By the end of this hour you can tell a sample from a population, pick a graph, and explain why correlation is not causation.
First-year course50 minutes
The plan
Three questions structure this hour
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
Statistics is the science of collecting, describing and drawing conclusions from data.
Where do the data come from?
Two branches: describe what we have, or infer beyond it
Descriptive statistics
Summarise the data we have: tables, graphs, averages. Most of this module.
Inferential statistics
Use a sample to draw conclusions about a larger population, with a measure of uncertainty. Later modules.
Story 1 · London, 1854
A simple picture of data pointed to the cause
During a cholera outbreak in Soho, John Snow marked each death on a street map. The deaths clustered around one water pump on Broad Street. The pump's handle was removed.
Image placeholder: John Snow's 1854 map of cholera deaths around the Broad Street pump.
Story 2 · The Literary Digest poll, 1936
A giant poll predicted the wrong winner
10Mballots mailed, about
2.4Mreturned, about
61%of the popular vote went to Roosevelt, about
46 of 48states won by Roosevelt
Takeaway
A huge biased sample is worse than a small well-chosen one
George Gallup, with a much smaller but better-chosen sample of about 50,000 people, correctly predicted Roosevelt's win. How data are collected matters more than how much there is.
Image placeholder: a 1936 newspaper front page about the Roosevelt landslide.
Where do the data come from?
A sample gives a statistic that estimates the population's parameter
Memory aid: Population – Parameter; Sample – 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?
Levels of measurement decide which arithmetic makes sense
Level
Order?
Equal intervals?
True zero?
Example
Nominal
no
no
no
eye colour
Ordinal
yes
no
no
race finishing position
Interval
yes
yes
no
temperature in °C
Ratio
yes
yes
yes
weight, income, time
Where do the data come from?
Only an experiment can show cause and effect
Observational study
Researchers observe and measure without assigning treatments, e.g. surveying students about sleep and grades. It can show an association, but not cause and effect.
Experiment
Researchers assign treatments and compare outcomes. Only a well-designed randomised experiment can establish cause and effect.
Where do the data come from?
A good experiment has four ingredients
Control group
for comparison
Random assignment
groups alike apart from the treatment
Placebo
where possible
Blinding
double-blind: neither subjects nor researchers know who got what
Where do the data come from?
Random sampling methods beat convenience
Method
How it works
Example
Simple random
every group of n has the same chance
draw 100 student IDs
Stratified
random sample from each stratum
25 students per year
Cluster
randomly choose whole groups
pick 5 lecture sections
Systematic
every k-th after a random start
every 20th name on a list
Convenience
whoever is easiest: avoid
friends in the cafeteria
Four ways a sample can mislead
Selection bias. Part of the population is left out.
Non-response bias. Non-responders differ from responders.
Response bias. Wording pushes people to an answer.
Voluntary response bias. Only strong opinions volunteer.
A bigger sample does not fix bias. It only makes a biased answer look more precise.
What do the data say?
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
direction, form, strength
A variable over time
line graph
trends
What do the data say?
Same summary numbers, four completely different pictures
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. Sketch of the four shapes.
What do the data say?
Describe every distribution by shape, centre, spread and unusual features
Skewed left is the mirror image of skewed right. Centre: the typical value. Spread: how much values vary. Unusual: outliers, gaps.
What do the data say? · Centre
Ten commute times: mean 26, median 21, mode 15
12, 15, 15, 18, 20, 22, 25, 27, 30, 76 minutes
26 minMean: sum 260 ÷ n = 10
21 minMedian: middle values 20 and 22
15 minMode: appears twice
What do the data say? · Centre
One outlier drags the mean but barely moves the median
Remove the 76-minute commute: the mean drops from 26 to about 20.4, the median only from 21 to 20.
Commute example, 10 students
With 76
Without 76
Mean
26
20.4
Median
21
20
What do the data say? · Spread
Range is fragile; IQR and SD tell the spread better
64 minRange: 76 − 12, very sensitive to outliers
12 minIQR: Q3 − Q1 = 27 − 15, the middle 50%
18.5 minStandard deviation s, about
What do the data say? · Spread
The standard deviation is the typical distance from the mean
s = √( Σ(x − x)² / (n − 1) )
x
the sample mean
n
how many values; we divide by n − 1 for a sample
What do the data say? · Outliers
The 1.5 × IQR rule flags the 76-minute commute as an outlier