STATS 101 · Module 1

Data, samples and describing data

By the end you can tell a population from a sample, choose a graph, describe a data set, find its outliers, and explain why correlation is not causation.

Lecture, 50 minutesFirst year, no statistics background

Three questions, then a quiz

1. Where do the data come from? about 15 min
2. What kind of data are they? about 7 min
3. What do the data say? about 22 min
4. Quiz and summary about 5 min

Why statistics?

Statistics is the science of collecting, describing and drawing conclusions from data.

It works in the presence of variation and uncertainty.

Part 1 · Where do the data come from?

A map of deaths pointed to the cause

In 1854 John Snow marked each cholera death on a street map. The deaths clustered around one water pump on Broad Street. The pump's handle was removed.

Part 1 · Where do the data come from?

A huge biased sample is worse than a small, good one

• 1936: about 10 million ballots mailed, about 2.4 million returned.
• Drawn largely from telephone directories and car-registration lists.
• It predicted Landon would beat Roosevelt. Roosevelt won about 61% of the popular vote.
• George Gallup's sample of about 50,000 correctly predicted Roosevelt's win.

Part 1 · Vocabulary

We want to know the population; we only measure a sample

Part 1 · Sampling

Random samples spread across the population; convenience samples clump

Part 1 · Sampling methods

Four methods use chance; convenience sampling is the one to avoid

Part 1 · Bias

Bias: four ways a sample misleads, and size fixes none of them

1. Selection bias

   The way the sample is chosen leaves out part of the population.

2. Non-response bias

   The people who do not respond differ from those who do.

3. Response bias

   Question wording or setting pushes people towards an answer.

4. Voluntary response bias

   Only people with strong opinions choose to take part.

5. A bigger sample does not fix bias

   It only makes a biased answer look more precise.

Part 1 · Study design

Only a randomised experiment can show cause and effect

1. Observational study

   Researchers observe without assigning treatments, e.g. surveying students about sleep and grades. It shows association, not cause and effect.

2. Experiment

   Researchers assign the treatments and compare outcomes, with a control group, random assignment, and a placebo and blinding where possible.

Part 2 · Types of variables

Every variable is either categorical or numerical

Part 2 · Levels of measurement

Only ratio data have order, equal intervals and a true zero

Why it matters

20 °C is not twice as hot as 10 °C. 20 kg is twice 10 kg.

Interval data have no true zero; ratio data do.

Part 3 · Graphs

Plot before you compute, using the graph that fits the variable

Part 3 · Always plot first

Same summary numbers, four completely different pictures

Anscombe's quartet (1973): four data sets of 11 points each

Part 3 · Shape and centre

Right skew pulls the mean above the median

Part 3 · Centre

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

Σx

add all the values

n

how many values

Median

middle of the ordered data; average the two middle values if n is even

Mode

the most frequent value

Worked example · commute times

Ten commutes give three different centres

Commute times in minutes, 10 students: 12, 15, 15, 18, 20, 22, 25, 27, 30, 76

Part 3 · Outliers and centre

One 76-minute commute moves the mean, but barely the median

Part 3 · Spread

The outlier stretches the range; the IQR covers the middle 50%

Quartiles: lower half 12, 15, 15, 18, 20 gives Q1 = 15; upper half 22, 25, 27, 30, 76 gives Q3 = 27

Part 3 · Standard deviation

One 76-minute outlier triples the standard deviation

x̄

the sample mean

n − 1

for a sample we divide by n − 1, not n; this corrects for the sample mean sitting closer to the data than the true mean

With the 76: s ≈ 18.5 min. Without it: s ≈ 6.1 min.

Part 3 · Outliers

The 1.5 × IQR rule flags the 76-minute commute

Part 3 · z-scores

A z-score says how many SDs a value sits from the mean

x

the value we are looking at

SD

the standard deviation of the data

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

Part 3 · Bell-shaped data

For bell-shaped data, about 68%, 95% and 99.7% lie within 1, 2 and 3 SDs

Part 3 · Correlation

Ice cream does not cause drownings; hot weather drives both

Quiz · 1 of 3

Check your understanding

1. Is "favourite colour" categorical or numerical, and at what level?

   Categorical, nominal

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

   Selection bias from a convenience sample: gym-goers exercise more than the population.

Quiz · 2 of 3

Check your understanding

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

   Mean 5, median 4

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

   The median

Quiz · 3 of 3

Check your understanding

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

   1.5, since (82 − 70) / 8 = 1.5

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

   No. Hot weather is a confounding variable.

Summary

Collect carefully, plot first, then describe with centre and spread

1. Design matters more than size

   Random sampling and randomised experiments. A bigger biased sample does not help.

2. Know the variable

   Categorical (nominal, ordinal) or numerical (discrete, continuous).

3. Plot before you compute

   Anscombe's quartet: the same summary numbers, four different pictures.

4. Centre and spread

   Mean (sensitive) or median (resistant); range, IQR, SD; outliers beyond 1.5 × IQR.

5. Correlation is not causation

   Look for confounders. Next module: probability, the language of uncertainty.