Connecting the Dots... · Lesson 13 of 13
Chapter Summary and Practice
“Revise the complete chapter with a quick recap, mixed examples, and connected practice.”
• Recall the chapter’s main concepts and choose suitable methods. • Calculate mean, median, and range accurately. • Interpret dot plots and clustered bar graphs. • Recognise common errors involving missing data and group averages. • Apply the chapter’s ideas in mixed questions and an optional reasoning puzzle.
Chapter Summary and Practice
We began by asking what data can tell us. We then described collections using mean, median, and range, displayed them using dot plots and clustered bars, and tested conclusions against the available evidence. Use this recap to choose a method before calculating; the mixed questions connect the ideas rather than introduce a new topic.
| What we studied | What it means | A useful check |
|---|---|---|
| Statistical questions | Investigate a collection with expected variation. | Specify group, quantity, and period. |
| Mean | Total divided by observation count; an equal-share value. | It lies between the minimum and maximum. |
| Median | Centre of sorted data; average the middle pair for even counts. | Keep repeated values and sort first. |
| Range | Maximum minus minimum. | It describes extremes, not every feature of spread. |
| Outliers | Observations far from most others in context. | Do not delete valid data automatically. |
| Dot plots | One dot per observation on a numerical scale. | Repeated values stack; time order may be lost. |
| Clustered graphs | Related bars together for each category. | Read units, scale, baseline, and legend. |
| Zero versus missing | Zero is observed; missing is not a known value. | Match the denominator to the stated question. |
| Monthly and daily values | Divide monthly totals by that month’s day count. | The result has a per-day unit. |
| Evidence and predictions | Distinguish observations, inferences, and estimates. | Restrict conclusions to the supplied data. |
| Data projects | Define, collect, organise, visualise, and interpret. | Report methods, missing data, and limits. |
The three main numerical summaries are mean = total ÷ count, median = middle of sorted data (or the mean of its middle pair), and range = maximum − minimum. They answer different questions. A graph lets us see features that one number can conceal, while the original table preserves exact labels and chronological order.
Problem
Describe 2, 3, 4, 5, 26 using mean, median, and range.
- 1.Total = 40 across 5 observations, so mean = 8.
- 2.The sorted middle value is 4, so median = 4.
- 3.Range = 26 − 2 = 24. The high value helps explain the gap between mean and median.
Problem
A player’s entries are 6, 0, 12, and “did not play”. Find mean per appearance.
- 1.There are three appearances and one non-appearance. Keep the recorded zero.
- 2.Mean = (6 + 0 + 12) ÷ 3 = 6 runs per appearance.
- 3.The result must say what its denominator counts.
Problem
City A has 310 daylight hours in a 31-day month; City B has 300 in a 30-day month. Compare daily averages.
- 1.City A: 310 ÷ 31 = 10 hours/day.
- 2.City B: 300 ÷ 30 = 10 hours/day.
- 3.The monthly totals differ, but their daily averages are equal. Plot the per-day quantities if that is your comparison.
Check four common mistakes: unsorted median data, an incorrect observation count, a graph ratio inferred from a nonzero baseline, and a claim about every individual made from a group average.
Connect the Dots…
The source closes with a three-digit lock puzzle. It is an optional logical-reasoning challenge: clues are constraints, and a proposed answer must satisfy every constraint. Treat each stated count of correct digits as exact; “wrongly placed” means the digit occurs in the code but at a different position.
| Attempt | Clue |
|---|---|
| 265 | One digit is correct and well placed. |
| 271 | One digit is correct but wrongly placed. |
| 542 | Two digits are correct but wrongly placed. |
| 036 | Nothing is correct. |
| 064 | One digit is correct but wrongly placed. |
The clue 036 excludes 0, 3, and 6. From 064, digit 4 occurs but not in the third position. In 265, either 2 is correct in the first position or 5 is correct in the third. If 2 were correct in the first position, 271 would contain a well-placed digit, contradicting its clue. Thus 2 is absent and 5 is in the third position. In 542, 5 and 4 are present but both wrongly placed, so 4 is not second and must be first. In 271, only 1 or 7 can remain; the available second position would place 7 correctly, which is not allowed. Therefore 1 is second. The code is 415. Check all five clues.
Quiz
For 1, 2, 3, 4, 20, which centre is most affected by the high value?
What is the median of 2, 4, 6, 10?
The range of 2, 4, 6, 10 is:
Which is the correct count for observations 4, 0, 8, missing?
Which graph best compares two towns month by month?
Which conclusion follows from a larger class mean height?
What is the mean daily amount from 270 hours in a 30-day month?
Practice Problems
- For 4, 4, 6, 8, 18, calculate centres and range; describe the plot.
- Create two five-value datasets with mean 10 but different ranges.
- Find mean and median of 0, 2, 2, 4, missing, 12.
- Compare onion prices using both means and medians.
- For paired values 20, 30, 40 and 25, 25, 45 across three categories, describe a double-bar graph.
- Use the supplied 2019 height table to test “Every girl aged 13 is taller than every girl aged 11”.
- Compare Aditi’s two weeks of Sudoku times. Week 1: 410, 400, 370, 340, 360, 400, 320, 330, 310. Week 2: 320, 290, 380, 280, 270, 230, 220, 240 seconds.
- Explain why an international height display starting at 145 cm can be useful but requires care.
- Give one supported observation and one unanswered question from the rocket-launch investigation.
- Choose a project and state how you will prevent a misleading conclusion.
- Solve the optional lock puzzle and verify its clues.
Total 40 and count 5 give mean 8. Median is 6; range is 18 − 4 = 14. Stack two dots at 4; 18 is distant from the lower cluster.
Key Takeaways
• Choose a measure and representation that fit the question. • Mean, median, and range describe different features. • Retain genuine zeros and identify missing observations. • Graph labels, scales, legends, and baselines are part of the information. • Inspect variation before relying on a representative value. • Keep conclusions within the observed group and period. • Explain the assumptions behind estimates, predictions, and projects.
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Data Projects
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