	Frequency / weight
60	1
80	3

Use the stated conditions and the labelled scales.

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.

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=4; weighted mean= $75$ .

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 median of $\{1,2,2,5\}$ . Show the relevant ordering or calculation.

Ordered data: $1,2,2,5$ ; number of values = ____.
The central positions are ____ and ____; their values are ____ and ____.
Median = $\frac{\square+\square}{2}=\square$ .

There are four observations, so one middle position is not enough.
Choose the second and third values in the ordered list, rather than the smallest and largest.
Both central values are 2; calculate their average as (2+2)/2.

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $2$ .

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.

Prerequisite3

Find the mean of $\{2,4,6\}$ . 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 $4$ .

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

Find the median of $\{5,1,3\}$ . Show the relevant ordering or calculation.

Identify the observations and any frequencies or weights.
Sort the data and locate the central position or positions.
Check the original data; do not discard a value merely because it is extreme.

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $3$ .

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.

Independent practice5

The table shows marks and their frequencies. Find the mean mark and the number of learners.

Value	Frequency / weight
60	1
80	2
100	1

Use the stated conditions and the labelled scales.

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.

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=4; weighted mean= $80$ .

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 mode or modes of $\{2,3,3,4,4\}$ . Show the relevant ordering or calculation.

Identify the observations and any frequencies or weights.
Compare all frequencies and retain every tied highest value.
Check the original data; do not discard a value merely because it is extreme.

Count all occurrences and retain every value tied for greatest frequency.
The listed modes all attain the highest repeated frequency.

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

Travel times are 1,2,3,4,100 minutes. Calculate the mean and median; recommend a typical-time summary while still reporting the long journey.

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 total is 110 over 5 observations, so the mean is 22.
The middle ordered value is 3. A small central summary should not hide the 100-minute journey.

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

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[10,20)	2
[20,30)	4
[30,40)	2

Use the stated conditions and the labelled scales.

Identify the observations and any frequencies or weights.
Form a weighted total and divide by the total weight.
Group midpoints give an estimate; original observations are not known exactly.

Group midpoints are $\{15,25,35\}$ .
Weight each midpoint by its group frequency and divide by the total count.

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 mode or modes of $\{3,1,3,5\}$ . Show the relevant ordering or calculation.

Identify the observations and any frequencies or weights.
Compare all frequencies and retain every tied highest value.
Check the original data; do not discard a value merely because it is extreme.

Count all occurrences and retain every value tied for greatest frequency.
The listed modes all attain the highest repeated frequency.

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 median of $\{2,6,8,10\}$ . Show the relevant ordering or calculation.

Identify the observations and any frequencies or weights.
Sort the data and locate the central position or positions.
Check the original data; do not discard a value merely because it is extreme.

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $7$ .

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 frequencies include zeros. Find the mean of the observations actually present.

Value	Frequency / weight
10	0
20	2
30	0

Use the stated conditions and the labelled scales.

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.

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=2; weighted mean= $20$ .

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

Two assessment components score 80 and 90 with weights 3 and 2. Find the weighted score; the weights are not already percentages.

Value	Frequency / weight
80	3
90	2

Use the stated conditions and the labelled scales.

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.

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=5; weighted mean= $84$ .

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

For 2,5,8, report the mode using the convention that a mode must be a repeated most-frequent value.

Identify the observations and any frequencies or weights.
Compare all frequencies and retain every tied highest value.
Check the original data; do not discard a value merely because it is extreme.

Count all occurrences and retain every value tied for greatest frequency.
No value is repeated; report no mode under the stated convention.

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

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[0,10)	1
[10,30)	2
[30,60)	1

Use the stated conditions and the labelled scales.

Identify the observations and any frequencies or weights.
Form a weighted total and divide by the total weight.
Group midpoints give an estimate; original observations are not known exactly.

Group midpoints are $\{5,20,45\}$ .
Weight each midpoint by its group frequency and divide by the total count.

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

One group of 3 learners has mean 10 and another group of 2 has mean 20. Find the combined mean and explain why simply averaging 10 and 20 is wrong.

Value	Frequency / weight
10	3
20	2

Use the stated conditions and the labelled scales.

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 group totals are 3×10=30 and 2×20=40.
Combined mean=(30+40)/5=14; the group sizes differ.

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

Starting with 1,2,3,4,10, replace only the last value by 1000. Find the new median and explain why it is unchanged.

Identify the observations and any frequencies or weights.
Sort the data and locate the central position or positions.
Check the original data; do not discard a value merely because it is extreme.

The central ordered position is still the third observation.
That observation is 3, although the mean changes greatly.

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

Fractions, ordering numbers and reading frequency tables.

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.

All reference answers

Worked example1

Use the given values and weights to find the weighted mean. State the total weight.

Value	Frequency / weight
60	1
80	3

Use the stated conditions and the labelled scales.

Reference answer

$75$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=4; weighted mean= $75$ .

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 median of $\{1,2,2,5\}$ . Show the relevant ordering or calculation.

Ordered data: $1,2,2,5$ ; number of values = ____.
The central positions are ____ and ____; their values are ____ and ____.
Median = $\frac{\square+\square}{2}=\square$ .

Reference answer

$2$

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $2$ .

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 $\{2,4,6\}$ . Show the relevant ordering or calculation.

Reference answer

$4$

Add every observation, then divide by the number of observations.
The result is $4$ .

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

Prerequisite4

Find the median of $\{5,1,3\}$ . Show the relevant ordering or calculation.

Reference answer

$3$

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $3$ .

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

Independent practice5

The table shows marks and their frequencies. Find the mean mark and the number of learners.

Value	Frequency / weight
60	1
80	2
100	1

Use the stated conditions and the labelled scales.

Reference answer

$80$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=4; weighted mean= $80$ .

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 mode or modes of $\{2,3,3,4,4\}$ . Show the relevant ordering or calculation.

Reference answer

Modes: $\{3,4\}$ .

Count all occurrences and retain every value tied for greatest frequency.
The listed modes all attain the highest repeated frequency.

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

Independent practice7

Travel times are 1,2,3,4,100 minutes. Calculate the mean and median; recommend a typical-time summary while still reporting the long journey.

Reference answer

Mean 22, median 3 minutes. The median can describe the central typical time, while the maximum 100 minutes must remain visible; another justified summary is accepted.

The total is 110 over 5 observations, so the mean is 22.
The middle ordered value is 3. A small central summary should not hide the 100-minute journey.

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

Independent practice8

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[10,20)	2
[20,30)	4
[30,40)	2

Use the stated conditions and the labelled scales.

Reference answer

Estimated mean $25$ ; the observations inside each group are unknown.

Group midpoints are $\{15,25,35\}$ .
Weight each midpoint by its group frequency and divide by the total count.

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 mode or modes of $\{3,1,3,5\}$ . Show the relevant ordering or calculation.

Reference answer

Modes: $\{3\}$ .

Count all occurrences and retain every value tied for greatest frequency.
The listed modes all attain the highest repeated frequency.

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 median of $\{2,6,8,10\}$ . Show the relevant ordering or calculation.

Reference answer

$7$

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $7$ .

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 frequencies include zeros. Find the mean of the observations actually present.

Value	Frequency / weight
10	0
20	2
30	0

Use the stated conditions and the labelled scales.

Reference answer

$20$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=2; weighted mean= $20$ .

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

Further practice12

Two assessment components score 80 and 90 with weights 3 and 2. Find the weighted score; the weights are not already percentages.

Value	Frequency / weight
80	3
90	2

Use the stated conditions and the labelled scales.

Reference answer

$84$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=5; weighted mean= $84$ .

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

Further practice13

For 2,5,8, report the mode using the convention that a mode must be a repeated most-frequent value.

Reference answer

No mode: every value occurs only once under the repeated-value convention used here.

Count all occurrences and retain every value tied for greatest frequency.
No value is repeated; report no mode under the stated convention.

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

Further practice14

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[0,10)	1
[10,30)	2
[30,60)	1

Use the stated conditions and the labelled scales.

Reference answer

Estimated mean $45/2$ ; the observations inside each group are unknown.

Group midpoints are $\{5,20,45\}$ .
Weight each midpoint by its group frequency and divide by the total count.

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

Extension15

One group of 3 learners has mean 10 and another group of 2 has mean 20. Find the combined mean and explain why simply averaging 10 and 20 is wrong.

Value	Frequency / weight
10	3
20	2

Use the stated conditions and the labelled scales.

Reference answer

$14$

The group totals are 3×10=30 and 2×20=40.
Combined mean=(30+40)/5=14; the group sizes differ.

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

Extension16

Starting with 1,2,3,4,10, replace only the last value by 1000. Find the new median and explain why it is unchanged.

Reference answer

$3$

The central ordered position is still the third observation.
That observation is 3, although the mean changes greatly.

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

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. 75–82 · textbook lessons 38–41

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

Data centres: weights, ordering and choosing a summary

Year 8, lower volume · text pp. 75–82 · textbook lessons 38–41 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Use the given values and weights to find the weighted mean. State the total weight.

Value	Frequency / weight
60	1
80	3

Use the stated conditions and the labelled scales.

Guided attempt2

Find the median of $\{1,2,2,5\}$ . Show the relevant ordering or calculation.

Ordered data: $1,2,2,5$ ; number of values = ____.
The central positions are ____ and ____; their values are ____ and ____.
Median = $\frac{\square+\square}{2}=\square$ .

Prerequisite3

Find the mean of $\{2,4,6\}$ . Show the relevant ordering or calculation.

Prerequisite4

Find the median of $\{5,1,3\}$ . Show the relevant ordering or calculation.

Data centres: weights, ordering and choosing a summary

Year 8, lower volume · text pp. 75–82 · textbook lessons 38–41 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

The table shows marks and their frequencies. Find the mean mark and the number of learners.

Value	Frequency / weight
60	1
80	2
100	1

Use the stated conditions and the labelled scales.

Independent practice6

Find the mode or modes of $\{2,3,3,4,4\}$ . Show the relevant ordering or calculation.

Independent practice7

Travel times are 1,2,3,4,100 minutes. Calculate the mean and median; recommend a typical-time summary while still reporting the long journey.

Independent practice8

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[10,20)	2
[20,30)	4
[30,40)	2

Use the stated conditions and the labelled scales.

Exit question9

Find the mode or modes of $\{3,1,3,5\}$ . Show the relevant ordering or calculation.

Exit question10

Find the median of $\{2,6,8,10\}$ . Show the relevant ordering or calculation.

Data centres: weights, ordering and choosing a summary — Further practice

Year 8, lower volume · text pp. 75–82 · textbook lessons 38–41

Further practice11

The frequencies include zeros. Find the mean of the observations actually present.

Value	Frequency / weight
10	0
20	2
30	0

Use the stated conditions and the labelled scales.

Further practice12

Two assessment components score 80 and 90 with weights 3 and 2. Find the weighted score; the weights are not already percentages.

Value	Frequency / weight
80	3
90	2

Use the stated conditions and the labelled scales.

Further practice13

For 2,5,8, report the mode using the convention that a mode must be a repeated most-frequent value.

Further practice14

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[0,10)	1
[10,30)	2
[30,60)	1

Use the stated conditions and the labelled scales.

Extension15

One group of 3 learners has mean 10 and another group of 2 has mean 20. Find the combined mean and explain why simply averaging 10 and 20 is wrong.

Value	Frequency / weight
10	3
20	2

Use the stated conditions and the labelled scales.

Extension16

Starting with 1,2,3,4,10, replace only the last value by 1000. Find the new median and explain why it is unchanged.

Data centres: weights, ordering and choosing a summary — Teacher answers

Year 8, lower volume · text pp. 75–82 · textbook lessons 38–41

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: Fractions, ordering numbers and reading frequency tables.

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

3′ Predict whether moving just one extreme value changes the mean and median equally.
8′ In the existing outlier inquiry keep 4,5,6,7 fixed and record fifth values 0,8,40. Sort each set: means 4.4,6,12.4; medians 5,6,6. Record every value, not only the preferred summary.
5′ Choose a summary for a stated purpose and explain its sensitivity. Include the extreme value when describing the data; do not remove it merely to improve a summary.
2′ Individually explain which of these summaries changes most and why, without claiming one is always best.

Worked example1

Use the given values and weights to find the weighted mean. State the total weight.

Value	Frequency / weight
60	1
80	3

Use the stated conditions and the labelled scales.

Reference answer

$75$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=4; weighted mean= $75$ .

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 median of $\{1,2,2,5\}$ . Show the relevant ordering or calculation.

Ordered data: $1,2,2,5$ ; number of values = ____.
The central positions are ____ and ____; their values are ____ and ____.
Median = $\frac{\square+\square}{2}=\square$ .

Reference answer

$2$

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $2$ .

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 $\{2,4,6\}$ . Show the relevant ordering or calculation.

Reference answer

$4$

Add every observation, then divide by the number of observations.
The result is $4$ .

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

Prerequisite4

Find the median of $\{5,1,3\}$ . Show the relevant ordering or calculation.

Reference answer

$3$

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $3$ .

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

Independent practice5

The table shows marks and their frequencies. Find the mean mark and the number of learners.

Value	Frequency / weight
60	1
80	2
100	1

Use the stated conditions and the labelled scales.

Reference answer

$80$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=4; weighted mean= $80$ .

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 mode or modes of $\{2,3,3,4,4\}$ . Show the relevant ordering or calculation.

Reference answer

Modes: $\{3,4\}$ .

Count all occurrences and retain every value tied for greatest frequency.
The listed modes all attain the highest repeated frequency.

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

Independent practice7

Travel times are 1,2,3,4,100 minutes. Calculate the mean and median; recommend a typical-time summary while still reporting the long journey.

Reference answer

Mean 22, median 3 minutes. The median can describe the central typical time, while the maximum 100 minutes must remain visible; another justified summary is accepted.

The total is 110 over 5 observations, so the mean is 22.
The middle ordered value is 3. A small central summary should not hide the 100-minute journey.

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

Independent practice8

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[10,20)	2
[20,30)	4
[30,40)	2

Use the stated conditions and the labelled scales.

Reference answer

Estimated mean $25$ ; the observations inside each group are unknown.

Group midpoints are $\{15,25,35\}$ .
Weight each midpoint by its group frequency and divide by the total count.

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 mode or modes of $\{3,1,3,5\}$ . Show the relevant ordering or calculation.

Reference answer

Modes: $\{3\}$ .

Count all occurrences and retain every value tied for greatest frequency.
The listed modes all attain the highest repeated frequency.

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 median of $\{2,6,8,10\}$ . Show the relevant ordering or calculation.

Reference answer

$7$

Sort the original observations from smallest to largest.
Use the middle observation, or the mean of the two middle observations, to obtain $7$ .

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 frequencies include zeros. Find the mean of the observations actually present.

Value	Frequency / weight
10	0
20	2
30	0

Use the stated conditions and the labelled scales.

Reference answer

$20$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=2; weighted mean= $20$ .

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

Further practice12

Two assessment components score 80 and 90 with weights 3 and 2. Find the weighted score; the weights are not already percentages.

Value	Frequency / weight
80	3
90	2

Use the stated conditions and the labelled scales.

Reference answer

$84$

Multiply each value by its weight, add all products, and divide by the sum of the weights.
Total weight=5; weighted mean= $84$ .

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

Further practice13

For 2,5,8, report the mode using the convention that a mode must be a repeated most-frequent value.

Reference answer

No mode: every value occurs only once under the repeated-value convention used here.

Count all occurrences and retain every value tied for greatest frequency.
No value is repeated; report no mode under the stated convention.

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

Further practice14

Only the grouped counts are known. Use group midpoints to estimate the mean, and explain why the answer is not necessarily the exact raw-data mean.

Group (lower inclusive, upper exclusive)	Frequency
[0,10)	1
[10,30)	2
[30,60)	1

Use the stated conditions and the labelled scales.

Reference answer

Estimated mean $45/2$ ; the observations inside each group are unknown.

Group midpoints are $\{5,20,45\}$ .
Weight each midpoint by its group frequency and divide by the total count.

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

Extension15

One group of 3 learners has mean 10 and another group of 2 has mean 20. Find the combined mean and explain why simply averaging 10 and 20 is wrong.

Value	Frequency / weight
10	3
20	2

Use the stated conditions and the labelled scales.

Reference answer

$14$

The group totals are 3×10=30 and 2×20=40.
Combined mean=(30+40)/5=14; the group sizes differ.

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

Extension16

Starting with 1,2,3,4,10, replace only the last value by 1000. Find the new median and explain why it is unchanged.

Reference answer

$3$

The central ordered position is still the third observation.
That observation is 3, although the mean changes greatly.

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