110i - Consolidation Central Tendency

Lesson 110 — Problem Solving and Consolidation: Measures of Central Tendency

Strand: Statistics | Descriptor: AC9M7ST01 | Duration: 45 minutes

Learning Intentions

• To consolidate calculating and choosing measures of centre and spread.
• To analyse a real data set and report findings with justification.

Success Criteria

I can:

1. Calculate range, median, mean and mode accurately from raw data.
2. Work backwards from a measure to a missing value.
3. Choose the most representative measure and justify it from the data.
4. Report findings clearly, with centre and spread together.

Warmup

(6 minutes — rapid recall, mini whiteboards)

Data: .

1. Range.
2. Median.
3. Mean.
4. Mode.
5. Is any value an outlier? Justify.

Answers: ordered : 1. ; 2. ; 3. ; 4. ; 5. No — the values are spread fairly evenly, with nothing far from the rest.

Q5 is the block’s real skill: deciding whether an outlier is present before choosing a measure.

Activities

Activity 1 — Mixed Skills Circuit (16 min)

Stations spanning Lessons 107–109.

Station A — Calculate all four.

Station B — Work backwards.

4. Five numbers have a mean of . Four are . Find the fifth.
5. A data set has range and largest value . Find the smallest.
6. The mean of eight numbers is . When one is removed, the mean of the remaining seven is . What was removed?
7. Six numbers have a median of . Five of them, in order, are . Where could the sixth go, and what values are possible?

Station C — Classify and choose.

8. Classify as discrete or continuous: number of tries scored; height of a plant; number of pages read; time spent on homework.
9. For each data set, name the best measure and justify: (a) (b) (c) favourite subject responses.

Station D — Interpret.

10. Two classes sit the same test. Class A: mean , range . Class B: mean , range . What does the difference tell you?
11. A data set’s mean is much larger than its median. What does this suggest about its shape?

Socratic scaffolding for Station B Q7:

Prompt	Purpose
With six values, where does the median sit?	Between the rd and th of the ordered set.
The five known values, in order?	.
If the sixth value is large (above ), what are positions and ?	and : median ✗
If is small (below ), positions and ?	The order becomes : positions and are and , median ✗
So must sit between and . Then?	Order: ; median .
Looking back	The median’s position rule drives the whole search — a good example of working backwards.

(Answers: 1. ordered : range , median , mean , mode . 2. ordered : range , median , mean , mode . 3. ordered : range , median , mean , mode . 4. total ; known ; fifth . 5. . 6. . 7. . 8. discrete; continuous; discrete; continuous. 9. (a) mean — evenly spread (b) median — is an outlier (c) mode — categorical. 10. Same centre, very different consistency: Class A’s marks cluster tightly, Class B’s scatter widely. 11. A few large values pull the mean up — the data is right-skewed, with a long tail of high values.)

Activity 2 — The Data Investigation (16 min)

Pairs. A complete small analysis, using the class’s own data from Lesson 107 if collected, or the set below.

Reaction times. Twenty students’ reaction times (milliseconds), measured with a ruler-drop test:

1. Is this variable discrete or continuous? Justify.
2. Calculate the range, median, mean and mode.
3. Identify any outlier. Suggest a plausible explanation for it.
4. Recalculate the mean and range without the outlier. Comment on the changes.
5. Which measure best represents a typical reaction time? Justify with reference to the data.
6. Write a two-sentence report of your findings, giving both a centre and a spread.

Socratic scaffolding:

Prompt	Purpose
Order the data first — how many values?	Twenty; the median sits between the th and th.
Spot anything unusual?	— more than three times any other value.
A plausible cause?	A distraction, a mistimed drop, or a recording error — but we cannot know, so we report rather than delete.
Compare the mean with and without it.	With: . Without: . A single value shifted the mean by about ms.
And the range?	versus — the range is almost entirely the outlier’s doing.
So what do you report?	The median () as the typical time, with the interquartile-free spread noted, and the outlier flagged separately.

(Answers: 1. Continuous — time can take any value in a range. 2. Ordered: . Range ; median ; mean ; mode — none (no value repeats). 3. . 4. Without: mean ; range . 5. The median — it barely moves, and sits among the bulk of the data, whereas the mean of exceeds all but one value. 6. E.g. “A typical reaction time was about ms, with most students between and ms. One reading of ms was far outside this range and may reflect a mistimed trial.“)

The teaching point to close on: the mode was useless here — with continuous data measured precisely, values rarely repeat. Naming which measures do not apply is part of the analysis.

Activity 3 — Quick Synthesis (5 min)

Whole class.

Complete the summary table from memory, then check.

Measure	What it tells you	Affected by outliers?	Works on categorical data?
Mean
Median
Mode
Range

Completed:

Measure	What it tells you	Affected by outliers?	Categorical?
Mean	The fair share; uses every value	Strongly	No
Median	The middle value by position	Barely	No
Mode	The most frequent value	No	Yes
Range	The total spread	Most of all	No

Checks for Understanding

(7 minutes — exit ticket, collected)

Data — masses of parcels (kg): .

1. Range.
2. Median.
3. Mean (2 d.p.).
4. Mode.
5. Identify the outlier and state its effect on the mean and the median.
6. Which measure best represents a typical parcel? Justify.
7. Reasoning. Two data sets have the same mean but very different ranges. What does that tell you, and why must both be reported?

Answers: ordered : 1. ; 2. ; 3. ; 4. No mode; 5. — it lifts the mean above every other value while moving the median hardly at all; 6. The median ( kg) — six of the seven parcels lie between and kg, and the mean of describes none of them; 7. They have the same centre but different consistency; a centre alone cannot distinguish tightly clustered data from widely scattered data, so both are needed for an honest summary.

Common Misconceptions

Misconception	How to pre-empt it
Calculating without ordering.	Warmup and every station begin with ordering.
Reporting a mode where no value repeats.	The reaction-time data has none — “no mode” is the answer.
Deleting outliers to tidy the data.	Investigation Q3–4: recalculate and report, do not delete.
Choosing a measure before examining the data.	Every justification must cite the data’s shape or outliers.
Reporting a centre without a spread.	Q6 of the investigation and exit Q7 both require both.
Assuming the mean is always the best summary.	The whole block, culminating in the reaction-time analysis.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Seven numbers have a mean of . Six of them are . Find the seventh.

Answer

Total ; known ; seventh .

E2 (AMC Junior style). A set of five numbers has mean , median and mode . Find a possible set.

Answer

Total ; median sits third; mode needs at least two s below it. Try : total ✓, median ✓, mode ✓

E3 (Challenge). Adding one number to a set of nine changes the mean from to . What number was added?

Answer

Old total ; new total ; added .

E4 (Challenge). In a data set of values with mean , every value is increased by . What happens to the mean, median, mode and range?

Answer

Mean, median and mode all increase by ; the range is unchanged, since both extremes shift equally. Measures of centre move with the data; measures of spread do not.

E5 (Challenge). Every value in a data set is doubled. Predict the effect on all four measures, and verify with .

Answer

All four double. Original: mean , median , mode , range . Doubled (): mean , median , mode , range ✓ (Contrast with E4: adding shifts centres only; multiplying scales everything.)

Homework

1. For each set, find all four measures: (a) (b) (c) .
2. Six numbers have a mean of . Five are . Find the sixth.
3. A data set has range and smallest value . Find the largest.
4. The mean of ten numbers is . One number, , is removed. Find the new mean.
5. Five numbers have a median of . Four of them, in order, are . Find the fifth value.
6. Rainfall (mm) over eight days: . (a) Find all four measures. (b) Identify the outlier and explain it. (c) Which measure best describes a typical day? (d) Which measure matters most to a farmer thinking about the month’s water? Justify.
7. Two athletes’ times over five races. A: . B: . (a) Find each mean and range. (b) Who is faster on average? (c) Who is more consistent? (d) Which athlete would you select, and why?
8. Classify as discrete or continuous: number of emails received; length of an email in words; time spent reading emails.
9. Reasoning. Explain why the mode is often useless for continuous data.
10. Reasoning. Explain why reporting a mean without a range can mislead, using Q7’s athletes as your example.
11. Challenge. A data set of six whole numbers has mean , median , mode and range . Find a set that satisfies all four, and verify each condition.

Answers: Q1 — (a) ordered : range , median , mean , mode (b) ordered : range , median , mean , mode (c) ordered : range , median , mean , mode . Q2 — total ; known ; sixth . Q3 — . Q4 — old total ; new total ; new mean . Q5 — the median of five values is the third; with and a fifth value , the median is only if and sits third: ordered ✓. Q6 — (a) ordered : range , median , mean , mode and (b) — a storm day (c) the median, since seven of eight days were under mm (d) the mean (or total) — the farmer cares about total water, and the storm genuinely delivered it. Q7 — (a) A: mean , range ; B: mean , range (b) A (c) A (d) A — faster on average and far more consistent. Q8 — discrete; discrete; continuous. Q9 — precisely measured continuous values rarely repeat, so either there is no mode or the mode is an accident of rounding. Q10 — both athletes average around – s, but B’s times swing by s while A’s vary by s; the means alone hide a large difference in reliability.