109i - Justifying the Best Measure

Lesson 109 — Justifying Which Measure Best Suits a Data Set

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

Learning Intentions

• To decide which measure of central tendency best represents a data set.
• To justify the choice by referring to the data’s shape and outliers.

Success Criteria

I can:

1. Identify an outlier and describe its effect on each measure.
2. Explain when the median is more representative than the mean.
3. Explain when the mode is the only sensible choice.
4. Justify a choice of measure with evidence from the data.

Warmup

(6 minutes — the misleading average, whole class)

Nine workers at a small business earn (in thousands): .

1. Calculate the mean.
2. Calculate the median.
3. The owner advertises “average salary 90,000$“. Is that true? Is it honest?
4. Which figure would you quote to a job applicant?

Answers: 1. Total ; mean , so about 90,600= $50,000$; 3. Arithmetically true — but eight of the nine workers earn far less, so it misrepresents a typical wage; 4. The median, which describes what a new employee could actually expect.

The lesson in one line: “average” is not one thing. Choosing which average to report is a decision — and sometimes a rhetorical one.

Activities

Activity 1 — Explicit Instruction: Outliers and Their Effects (14 min)

Definition. An outlier is a value far from the rest of the data.

Test each measure against the salary data, with and without the :

Measure	With outlier	Without	Change
Mean			huge
Median			small
Mode			none
Range			huge

The rule that falls out:

• The mean uses every value, so a single extreme value drags it. Sensitive to outliers.
• The median depends only on position, so extremes barely move it. Resistant to outliers.
• The mode ignores everything but frequency. Unaffected by outliers.
• The range is defined by the extremes, so it is the most sensitive of all.

When to choose which — the decision guide:

Situation	Best measure	Why
Data is fairly even, no extremes	Mean	Uses all the data; most informative
Outliers present, or a skewed spread	Median	Unmoved by extremes; describes the typical case
Categorical or “most popular” questions	Mode	The only measure that works for non-numerical data
Data where every value is different	Mean or median	The mode is useless — no value repeats

I do — three quick judgements:

1. House prices in a suburb (a few mansions among many units) → median. (This is why real estate reports quote median prices.)
2. Shoe sizes a shop should stock → mode. The shop wants the size that sells most, not an average of .
3. Marks on a class test with no unusual scores → mean. It uses every mark.

We do — decide and justify:

1. Times for ten students to run m, one of whom stopped to tie a shoelace.
2. The number of pets owned by students in a class.
3. Daily rainfall over a month with one storm day.
4. Favourite lunch option among students.

(Answers: 1. median — the interrupted run is an outlier; 2. mode or median — small whole numbers, and the mode says what is most common; 3. median for a typical day, though the mean matters for total water; 4. mode — the data is categorical, so mean and median do not exist.)

Activity 2 — Justification Practice (14 min)

Pairs. Every answer states the chosen measure, its value, and a one-sentence justification referring to the data.

Data 1 — Sleep hours (ten students): .

Data 2 — Prices of items in a shop (3, 4, 4, 5, 6, 6, 7, 250$.

Data 3 — Number of siblings: .

Data 4 — Test marks out of : .

Data 5 — Shirt sizes sold: S, M, M, L, M, S, L, M, XL, M.

For each: calculate all applicable measures, choose the best representative, and justify.

Socratic scaffolding for Data 4:

Prompt	Purpose
Calculate the mean and the median.	Mean ; median .
Which is closer to most of the marks?	The median — six of eight marks lie between and .
What is pulling the mean up?	The — an outlier.
Is the an error, or real?	Both are possible. Real outliers stay in the data but are reported; errors are corrected.
So what do you report?	The median as the typical mark, and note that one student scored far higher.
Looking back	Choosing the median does not mean hiding the outlier — it means not letting it distort the summary.

(Answers: 1. mean , median , mode — all similar; the mean is fine, since there are no outliers. 2. median — the 250$35.6= 1.5= 18= 2.25= 15.5=$ M — categorical data, so no mean or median exists.)

Activity 3 — Inquiry: whose Average? (9 min)

Pairs, then class discussion.

A school reports on a fundraising day. Donations (in dollars) from ten families:

1. Calculate the mean, median and mode.
2. The principal wants to report a strong result. Which would she quote?
3. A parent group argues most families gave modestly. Which would they quote?
4. Which is the most honest single figure, and what should be reported alongside it?

Socratic scaffolding:

Prompt	Purpose
Compute all three.	Mean ; median ; mode .
Which flatters the result?	The mean — inflated by one 500$ donation.
Which understates the total effort?	The mode — most donations were near it, but it ignores the large gift entirely.
Is any of them a lie?	No. Each is correctly calculated — they answer different questions.
What is the honest report?	The median as the typical donation, plus the total raised, plus a note about the large single gift.
Looking back	Honesty in statistics is usually about what you report alongside the number, not about which number is “true”.

Closing point (board): whenever you meet an “average” in the media, ask three questions — which average, what the spread is, and whether outliers are present.

Checks for Understanding

(5 minutes — exit ticket, collected)

Data: .

1. Calculate the mean and the median.
2. Which better represents this data? Justify in one sentence.
3. Name the outlier and its effect on each measure.
4. For “favourite sport” data, which measure can be used, and why only that one?
5. Reasoning. A report quotes a mean without mentioning the spread. Why is that a problem?

Answers: 1. Mean ; median ; 2. The median — six of the seven values lie between and , while the mean is pulled above every one of them by the outlier; 3. ; it inflates the mean and the range, barely moves the median, and leaves the mode ( and ) untouched; 4. The mode — the data is categorical, so it cannot be ordered or added; 5. Two very different data sets can share a mean; without the spread, the reader cannot tell whether the values cluster tightly or scatter widely.

Common Misconceptions

Misconception	How to pre-empt it
”The average” means the mean, always.	The warmup’s salary data; three averages, three answers.
The mean is the most accurate measure.	It is the most complete, but completeness is a liability when outliers are present.
Outliers should be deleted.	Real outliers stay; they are reported, not hidden. Only errors are corrected.
The mode is a beginner’s measure.	It is the only measure available for categorical data.
A measure can be chosen without looking at the data.	Every justification must cite the data’s shape or outliers.
One number can summarise a data set fully.	The closing point: always report spread alongside centre.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). In the set , which is larger, the mean or the median, and by how much?

Answer

Mean ; median ; the mean is larger by — the outlier’s doing.

E2 (AMC Junior style). A data set has mean and median . One value of is added. Describe what happens to each.

Answer

The mean rises noticeably (by an amount depending on the set’s size); the median rises by at most a little, shifting one position up the ordered list.

E3 (Challenge). Construct two different data sets of five whole numbers that both have a mean of , but where one is well represented by the mean and the other is not. Explain.

Answer

E.g. — tightly clustered, the mean describes every value well. And — the mean of describes none of the values; the median of is far more representative.

E4 (Challenge). A cricketer’s scores are . Which measure would the player quote? Which would a selector prefer, and why might they disagree?

Answer

The player quotes the mean (), flattered by one big innings. A selector wants the median () or the whole distribution, since consistency matters for team selection. Both figures are correct; they answer different questions — which is precisely why the spread should accompany any average.

E5 (Challenge). Explain why a data set of house prices almost always has a mean greater than its median.

Answer

Prices have a floor near zero but no ceiling, so the expensive tail stretches far to the right. Those extreme values lift the mean while leaving the median — a positional measure — near the bulk of the data. This right-skew is why property reports quote medians.

Homework

1. For each data set, calculate the mean and median, then state which better represents the data and why: (a) (b) (c) (d)
2. Name the outlier in each of 1(b) and 1(d), and state its effect on the mean, median and range.
3. Which measure would you use, and why: (a) the most popular ice cream flavour (b) typical time to walk to school, where one student was delayed by a train (c) average marks on an evenly-spread test (d) shoe sizes a shop should order?
4. Donations: . (a) Calculate all three measures. (b) Which would a fundraiser quote? (c) Which is most representative? (d) What should be reported alongside it?
5. Explain why the mode is the only usable measure for data such as favourite colour.
6. A set of seven numbers has a mean of and a median of . Suggest a possible data set and explain the difference between the two measures.
7. Reasoning. Explain why the median is described as “resistant to outliers” but the mean is not.
8. Reasoning. A news report says “average household debt has risen to 95,000$.” What three questions should a careful reader ask?
9. Challenge. Construct a data set of six values where the mean, median and mode are all different, and state each.

Answers: Q1 — (a) mean , median ; either, no outliers (b) mean , median ; median, since is an extreme outlier (c) mean , median ; either (d) mean , median ; median. Q2 — (b) (d) ; each inflates the mean and range hugely, moves the median barely, and leaves the mode unchanged. Q3 — (a) mode (b) median (c) mean (d) mode. Q4 — (a) mean , median , mode and (b) the mean (c) the median (d) the total raised and a note about the 3001,2,3,4,5,6,491042, 3, 3, 5, 8, 15643$.