Year 7StatisticsInterpretation of dataYear 7 Maths

Reading graphs, shape and skew

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.
Interpreting a data visualisation includes describing its variables and their units, the context for the data, and the visualisation's key 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

What does this graph tell you?

You can see one value is much bigger than the rest.
And that is genuinely all you can say.
Bigger what? Of whom? Measured in what?

Key idea

What reading a graph actually means

Name the variable and its units.
Say who the data came from, and when.
Then point at what stands out.

Say	Meaning	Here
The variable	what was measured	rainfall
The units	in what	millimetres
Who or where	the context	one town, one week
Key features	what stands out	one very wet day

Key idea

Now the same graph means something

Daily rainfall in millimetres, one town, one week.
Six days near zero and one at 46.
Same dots. Completely different amount of information.

Key idea

Look at the shape before anything else

The sides of a data set are called its tails.
If both taper the same, it is symmetrical.
If one runs on much further, it is skewed.

Shape	Tails	Averages
Symmetrical	both tails taper the same	mean ≈ median
Skewed right	a long tail to the right	mean above median
Skewed left	a long tail to the left	mean below median

Key idea

A symmetrical set

It falls away at the same rate on both sides.
The mean and the median land in the same place.
Both are 5. Either one describes it fairly.

Key idea

A skewed set

Most values are low, with a long tail running right.
The median is 3.5. The mean is 4.65.
The tail has dragged the mean towards it.

Key idea

The tail tells you which way

Tail to the right: the mean sits above the median.
Tail to the left: the mean sits below it.
You can say that by eye, before calculating.

Set	Mean	Median
Symmetrical set	mean 5	median 5
Right-skewed set	mean 4.65	median 3.5
The gap	the tail pulled the mean	+1.15

Key idea

So which average do you report?

Symmetrical: either. They agree.
Skewed: the median is the honest one.
Or report both and say why they differ.

Shape	Tails	Averages
Symmetrical	both tails taper the same	mean ≈ median
Skewed right	a long tail to the right	mean above median
Skewed left	a long tail to the left	mean below median

Key idea

The range is the fragile one

It is worked out from the two most extreme values.
And from nothing else at all.
So one outlier controls it completely.

Worked example

Range 46, on a week that mostly spanned 3

Six days between 0 and 3 mm.
The range says 46.
That is a fact about Friday, not about the week.

Data	Working	Spread
Six ordinary days	span 0 to 3 mm	spread 3
Add the storm	range is 46 − 0	spread 46
So the range	describes one day	not the week

Key idea

What to do about it

Do not delete the outlier. Say what it is doing.
"The range is 46, but that is one storm."
"The other six days spanned only 3 mm."

Data	Working	Spread
Six ordinary days	span 0 to 3 mm	spread 3
Add the storm	range is 46 − 0	spread 46
So the range	describes one day	not the week

Key idea

A finished answer is a sentence

Not a circled number.
Say what was measured, of whom, and what you found.
And quote the numbers as your evidence.

Part	Here
What	daily rainfall in millimetres
Who and when	one town, one week
Evidence	median 1 mm, mean 7.6 mm
The finding	six dry days and one storm

Worked example

Writing it out

"Most days that week were nearly dry."
"The median rainfall was 1 mm."
"One storm on Friday brought 46, lifting the mean to 7.6."

Part	Here
What	daily rainfall in millimetres
Who and when	one town, one week
Evidence	median 1 mm, mean 7.6 mm
The finding	six dry days and one storm

Key idea

Graphs in the news

The same rules apply, and they get broken more often.
Check the axis. Check the units. Check the source.
A graph with pieces missing may be missing them on purpose.

Missing	So you cannot tell
No axis numbers	you cannot tell how big anything is
No units	millimetres or metres: no way to know
Axis not from zero	small differences look enormous
No title	you do not know what it is about
No source	you do not know who was asked

Key idea

The axis that does not start at zero

On a time-series graph that is often the right choice.
On a bar graph comparing amounts, it is not.
It makes a small gap look like a huge one.

Key idea

Sometimes the honest answer is "I cannot tell"

If the units are missing, you cannot tell.
If the source is missing, you cannot tell who it is about.
Saying so is a correct answer, not a failure.

Missing	So you cannot tell
No axis numbers	you cannot tell how big anything is
No units	millimetres or metres: no way to know
Axis not from zero	small differences look enormous
No title	you do not know what it is about
No source	you do not know who was asked

Your turn

Your turn: four questions

Pause here and try all four.
Question 4 wants an actual sentence, written out.
Include the numbers as evidence.

	Question
1	A dot plot has a long tail to the right. Mean bigger or smaller than median?
2	A bar graph has no numbers on its vertical axis. What can you not tell?
3	Find the range of 5, 6, 6, 7, 8, 45. Is it a fair measure of spread?
4	Write one sentence reporting: median 1 mm, mean 7.6 mm, one day of 46 mm.

Show answer

1. Bigger. The tail drags the mean towards it, and the median barely moves. 2. How big anything is, or how big the differences between the bars are. You can see which is tallest and nothing more. 3. 40. It is not fair: 45 is an outlier, and without it the range is only 3. Report both, or say the spread is about 3 with one exceptional value. 4. Something like: "Most days that week were nearly dry (the median was 1 mm), but a single storm on Friday brought 46 mm, which lifted the mean to 7.6 mm."

Worked example

The answers

A right tail pulls the mean above the median.
With no axis numbers you cannot tell how big anything is.
The range is 40, and it is not fair: without the 45 it is 3.

	Answer
1	bigger: the tail drags the mean
2	how big anything is
3	40, and not fair: without the outlier it is 3
4	a sentence with the numbers in it as evidence

Summary

What to take away

Look at the shape first; it decides which average is honest.
A tail to the right pulls the mean above the median.
Finish with a sentence, and put the numbers in it.

	Remember
A finished answer	is a sentence, with evidence
Look at the shape first	it decides which average is honest
A tail to the right	pulls the mean above the median
The range	is the measure an outlier ruins
A graph missing labels	cannot be interpreted at all

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