Year 8StatisticsInterpretation of dataYear 8 Maths

Reading a graph properly: shape, centre and spread

What this lesson covers

The response to a statistical question includes findings that are summarised and interpreted in context and using evidence.
The tapering sides of a data visualisation are known as tails and may taper at the same rate, producing a symmetrical shape, or an uneven rate, producing a skewed shape. In positively skewed data, the right-tail tapers more slowly than the left tail. In negatively skewed data, the left tail tapers more slowly than the right tail.
Interpreting a data visualisation includes describing its variables and their units, the context for the data, and the visualisation's key features: its shape, its central tendency, its spread, and other features.
A graph that is missing parts (e.g. title, axis labels, axis scales) or has errors may have been made to be misleading or to hide information.

Key idea

Two different jobs

"The median is 24.5." That is a summary. It is a number.
"A typical foot is about 24.5 centimetres." That is an interpretation.
Same fact. Only the second one says anything.

Job	Looks like	Produces
Summarising	the median is 24.5	a number
Interpreting	a typical foot is about 24.5 cm	a sentence
The outcome	asks for the second

Key idea

What to say about any graph

What was measured, in what units, and who from.
Then the shape, the centre and the spread.
Work through that list and you have an interpretation.

Ask	Which is
What	the variable, and its units
Who	who the data came from
Shape	symmetrical, or skewed which way
Centre	where the typical value sits
Spread	how far it runs, and any outliers

Key idea

In context means with the noun attached

"The range is 7" is not a finding. Seven what?
"Foot lengths vary by up to 7 cm across our class" is.
Units and context, every time, or it does not count.

Written	Verdict
The range is 7	no units, no context	not a finding
The mean is 25	the mean of what?	not a finding
Feet vary by up to 7 cm	says what and how much	a finding

Key idea

Now the shape of the data

Plot it and the dots make a shape.
The thin ends are called the tails.
How those tails taper is what you describe.

Key idea

Both tails the same: symmetrical

Fold it down the middle and the halves match.
2, 3, 4, 4, 4, 5, 6. One peak, even sides.
Mean 4, median 4. They agree, which is the giveaway.

Key idea

One tail longer: skewed

Now 1, 2, 2, 3, 3, 3, 4, 5, 8, 12.
Most of it is bunched low, with a few stragglers high up.
The right tail tapers slowly. This is positive skew.

Key idea

And the other way round

1, 5, 8, 9, 10, 10, 10, 11. Bunched high this time.
The long tail runs off to the left.
Left tail tapers slowly, so this is negative skew.

Key idea

The name comes from the tail

Positive skew has most of its values on the LEFT.
That feels backwards, and it catches almost everybody.
Name it from where the tail points, never from the bulk.

Feature	Where	Name
Most values	on the left
Long tail	to the right
Named after	the tail, never the bulk	positive skew

Key idea

You can check it with arithmetic

A long right tail drags the mean up but not the median.
So mean above median means skewed right.
Mean below median means skewed left. Same idea as an outlier.

Data	Summaries	Shape
2,3,4,4,4,5,6	mean 4, median 4	symmetrical
1,2,2,3,3,3,4,5,8,12	mean 4.3, median 3	positive skew
1,5,8,9,10,10,10,11	mean 8, median 9.5	negative skew

Key idea

Which centre to report, then

On our skewed set, the mean is 4.3 and the median is 3.
But 4.3 is bigger than seven of the ten values.
So the median describes a typical value better. Use it.

Key idea

Then the spread

The range is the highest value minus the lowest.
For the skewed set that is 12 − 1, which is 11.
But look at how it was built: only two values decided it.

Key idea

Which makes it fragile

Take 3, 4, 4, 5 and 40. The range is 37.
Drop the 40 and the range is 2.
So 37 is describing the outlier, not the rest of the data.

Data	Working	Range
3, 4, 4, 5, 40	range 40 − 3	37
Without the 40	range 5 − 3	2
So the range	describes the outlier	not the data

Key idea

So say what you saw

Spot outliers by eye before you quote a range.
Then say: the range is 37, but one value is far out.
Most of the data runs from 3 to 5. Both facts, honestly.

Report	Verdict
The range is 37	true and misleading	incomplete
Most run 3 to 5, one is 40	both facts	honest

Key idea

Sometimes you cannot interpret at all

No scale on the axis? The bars cannot be compared.
No units? The numbers do not mean anything.
No sample size? You cannot weigh the evidence.

Missing	Consequence
No scale on the axis	bars cannot be compared
No units	the numbers mean nothing
No sample size	you cannot judge the evidence
Axis not from zero	differences are exaggerated

Key idea

And saying so is the right answer

Students treat "I cannot tell" as failing to answer.
It is not. Name what is missing and why it matters.
That is a full answer, and it is the honest one.

Response	Verdict
Guessing anyway	invents information	wrong
"There is no scale, so the bars cannot be compared"	names the gap	right

Key idea

Which is worth doing outside class

Graphs turn up in the news constantly.
Run the same checks: units, scale, who was asked, how many.
A graph that will not survive those questions is not evidence.

Ask	Which is
What	the variable, and its units
Who	who the data came from
Shape	symmetrical, or skewed which way
Centre	where the typical value sits
Spread	how far it runs, and any outliers

Your turn

Your turn: three questions

The data is 1, 2, 2, 3, 3, 3, 4, 5, 8, 12. Pause here.
Question 1 wants a direction, so check which way the tail runs.
Question 3 has a short honest answer.

	Question
	The data: 1, 2, 2, 3, 3, 3, 4, 5, 8, 12
1	Describe its shape.
2	Mean 4.3, median 3. Which summarises it better, and why?
3	A bar graph has no scale on its vertical axis. What can you say?

Show answer

1. Positively skewed. Most values are bunched at the low end and there is a long tail running right to 12. 2. The median, 3. The mean of 4.3 is above almost every value in the set, because the 8 and the 12 have pulled it up. 3. Very little. You can see which bar is tallest, and nothing about by how much. A graph with no scale cannot be interpreted, and saying so is the correct answer rather than a refusal to answer.

Worked example

The answers

Positively skewed: bunched low, long tail out to 12.
The median, 3. The mean of 4.3 is above seven of the ten values.
And with no scale, you can say which bar is tallest and no more.

	Answer
1	positively skewed
2	the median, 3
3	it cannot be interpreted

Summary

What to take away

A number is a summary. A sentence with units is a finding.
Skew is named after the tail, never after where the bulk sits.
And a graph you cannot check is a graph you should not believe.

	Remember
Summarise then interpret	a number, then a sentence
Say the shape	symmetrical, or skewed which way
Skew is named	after the tail, not the bulk
Mean above median	means a tail to the right
A range	is built from the two most extreme values

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