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.

Why statistics?

Statistics draws conclusions from data that vary

Descriptive statistics summarise the data we have: tables, graphs, averages. Most of today.
Inferential statistics use a sample to draw conclusions about a larger population, with a measure of uncertainty. Later modules.

Image: everyday statistics: an opinion poll, a weather forecast, a clinical trial

Today

Three questions run through today's lecture

Where do the data come from?
Populations, samples and study design
What kind of data are they?
Types of variables
What do the data say?
Graphs, centre, spread and shape

London, 1854

A map of deaths pointed to one water pump

During a cholera outbreak in Soho, John Snow marked each death on a street map.

The deaths clustered around one pump on Broad Street. The pump's handle was removed.

Image: John Snow's 1854 map of cholera deaths around the Broad Street pump

United States, 1936

2.4 million replies lost to 50,000 well-chosen people

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

The lesson of 1936

A huge biased sample is worse than a small, well-chosen one.

Where do the data come from?

We measure a sample to learn about a population

Memory aid: Population goes with Parameter; Sample goes with Statistic.

What kind of data are they?

Every variable is categorical or numerical

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

What kind of data are they?

20 kg is twice 10 kg, but 20 °C is not twice as hot as 10 °C

	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

Only a randomised experiment can show cause and effect

Observational study

Observe without assigning treatments. Association, not causation.

Experiment

Assign treatments; compare outcomes
Control group + random assignment

Where do the data come from?

Random methods give everyone a fair chance; convenience does not

Each panel: the same 20 people in 4 groups. Filled dots are the sample.

How do we choose a sample? Each panel shows the same population of 20 people in four groups, think of them as the four years of study. The filled dots are the people we pick.

Simple random sample: every group of n individuals has the same chance of being chosen, like drawing 100 student IDs at random.

Stratified sample: split the population into groups, called strata, and take a random sample from each, like 25 students from each year of study.

Cluster sample: split into groups and randomly choose whole groups, like picking 5 lecture sections at random and surveying everyone in them.

Systematic sample: choose every k-th individual from a list after a random start, like every 20th name on the class list.

Convenience sample: whoever is easiest to reach, like asking your friends in the cafeteria. It is not random. Avoid it.

A bigger sample does not fix bias

Selection bias. The way the sample is chosen leaves out part of the population.
Non-response bias. The people who don't respond differ from those who do.
Response bias. Wording or setting pushes an answer: "Don't you agree that…?"
Voluntary response bias. Only people with strong opinions take part.
Size is no cure. A bigger biased sample only looks more precise.

The type of data picks the graph

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	Relationship: direction, form, strength
A variable over time	Line graph	Trends

What do the data say?

Always plot the data before computing anything

Anscombe's quartet, 1973: 11 points each, mean of x 9, mean of y about 7.50, correlation about 0.816, line y = 3 + 0.5x.

What do the data say?

Describe a distribution by its shape, centre and spread

Then add unusual features: outliers and gaps.

What do the data say?

The mean shares out the total; the median is the middle

x̄ = Σx / n

x̄: the mean of the sample
Σx: the sum of all the values
n: how many values there are
Median: the middle of the ordered values
Mode: the most frequent value

Commutes (min): 12, 15, 15, 18, 20, 22, 25, 27, 30, 76. Sum 260, n = 10: mean 26, median 21, mode 15.

What do the data say?

One long commute pulls the mean, but barely moves the median

Use the median for skewed data such as house prices and incomes; the mean for roughly symmetric data.

What do the data say?

The range is 64 minutes, but the middle half spans only 12

64 minRange: 76 − 12

12 minIQR: 27 − 15, the middle 50%

18.5 minStandard deviation

What do the data say?

One outlier triples the standard deviation

s = √( Σ(x − x̄)² / (n − 1) )

s: the sample standard deviation
x − x̄: a value's distance from the mean
s²: the variance (before the root)
n − 1: divide by n − 1 for a sample

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

What do the data say?

The 76-minute commute lies beyond the cut-off: it is an outlier

Five-number summary 12, 15, 21, 27, 76. 1.5 × IQR = 18, so the cut-offs are −3 and 45.

What do the data say?

A z-score counts standard deviations from the mean

z = (x − mean) / SD

z: SDs above (+) or below (−) the mean
x: the value
mean: the mean of the data
SD: the standard deviation

Exam with mean 70, SD 8. A score of 86: z = (86 − 70) / 8 = 2, two SDs above the mean.

What do the data say?

In a bell curve, 68% lie within 1 SD, 95% within 2, 99.7% within 3

Exam scores, mean 70, SD 8. The rule needs roughly bell-shaped data; it does not fit skewed data like the commutes.

Correlation is not causation

Ice-cream sales and drownings rise together, but hot weather drives both

Other reasons for a correlation without causation: reverse causation, and coincidence 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 on exercise only asks people leaving a gym. What is the problem?

Response bias from the wording
Selection bias from a convenience sample
Non-response bias
The sample is too small

Check your understanding · 3 of 6

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

Mean 4, median 5
Mean 5, median 5
Mean 5, median 4
Mean 4, median 4

Check your understanding · 4 of 6

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

The mean
The median
Both
Neither

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 is strong
No: hot weather drives both
Yes: sales come first
No: drowning causes sales

You can now turn data into conclusions, despite variation

Collecting data

Population → parameter (μ, σ); sample → statistic (x̄, s)
Categorical or numerical; variable types and levels
Random samples + randomised experiments; bias matters

Describing data

Always plot first (Anscombe)
Centre: mean, median, mode. Spread: range, IQR, SD
Outliers: below Q1−1.5×IQR or above Q3+1.5×IQR
z = (x − mean) / SD; bell-shaped: 68–95–99.7
Correlation ≠ 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.

Where do the data come from: a population described by parameters, a sample described by statistics, and whether the sample and the study were random. What kind of data are they: categorical or numerical, and at what level.

What do the data say: plot first, then centre and spread, outliers with the 1.5 times IQR rule, z-scores, and the 68, 95, 99.7 rule for bell-shaped data. And never read cause into a correlation without looking for a confounder.

Above all: how data are collected matters more than how much there is. A huge biased sample loses to a small, well-chosen one.

Next time, Module 2: probability, the language of uncertainty. You came in today with no statistics, and you leave able to question the next poll you read. Well done, all of you.