STATS 101 · Module 1

Data, samples and describing data

By the end of today you can say where data come from, what kind they are, and what they say: with a graph and a few numbers.

STATS 101Module 1 of the course50-minute lecture

Today we ask three questions of any data set

Where do the data come from? populations, samples, study design
What kind of data are they? types of variables
What do the data say? graphs, centre, spread, shape
Check your understanding six-question quiz

Image: a collage of everyday statistics: an opinion poll, a clinical trial of a new medicine, sports analytics, a weather forecast, a website A/B test

1 · Why statistics?

Statistics draws conclusions from data despite variation

Descriptive: summarise the data we have. Most of today.
Inferential: from a sample to a population, with a measure of uncertainty. Later modules.

Image: John Snow’s 1854 street map of Soho, with each cholera death marked and the Broad Street pump at the centre of the cluster

2 · Two stories about data

A map of deaths found the source of cholera

London, 1854. John Snow marked each cholera death in Soho on a street map.

The deaths clustered around one water pump on Broad Street. Its handle was removed.

2 · Two stories about data · 1936

A huge biased sample lost to a small well-chosen one

Result: Roosevelt won about 61% of the popular vote and 46 of the 48 states. Bars drawn to scale.

The bad story. 1936, the Literary Digest magazine. Click one: it mailed about 10 million ballots, drawn largely from telephone directories and car-registration lists, and about 2.4 million came back. That is the shaded part of the bar.

Click two: it predicted that Alf Landon would beat Franklin D. Roosevelt. Roosevelt won with about 61% of the popular vote and 46 of the 48 states.

Click three: George Gallup used a much smaller but better-chosen sample, about 50,000 people. That is the thin sliver on the second row, drawn to the same scale. He correctly predicted Roosevelt's win.

This is the main idea of the first half of the lecture: a huge biased sample is worse than a small well-chosen one. How data are collected matters more than how much there is. We will see exactly what went wrong when we reach bias.

3 · Population, sample, parameter, statistic

We study a sample to learn about a population

Memory aid: Population – Parameter; Sample – Statistic.

4 · Types of variables

Every variable is categorical or numerical

Watch out: a postcode or a student ID is categorical. Averaging them means nothing.

5 · Levels of measurement

Only ratio data have a true zero

	Nominal	Ordinal	Interval	Ratio
Order?	no	yes	yes	yes
Equal intervals?	no	no	yes	yes
True zero?	no	no	no	yes
Example	eye colour	race finishing position	temperature in °C	weight, income, time

20 °C is not “twice as hot” as 10 °C; 20 kg is twice 10 kg.

Where do the data come from?

How data are collected matters more than how much

Study design, sampling methods, and the biases that mislead.

6 · Observational studies and experiments

Only a randomised experiment can show cause and effect

An observational study only watches and measures, so it can show an association, not cause and effect.

Two ways to collect data. In an observational study, researchers observe and measure without assigning treatments, for example surveying students about sleep and grades. It can show an association, but not cause and effect.

In an experiment, researchers assign the treatments and compare outcomes. Click by click, the parts of a good experiment: subjects are assigned to groups at random, so the groups are alike apart from the treatment. One group is a control group for comparison, given a placebo where possible.

Then we compare the outcomes. Last click: blinding. Double-blind means neither the subjects nor the researchers know who got what.

The claim on the slide: only a well-designed randomised experiment can establish cause and effect. Hold on to that; it comes back at the very end with ice cream.

7 · Sampling methods

Pick a random method; avoid convenience samples

Every group of n has the same chance. Draw 100 student IDs at random.
A random sample from each group. 25 students from each year of study.
Randomly choose whole groups. Pick 5 lecture sections, survey everyone in them.
Every k-th after a random start. Every 20th name on the class list.
Whoever is easiest to reach: not random, avoid. Asking friends in the cafeteria.

A bigger sample does not fix bias

Selection: the sampling leaves people out (1936: poorer voters).
Non-response: those who don’t reply differ.
Response: “Don’t you agree that…?”
Voluntary response: online polls, call-in shows.
More data only makes a biased answer look more precise.

Bias is a systematic error that pushes results away from the truth in one direction. Four kinds to recognise.

Selection bias: the way the sample is chosen leaves out part of the population. This is what sank the Literary Digest: its phone and car-registration lists left out poorer voters in 1936.

Non-response bias: the people who don't respond differ from those who do. Remember the Digest got about 2.4 million of 10 million back. Response bias: the question wording or setting pushes people towards an answer, like 'Don't you agree that...?'. Voluntary response bias: only people with strong opinions choose to take part; online polls and call-in shows. Think back to the online poll some of you answered.

The last point is the thesis again, a second way: a bigger sample does not fix bias; it only makes a biased answer look more precise. 2.4 million biased replies lost to about 50,000 well-chosen ones.

What do the data say?

Plot first, then measure centre, spread and shape

Graphs, the mean and median, IQR and standard deviation, outliers, z-scores.

9 · Graphs

Match the graph to the data, and plot first

Data	Graph	What it shows
One categorical variable	bar chart (or pie chart)	counts or percentages per category
One numerical variable	histogram	shape of the distribution
One numerical variable	box plot	five-number summary and outliers
Two numerical variables	scatter plot	direction, form, strength
A variable over time	line graph	trends

9 · Anscombe’s quartet, 1973

Same summary numbers, four different pictures

Each set: 11 points, mean of x 9, mean of y ≈ 7.50, correlation ≈ 0.816, line y = 3 + 0.5x.

10 · Describing a distribution

Describe shape, centre, spread and anything unusual

Centre: where the typical value is. Spread: how much values vary. Unusual: outliers, gaps.

11 · Measures of centre

The outlier pulls the mean; the median resists

Mean x̄ = Σx / n = 260 / 10 = 26. Median: the middle of the ordered data, (20 + 22) / 2 = 21. Mode: most frequent.

Our data set from here on: commute times in minutes for 10 students: 12, 15, 15, 18, 20, 22, 25, 27, 30 and 76. Each dot is a student.

Click: the mode is the most frequent value, 15 minutes, which appears twice.

Click: the median is the middle value of the ordered data. With n = 10, an even number, we average the two middle values, 20 and 22, so the median is 21 minutes.

Click: the mean, x-bar equals sum of x over n. The sum is 260, n is 10, so the mean is 26 minutes. Notice it sits to the right of nine of the ten students: the 76 has dragged it.

Last click: remove the 76-minute commute. The mean drops to about 20.4; the median only to 20. The median is resistant to outliers; the mean is not. In a right-skewed distribution the mean is pulled above the median; in a left-skewed one, below. So use the median for skewed data such as house prices and incomes, and the mean for roughly symmetric data.

12 · Measures of spread

The IQR measures the middle half: 27 − 15 = 12 minutes

Commute times, ordered. Q1 is the median of the lower half; Q3 the median of the upper half.

12 · Standard deviation

One outlier triples the standard deviation

s = √( Σ(x − x)² / (n − 1) )

s: roughly the typical distance of values from the mean
n − 1: for a sample; the sample mean sits closer to the data than the true mean

Commute: variance ≈ 341.3, s ≈ 18.5 min. Without the 76: s ≈ 6.1 min.

13 · Five-number summary, box plots and outliers

Beyond the 45-minute fence, the 76-minute commute is an outlier

Five-number summary: 12, 15, 21, 27, 76. IQR 12, so 1.5 × IQR = 18: fences 15 − 18 = −3 and 27 + 18 = 45.

14 · z-scores

A z-score counts standard deviations from the mean

z = (x − mean) / SD

SD: standard deviation; z > 0 above the mean, z < 0 below
use: compare a hard test with an easy one

Exam mean 70, SD 8: z for 86 = (86 − 70) / 8 = 2

15 · The 68–95–99.7 rule

Bell-shaped data: 68% within 1 SD, 95% within 2, 99.7% within 3

Exam: mean 70, SD 8. Only for roughly bell-shaped data; not for skewed data like the commute times.

16 · Correlation is not causation

Ice cream does not cause drowning; hot weather drives both

Other non-causal reasons: reverse causation, coincidence (especially when many variables are searched).

Check your understanding · 1 of 6

Is “favourite colour” categorical or numerical, and at what level?

Categorical, nominal
Categorical, ordinal
Numerical, discrete
Numerical, continuous

Check your understanding · 2 of 6

A survey about exercise habits only asks people leaving a gym. What is the problem?

The sample is too small
Selection bias: gym-goers exercise more
Response bias from the wording
No problem: it is random

Check your understanding · 3 of 6

Find the mean and median of 2, 4, 4, 5, 10.

mean 5, median 4
mean 4, median 5
mean 5, median 5
mean 4, median 4

Check your understanding · 4 of 6

Which is resistant to outliers: the mean or the median?

The mean
The median

Check your understanding · 5 of 6

An exam has mean 70 and SD 8. What is the z-score of 82?

1
1.5
2
12

Check your understanding · 6 of 6

Ice-cream sales and drownings are correlated. Does ice cream cause drowning?

Yes: the correlation proves it
No: hot weather is a confounding variable
Yes, if the sample is big enough

Summary · next: Module 2, Probability

You can now ask three questions of any data set

Where from?

Population → parameter (μ, σ); sample → statistic (x̄, s)
Collect at random; more biased data doesn’t help

What kind?

Categorical: nominal, ordinal
Numerical: discrete, continuous

What do they say?

Plot first; mean vs median; IQR, SD
1.5 × IQR; z; 68–95–99.7
Correlation is not causation

Here is what you now have. Statistics turns data into conclusions in the presence of variation, and you can now ask the three questions of any data set.

Where from: population goes with parameter, mu and sigma; sample goes with statistic, x-bar and s. How data are collected matters most: random sampling and randomised experiments. A bigger biased sample does not help: the Literary Digest, 1936. That is the main idea a third time.

What kind: categorical, nominal or ordinal; numerical, discrete or continuous.

What they say: always plot first, remember Anscombe. Centre: the mean is sensitive, the median resistant. Spread: range, IQR, SD. Outliers lie beyond Q1 minus 1.5 IQR or Q3 plus 1.5 IQR. z equals x minus the mean over the SD; for bell-shaped data, 68, 95, 99.7. And correlation is not causation: look for confounders.

Next module: probability, the language of uncertainty. You now read data the way John Snow read his map. Well done today.