108i - Range Median Mean and Mode

Lesson 108 — Calculating Range, Median, Mean and Mode

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

Learning Intentions

• To calculate the range, median, mean and mode of a data set.
• To understand what each measure describes.

Success Criteria

I can:

1. Calculate the range and explain what it measures.
2. Find the median, including when there is an even number of values.
3. Calculate the mean and explain it as a fair share.
4. Identify the mode, including when there is more than one or none.

Warmup

(6 minutes — order first, mini whiteboards)

The data: .

1. Write the values in order from smallest to largest.
2. What is the largest? The smallest?
3. Which value appears most often?
4. Which value sits exactly in the middle of your ordered list?

Answers: 1. ; 2. and ; 3. ; 4. .

The habit to establish now: order the data first, every time. Three of today’s four measures depend on it, and unordered data is the single biggest source of error in this topic.

Activities

Activity 1 — Explicit Instruction: the Four Measures (16 min)

Three of these describe the centre; one describes the spread.

1. Range — a measure of spread.

For : range .

Note: the range is a single number, not an interval. “The range is to ” is wrong; “the range is ” is right.

2. Median — the middle value of the ordered data.

Odd number of values: the middle one. Seven values → the th → .

Even number of values: the mean of the middle two. For (six values), the middle two are and :

The position rule. For values, the median sits at position . With : position ✓. With : position — between the rd and th, hence the average.

3. Mean — the fair share.

For :

The fair-share image: if all seven people pooled their sweets and redistributed them equally, each would get . The mean need not be a value in the data set, and need not be a whole number even for discrete data — a class averaging siblings is meaningful.

Connect to Lesson 91: finding a missing value from a mean means recovering the total first. Mean count total.

4. Mode — the most frequent value.

For : mode .

Three cases to name:

• One mode — the usual case.
• Two or more modes — e.g. has modes and (bimodal). Report both.
• No mode — if every value appears once, e.g. . Say “no mode”, not “zero”.

We do — full analysis of one set: .

Activity 2 — Calculation Circuit (14 min)

Pairs. Order first; show working; state all four measures.

Set A — small sets.

Set B — the missing value.

5. Four numbers have a mean of . Three are , and . Find the fourth.
6. Five test scores have a mean of . Four are . Find the fifth.
7. A data set of six values has a range of and a smallest value of . What is the largest?

Set C — in context.

8. Goals scored in seven matches: . Find all four measures and say what each tells the coach.
9. Class test scores out of : . Find all four measures.
10. Daily maximum temperatures (°C): . Find the range, median and mean.

Socratic scaffolding for Set B:

Prompt	Purpose
What does a mean of over four numbers tell you about the total?	Total .
What do the three known values total?	.
So the fourth?	.
Why does the mean hide the total?	It reports a share, not a sum — recovering the total is the first move every time.

(Answers: 1. ordered : range , median , mean , no mode. 2. ordered : range , median , mean , mode . 3. range , median , mean , mode . 4. ordered : range , median , mean , mode . 5. . 6. . 7. . 8. ordered : range , median , mean , mode . 9. ordered : range , median , mean , mode . 10. range , median , mean .)

Activity 3 — Inquiry: Build a Data Set (9 min)

Pairs — construction rather than calculation.

Create a data set of five whole numbers that satisfies each set of conditions. Some may be impossible — say so and explain.

1. Mean , median , mode .
2. Mean , median .
3. Mean , median , no mode.
4. Mean , range .
5. Median , mean , mode .
6. Mean , mode , range .

Socratic scaffolding:

Prompt	Purpose
Start from the total: what must five values sum to for a mean of ?	— always the first move.
For Q2, place the median first.	Middle value ; then arrange the rest to reach , e.g. .
For Q4, what does range force?	Every value identical: .
For Q6, range with mean — how much room is there?	All five values lie within a span of ; with mode the set must reach at least , so values sit in — total needs e.g. : mean ✓, mode ✓, range ✓.
Q5: median but mean — possible?	Yes: a large outlier drags the mean up, e.g. .

(Answers: 1. e.g. ; 2. e.g. ; 3. e.g. ; 4. only; 5. e.g. ; 6. e.g. .)

Checks for Understanding

(5 minutes — exit ticket, collected)

Data: .

1. Range.
2. Median.
3. Mean.
4. Mode.
5. Reasoning. Three numbers have a mean of . Two of them are and . Find the third, showing your reasoning.

Answers: ordered : 1. ; 2. ; 3. ; 4. ; 5. Total ; known ; third .

Common Misconceptions

Misconception	How to pre-empt it
Finding the median without ordering.	”Order first” as a stated habit; the warmup drills it.
Reporting the range as an interval.	The range is one number — a subtraction result.
For even , taking one of the middle two.	The averaging step is modelled and required.
Assuming the mean must be a data value.	The fair-share image; siblings is meaningful.
Saying the mode is when there is no mode.	”No mode” is the correct report.
Missing a second mode.	Bimodal sets included in the circuit; report all.
Trying to find a missing value without recovering the total.	Set B’s scaffolding — mean count total, every time.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Five numbers have a mean of . Four of them are . Find the fifth.

Answer

Total ; known ; fifth .

E2 (AMC Junior style). The mean of six numbers is . When one number is removed, the mean of the remaining five is . What number was removed?

Answer

Original total ; new total ; removed .

E3 (Challenge). A set of five whole numbers has median , mode , mean and range . Find a set that works.

Answer

Total ; median and mode both suggest at least two s with in the middle. Try : median ✓, mode ✓, total ✓, range ✓

E4 (Challenge). Adding the value to the set leaves the mean unchanged. Find .

Answer

Current mean . For the mean to stay at with five values, the total must be ; current total is , so — adding the mean itself never shifts the mean.

E5 (Challenge). A class of students has a mean test score of . A new student scores . Find the new mean, and comment on the effect of the outlier.

Answer

Old total ; new total ; new mean . One extreme score lifted the mean by a full mark — the mean is sensitive to outliers, which is exactly the issue Lesson 109 examines.

Homework

1. For each set, find the range, median, mean and mode: (a) (b) (c) (d) .
2. Six numbers have a mean of . Five of them are . Find the sixth.
3. Four test scores have a mean of . Three are . Find the fourth.
4. A data set has a range of and a largest value of . What is the smallest?
5. Goals in eight matches: . Find all four measures and say what each tells you.
6. Heights (cm): . Find the range, median, mean (1 d.p.) and mode.
7. Create a set of five whole numbers with: (a) mean , median , mode (b) mean , median (c) mean , no mode.
8. The mean of seven numbers is . When one is removed, the mean of the remaining six is . What was removed?
9. Reasoning. Explain why the mean of a discrete data set can be a decimal, using “number of siblings” as your example.
10. Reasoning. Explain why data must be ordered before finding the median but not before finding the mean.
11. Challenge. A set of five whole numbers has mean , median , mode and range . Find a set that works, and show it satisfies all four conditions.

Answers: Q1 — (a) ordered : range , median , mean , mode (b) ordered : range , median , mean , mode (c) range , median , mean , mode (d) ordered : range , median , mean , mode . Q2 — total ; known ; sixth . Q3 — total ; known ; fourth . Q4 — . Q5 — ordered : range , median , mean , modes and (all appear twice). Q6 — ordered : range , median , mean , mode . Q8 — original total ; new total ; removed . Q9 — the mean is a fair share, not a possible count; siblings describes the group, not any one family. Q10 — the median is defined by position in the ordered list; the mean depends only on the total, which order cannot change. Q11 — total ; e.g. : mean ✓, median ✓, mode ✓, range ✓