Statistical Questions & Distributions

A statistical question is one that expects a spread of answers, not a single one.

"How tall am I?" and "How tall is a sixth-grader?"

The first has one answer. Measure once and you are finished.

two questions that look alike how tall am I? 148 cm one answer how tall is a sixth-grader? a spread of answers only the second one needs statistics

The second does not. Sixth-graders come in many heights, and no single number is the answer.

You could measure a hundred of them and still not have one number — you would have a shape.

That shape is a distribution, and questions that produce one are called statistical. Everything in this unit exists because the answer is a spread rather than a value.

A small random sample drawn from a large population population — 900 students sample — 60 students 18 of 60 walk → about 30%

Spotting a statistical question

Ask whether the answers would vary. If they would, the question is statistical.

"How old am I?" is not. "How old are the people in my street?" is.

The test is about the answers, not about how the question is worded.

Would the answers differ?

One answer means arithmetic; many answers mean statistics. That single question sorts them every time.

Describing a distribution

Three things are worth saying: where the centre is, how spread out it is, and what shape it has.

Shape covers whether it is symmetric, lopsided, or has two humps.

Anything unusual — a gap, or a value far from the rest — deserves a mention too.

Centre, spread, shape

Three words cover almost every description question. Answering with only the average leaves two-thirds unsaid.

Dot plots and histograms

A dot plot puts one dot per value above a number line, so nothing is hidden.

A histogram groups values into intervals and draws a bar for each, which suits larger data sets.

Histogram bars touch, because the intervals run into each other; bar-chart bars do not.

Not the same as a bar chart

A bar chart compares separate categories; a histogram shows a distribution across a number line. The touching bars are the visible difference.

Shape

Symmetric means the two halves roughly mirror each other.

Skewed means a long tail to one side, which drags the mean towards it.

Two humps usually mean two different groups have been mixed together.

What two humps mean

A double-humped plot of heights often means two age groups in one data set. The shape is telling you the data should be split.

Outliers

An outlier sits far from everything else, and it can be a real value or a recording error.

Either way it should be noticed and mentioned, not silently deleted.

It pulls the mean noticeably while barely moving the median.

Investigate, do not delete

A height of 2 cm is a typing slip; a salary ten times the rest may be perfectly real. The data cannot tell you which — only checking can.

Worked examples

Statistical or not

Which is a statistical question: "How many pets do I have?" or "How many pets do students in my class have?"

  1. The first has one answer, found by counting once. No variability.
  2. The second varies from student to student. Different answers from different people.
  3. So the second is statistical. It produces a distribution rather than a number.

The second

Describing a plot

A dot plot of test scores clusters near 80, spreads from 70 to 90, and has one dot at 30. Describe it.

  1. Centre: around 80, where most dots sit. Where the data piles up.
  2. Spread: most from 70 to 90, so about 20 points wide. Ignoring the stray value for now.
  3. The dot at 30 is an outlier, far from the rest and worth checking. Centre, spread, shape, then anything unusual.

Centred near 80, spread 70 to 90, with an outlier at 30

Reading a histogram

A histogram has bars of height 2, 5, 9, 5, 2 across five intervals. What shape is it?

  1. The tallest bar is in the middle. That is where the centre sits.
  2. The heights fall away evenly on both sides: 5, 2 each way. The halves mirror each other.
  3. So the distribution is symmetric. No long tail in either direction.

Symmetric

Words to know

Statistical question
One that anticipates variability in the answers.
Data
The collected answers or measurements.
Distribution
The pattern of how the values are spread out.
Dot plot
A chart with one dot per value, stacked above a number line.
Outlier
A value far away from the rest.

Mistakes to watch for

Calling any data question statistical. "How tall am I?" has one answer, so it is not — the test is whether the answers would vary.

Describing a distribution with only its average. Centre, spread and shape are three separate things, and two of them are missing from a lone mean.

Confusing a histogram with a bar chart. Histogram bars touch because the intervals are next to each other on a number line; bar-chart categories are separate.

Deleting an outlier because it looks wrong. It may be a recording error or a genuine extreme value, and only checking the source can tell you which.

Ignoring two humps. A double-peaked plot usually means two different groups have been combined, and the average of the whole describes neither of them.

Florence Nightingale invented a new kind of chart in 1858 to show that most soldiers in the Crimea died of disease rather than wounds. The picture changed policy where tables of numbers had not.

A dot plot is often the most honest chart there is: every single data point is visible, and nothing is averaged away.

The word "statistics" comes from "state" — the subject began as the counting of populations for governments.

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