114i - Consolidation Data Displays

Lesson 114 — Problem Solving and Consolidation: Data Displays

Strand: Statistics | Descriptor: AC9M7ST02 | Duration: 45 minutes

Learning Intentions

• To consolidate constructing, reading and comparing data displays.
• To produce a complete display-and-description of a data set.

Success Criteria

I can:

1. Choose and construct an appropriate display for a given data set.
2. Read values and summary statistics from a display.
3. Describe a distribution using shape, centre, spread and outliers.
4. Compare two distributions and justify a conclusion.

Warmup

(6 minutes — display triage, pairs)

For each, name the best display and give one reason.

1. Twenty-five exam marks out of , keeping every individual mark.
2. Number of siblings for students.
3. Heights of adults to the nearest centimetre.
4. Two classes’ marks, to be compared directly.
5. Monthly rainfall across a year.

Answers: 1. Stem-and-leaf — retains individuals; 2. Dot plot or column graph — few distinct values; 3. Grouped column graph — too many distinct values otherwise; 4. Back-to-back stem-and-leaf — shared scale; 5. Column graph in time order — sequence matters.

Activities

Activity 1 — Mixed Skills Circuit (16 min)

Stations spanning Lessons 111–113.

Station A — Construct.

1. Build a stem-and-leaf plot with a key for: .
2. Build a dot plot for: .

Station B — Read. From this plot:

 Stem | Leaf
    3 | 2 6 8
    4 | 0 1 4 4 7 9
    5 | 3 5
    6 |
    7 | 2

Key: 4 | 1 means 41

3. How many values?
4. The range.
5. The median.
6. The mode.
7. Any outlier?

Station C — Describe. Using Station B’s plot, write a four-feature description (shape, centre, spread, outliers).

Station D — Compare. Two teams’ scores across ten games:

• Team X:
• Team Y:

8. Median and range for each.
9. Which team is more consistent? Which has the higher typical score?
10. Write a two-sentence comparison.

Socratic scaffolding for Station B Q5:

Prompt	Purpose
Count the leaves.	values.
With twelve values, where is the median?	Between the th and th.
List in order and count.	: the th is , the th is .
So the median?	.
Note the empty stem.	The s row is empty — the gap before is what marks it as an outlier.

(Answers: 1. stems (), (), (); 3. ; 4. ; 5. ; 6. ; 7. yes — , separated by an empty stem. 8. X: median , range ; Y: median , range . 9. X is far more consistent; Y has a slightly higher median. 10. E.g. “Team Y has a marginally higher typical score ( vs ), but Team X’s scores are much more consistent, spanning only compared with Y’s . Team X’s reliability may matter more than Y’s occasional high scores.“)

Activity 2 — The Display Task (16 min)

Pairs. One data set, taken through the full process.

Bus waiting times. A student records how long she waits for the bus (minutes) on school mornings:

1. Is the variable discrete or continuous? Justify.
2. Construct a stem-and-leaf plot with a key.
3. Calculate the median, mean and range.
4. Describe the distribution using all four features.
5. Identify the outlier and suggest a plausible cause.
6. Which measure best represents a typical wait? Justify.
7. Write a two-sentence report the student could send to the bus company.

Socratic scaffolding:

Prompt	Purpose
Q1: is waiting time counted or measured?	Measured — continuous, though recorded to the nearest minute.
Q2: what stems?	, , (tens), with a key like means minutes.
Q3: order first. Twenty-four values — the median sits where?	Between the th and th: both , so median .
The mean?	Total ; mean .
Q4: shape?	Right-skewed — a single peak around – minutes, with a long thin tail to one extreme value.
Q5: a plausible cause?	A cancelled or broken-down bus. We report it; we do not delete it.
Q6: median or mean?	Median — the mean of is inflated by the -minute wait.
Q7: what does the bus company need to hear?	Both the typical wait and the failure: “Most mornings I wait about minutes, but on one occasion I waited .”

(Answers: 1. Continuous. 2. stems (), (), (). 3. median , mean , range . 4. Right-skewed, peak at – min, median min, range min, one outlier at min. 5. As scaffolded. 6. Median. 7. As scaffolded.)

Activity 3 — Quick Synthesis (5 min)

Whole class, closing the ST02 block.

Complete from memory, then check.

Display	Best for	Keeps individual values?
Stem-and-leaf
Dot plot
Column graph
Back-to-back stem-and-leaf

Completed:

Display	Best for	Individuals?
Stem-and-leaf	Many values, moderate spread	Yes
Dot plot	Few distinct values	Yes
Column graph	Frequencies, grouped or discrete	No
Back-to-back stem-and-leaf	Comparing two data sets	Yes

And the description order: shape, centre, spread, outliers — every time.

Checks for Understanding

(2 minutes — one question, collected with the work)

Reasoning. Two data sets have the same median and the same range. Explain, with reference to shape, why this is not enough to say they are similar — and name the display you would use to reveal the difference.

Expected answer: Median and range describe only the centre and the total width; they say nothing about how the values are arranged between the extremes. One set could have a single central peak while the other has two clusters with a gap. A dot plot or stem-and-leaf plot shows the shape directly and reveals the difference immediately. (Lesson 113’s Sets P and Q.)

Common Misconceptions

Misconception	How to pre-empt it
Choosing a display by habit rather than by data type.	The warmup’s triage; the synthesis table.
Omitting keys or axis labels.	Marked at every construction station.
Skipping empty stems.	Station B’s Q7 depends on the empty s row.
Describing only the centre.	The four-feature order is required.
Comparing on one feature only.	Station D Q9 and Q10.
Deleting an outlier rather than reporting it.	The bus report must mention the -minute wait.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A stem-and-leaf plot has leaves on stem , on stem , and on stem . Where does the median lie?

Answer

Fifteen values; the median is the th, which falls in the stem (values through ).

E2 (AMC Junior style). In a dot plot, the tallest stack is at with dots, and there are dots in total. What is the mode, and what is the least possible number of distinct values?

Answer

Mode . With dots left and no stack allowed to reach , at least further values are needed, so at least four distinct values in total.

E3 (Challenge). A data set of values has median and range . Sketch two possible stem-and-leaf plots with visibly different shapes, both satisfying these figures.

Answer

One tightly clustered around with two extreme values at and ; one with two clusters, near and near , and few values in the middle. Both give median and range — the lesson of Sets P and Q.

E4 (Challenge). Why does a back-to-back stem-and-leaf plot make comparison more reliable than two separate column graphs?

Answer

The shared stem forces a single common scale, so shape, centre and spread are directly comparable. Two separate charts can be drawn with different axis ranges, making genuinely different distributions look alike (or alike ones look different) — the scale trap from Lessons 83 and 112.

Homework

1. Construct a stem-and-leaf plot with a key for: .
2. From your Q1 plot, state: (a) the number of values (b) the range (c) the median (d) the mode, if any.
3. Write a four-feature description of your Q1 distribution.
4. Construct a dot plot for: .
5. Two classes’ spelling test scores out of :
   • Class P:
   • Class Q: (a) Build a back-to-back stem-and-leaf plot. (b) Find each median and range. (c) Write a two-sentence comparison.
6. A data set: . (a) Describe the shape. (b) Which measure of centre would you use, and why? (c) Identify the outlier.
7. Name the best display for each: (a) students’ exact test marks (b) the number of pets owned by students (c) comparing two years’ rainfall by month (d) the masses of parcels to the nearest gram.
8. Reasoning. Explain why an empty stem must be shown in a stem-and-leaf plot, using an outlier as your example.
9. Reasoning. A report gives only a mean. Name two things the reader still does not know, and say which display would supply them.
10. Challenge. Design a data set of values with: median , range , one clear outlier, and a right-skewed shape. Present it as a stem-and-leaf plot and verify all four features.

Answers: Q1 — stems (), (), (); key means . Q2 — (a) (b) (c) between the th () and th (): (d) no mode. Q5 — (b) P: median , range ; Q: median , range (c) e.g. “The two classes have almost the same typical score, but Class P’s results are far more consistent, spanning marks against Class Q’s . Class Q contains both the highest and the lowest scores.” Q6 — (a) right-skewed (b) median , since the mean of is inflated by the (c) . Q7 — (a) stem-and-leaf (b) dot plot or column graph (c) column graph in month order, or back-to-back if only two years (d) grouped column graph. Q8 — the empty row makes the gap between the bulk and the outlier visible; without it the outlier looks like the next value in sequence. Q9 — the spread and the shape; a stem-and-leaf or dot plot shows both.