Class 8 · English Version · Mathematics
2026 Class 8 Mathematics · English Version · Chapter 11 · Chapter companion part
Organise observations without losing records, choose non-overlapping groups and explain what each count represents.
Class 8 English Version: part A of the Chapter 11 companion. Eight teaching sections and 30 original written tasks. Selected companion coverage, not complete textbook coverage; use all four parts. Written interpretation, chart drawing and complete working need guardian or teacher review.
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Data are recorded observations. Statistics is the practice of organising, describing and learning from them. Our invented ten reading-session lengths are 12, 14, 14, 16, 18, 18, 18, 20, 22 and 28 minutes. Each number records one session, not one child’s ability. The observation count n=10 says how many sessions are listed. A table of these fictional records is a teaching example, not a report about real pupils.
Primary data are collected directly for your question; secondary data come from an existing source. The source says direct data are more reliable, but that is not guaranteed. A poorly measured direct record can be worse than a checked published dataset. Ask who collected it, how, when, which cases were included and whether units agree. Do not upload names, ages, heights or family details to practise this lesson; anonymous object counts work.
Ascending order goes from smallest to largest; descending order reverses it. Sorting changes positions, not the number or total of the observations. In our list, three separate 18-minute sessions remain three records. Equal values are not accidental duplicates to delete. Recheck the count after sorting and compare with the raw list. Missing and recorded-zero values are different: a missing measurement is not automatically zero.
The usual range measures the spread from the smallest to largest value: maximum minus minimum. Our maximum 28 and minimum 12 give a range of 16 minutes. Printed page 172 adds one and calls the result range. For consecutive whole-number possibilities, 28−12+1=17 counts the possible integer values; it is an inclusive span, not a 17-minute measured distance. The plus-one rule does not apply to arbitrary decimals or continuous lengths.
A frequency is the number of observations in a named value or group. For whole-minute records, inclusive classes 10–14, 15–19, 20–24, 25–29 each cover five integers. For measurements rounded to the nearest minute, their continuous boundaries can be 9.5, 14.5, 19.5, 24.5, 29.5; choose lower-inclusive, upper-exclusive intervals to prevent overlap. Alternatively use directly defined intervals such as 10≤x<15. State the convention instead of counting a shared endpoint twice.
For integer classes of width w beginning at lower limit L, enough classes are needed to cover the maximum M: round (M−L+1)/w upwards to the next whole number. This is a coverage count, not rounding to the nearest integer. For L=10, M=28, w=5, the quotient is 19/5=3.8, so four classes are needed. Changing the start can change the number and grouping. The book’s suggestion about five to fifteen is not a universal statistical rule.
Tally marks are counting strokes, commonly grouped in fives by crossing the first four with the fifth. Seven records make one complete five and two additional strokes. For our four classes the frequencies are 3, 4, 2, 1. Their sum is 10, matching all ten sessions. Frequency is a count of records, not their sum in minutes. A class with no records still has frequency zero and should remain in a continuous interval sequence.
A frequency table groups observations so patterns are easier to see, but intervals hide individual values. In our example 15–19 contains four sessions; it does not mean each lasted 17 minutes. Check the total, boundaries and missing entries before interpreting. A small convenient set of ten sessions does not establish what every learner does. The table describes these records; extending its conclusions needs a justified collection plan.
For our ten invented sessions, the inclusive classes 10–14,15–19,20–24,25–29 have frequencies 3,4,2,1. The total 10 confirms every listed session has been counted; it does not restore exact durations hidden by grouping.
Printed page 172 calls maximum−minimum+1 the range. We distinguish that integer inclusive span from the usual maximum−minimum spread.
Primary and secondary describe provenance. Reliability depends on collection and checking, not the name alone.
Whole-minute observations; each listed repeat is a separate record.
| Inclusive class (minutes) | Tally count | Frequency (sessions) |
|---|---|---|
| 10–14 | Three strokes | 3 |
| 15–19 | Four strokes | 4 |
| 20–24 | Two strokes | 2 |
| 25–29 | One stroke | 1 |
| Total | Ten strokes | 10 |
Read 15–19 across: four records fall in that class. It does not mean the four durations total 4 minutes or that each lasted 17 minutes.
Nearest-minute records; boundaries use a lower-inclusive, upper-exclusive convention.
| Integer limits | Continuous boundaries (minutes) | Width |
|---|---|---|
| 10–14 | 9.5≤x<14.5 | 5 minutes |
| 15–19 | 14.5≤x<19.5 | 5 minutes |
| 20–24 | 19.5≤x<24.5 | 5 minutes |
| 25–29 | 24.5≤x<29.5 | 5 minutes |
14.5 belongs to the second interval under this stated rule. For already continuous measurements, choose appropriate original intervals instead of automatically adding half-unit corrections.
These are illustrative collection plans, not measured results.
| Source type | Example | What still needs checking |
|---|---|---|
| Primary | Measure anonymous pencils directly | Zero alignment, instrument unit, which pencils, missing readings |
| Secondary | Use an existing checked table | Original method, date, units, coverage and copying errors |
Primary means direct collection; secondary means reuse. Either can contain mistakes. The source’s automatic reliability comparison is too broad.
Classes of width w begin at integer lower limit L and must include observed maximum M.
k = ceiling((M−L+1)/w) L=10, M=28, w=5 ⇒ k=ceiling(3.8)=4
The usual numerical range is M−minimum. The source’s plus-one expression instead counts inclusive integer possibilities. Non-integer or continuous classes require their own boundary convention.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original diagram using fictional classroom records, not measured personal information or current statistics. Read the quantities, units and conventions beside the figure; colour alone does not carry the meaning.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original diagram using fictional classroom records, not measured personal information or current statistics. Read the quantities, units and conventions beside the figure; colour alone does not carry the meaning.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original diagram using fictional classroom records, not measured personal information or current statistics. Read the quantities, units and conventions beside the figure; colour alone does not carry the meaning.
Show your working on paper. Check your answer, then choose Next.
10
There are ten session records; repeated lengths still count separately.
3
The value 18 occurs three times, representing three sessions.
16
28−12=16 minutes measures the spread.
17
28−12+1=17 possible integer values; this is an inclusive count.
5
The values 10,11,12,13,14 make five integers.
14
Five integers from 10 end at 14; the next starts at 15.
4
Round (28−10+1)/5=3.8 upwards to four classes.
3
The fixed classes are 10–14,15–19,20–24. A minimum-anchored calculation would answer a different question.
7
5+2=7 records; the crossing stroke is the fifth, not an extra sixth.
3
12,14,14 give three records.
4
16,18,18,18 give four records.
2
20 and 22 give two records.
10
3+4+2+1=10 observations, not ten minutes.
14.5
Half-unit boundaries put the class between 14.5 and 19.5 minutes under the stated convention.
2
The mismatch 20−18=2 requires a recount; do not guess their values.
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Each value is one session duration in minutes. n=10 counts sessions, including repeats; it does not count marks, ability or necessarily ten distinct children. These are fictional teaching records.
Name the observation, unit and count, then state the limits.
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Primary describes direct collection, not guaranteed quality. Check instruments, selection, missing records, date and independent validation. A checked secondary table may be more dependable than a careless direct survey.
Give a concrete collection-error example without inventing real evidence.
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Ascending order is 2,3,5,8,8. Both lists have five values and total 26; both 8s remain because they are separate observations.
Keep repeated values and verify both count and total.
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Sixteen minutes is the usual range, the distance between the extremes. Seventeen is the count of consecutive integer possibilities including both ends. Printed page 172 uses the second convention under the word range. State which question is intended.
Do not apply plus one to a continuous distance.
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One choice is 10–14,15–19,20–24,25–29, with both integer endpoints included. These cover all records once. Another start is possible, but it must be stated and checked.
List every interval and test the largest value and shared-edge cases.
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Use an explicit convention such as 10≤x<20 and 20≤x<30; then 20 belongs only to the second. If both endpoints were included in both classes, the table would overlap.
State all endpoint inequalities; final endpoints need an explicit rule too.
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A partial last class still needs a row to hold its largest records. Taking only three would leave some values outside. This is upward coverage rounding, not nearest-integer rounding.
Connect the decision to covering the maximum.
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Starting at 14 gives 14–18 and 19–23: two classes. Starting at 10 gives 10–14,15–19,20–24: three classes. The observed spread alone does not specify the origin.
Use two valid complete partitions and name each start.
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Draw four upright strokes and cross them with a fifth, then draw two more upright strokes. The total is 7. A text explanation or labelled drawing should distinguish groups and remaining strokes.
Check that the crossing mark is counted once.
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The four frequencies are 3,4,2,1 and total 10. The first group includes both 14 records, while the second includes all three 18 records. Every record is placed exactly once.
Show headings, inclusive integer convention and a total row.
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A zero row marks a real interval with no observed records. Deleting it and sliding later groups left changes the horizontal meaning and falsely suggests adjacency of different values.
Keep the interval at its correct horizontal position.
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For example count visible blocks in ten labelled fictional bags, or measure anonymous pencils in centimetres. State the question, one record per object, unit, instrument and missing-value rule. Report only what that collection supports.
Do not upload names, family ages or body measurements.
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There are four records in that interval. Their exact durations and order are hidden; they need not all equal the midpoint 17. The class count cannot alone describe all learners or establish a cause.
Name the lost information and avoid a wider claim.
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Zero is an observed numerical value. Missing means no usable observation was recorded. Treating missing as zero changes counts and summaries. State a checking and reporting rule instead.
Distinguish a measurement from the absence of a measurement.
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Check the question and unit, count every raw record, keep repeats, define non-overlapping boundaries, include zero intervals, verify frequency total and label fictional data. Note collection limits before interpreting.
Apply the checklist to the ten-session example.
Build and explain one checked frequency table.
Keep your work on paper or in a local file. This page does not upload it or automatically grade the project.
This original Kidemy practice follows the topic order of 2026 Class 8 Mathematics · English Version, chapter 11, printed pages 171–187. It does not replace the complete chapter or its practical teaching.
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