- Definitions and Concepts

Statistics - Basic definitions and concepts

Use of statistics

• Design of experiment / sampling methods

• Data collection

• Organization of data (and graphing)

• Analysis

• Estimation of uncertainties

• Interpretation

Incorrect data causes

• Sampling bias

• Sample too small

• Sample not representative

• Leading question (also, people lie)

Data is correct, but presentation is misleading

• Cherry picking data

• Misleading graphs

Presentation of data

• Percentages can be misleading with small numbers

• Changes in absolute numbers can be misleading with large numbers

Leading question affects the data

Asking what makes a situation worse first is more likely to be voted first

Population

Collection of the entire set of all measurements of interest in a study

Ex. StatsCanada census where all Canadian households must respond to a survey

Parameter (relating to a population)

A numerical measurement that describes a population

Ex. Class average if you consider a class as an entire population

Sample

A smaller set of measurements taken from a population

Statistic (hint: relation to sample)

A numerical measurement describing a sample

Two types of stats

Descriptive statistics:

• A description of the data (sample or population)

• Find averages and dispersion

• Graph the data

Inferential statistics:

• An interpretation of sample data

• Given a sample, what are the properties of a population?

Inference

Inferring data when a population is inaccessible and the parameters are unknown. (guessing/determining results)

Types of Data

• Defined when experiment is designed

• Qualitative and Quantitative data

Qualitative Data

not numerical, also called categorical data

Quantitative Data

• numerical

• can be discrete: counting, gaps b/w values

• or continuous: measured, no gaps (size of gaps depends on precision of instruments)

Level of measurement

• Nominal: categorical data that cannot be ordered

  • Ex. colours (red, blue, purple)

  • brands (Ford, GM, Toyota)

• Ordinal: categorical data that has a natural order

  • Ex. high, low

  • large, medium, small

Level of measurements

• Interval: numerical data for which differences are significant but not ratios

  • Zero is arbitrary, it has no special meaning

  • Ex. temperature: 20C is not twice as hot as 10C

• Ratio: numerical data, differences and ratios are significant

  • Zero really means that nothing is measured

  • Ex. weight: 20 kg is twice as heavy as 10 kg

Level of measurement (Image order)

Identifying types of data (Image order)

Identifying types of data (examples)

•If we record the colour of the cars passing on a bridge in a day, the variable is qualitative nominal

•If we record the age of the cars in categories such as ‘new’, ‘fairly new’, ‘used’, ‘old’, and ‘antique’, the variable is qualitative ordinal

•The time between each car measured by a chronometer is quantitative continuous ratio

•The year of construction of the cars is quantitative discrete interval

Designing (a good) experiment

• Random sampling

  • All subgroups represented

  • No arbitrary selection

• Controlling effects

  • Placebo effect

  • Double blind experiment

• Repeatable

  • Any sample can be unrepresentative of the population by fluke

Simple sampling schemes

Complex sampling schemes

• Stratified sampling

  • Break the population in subgroups, f.ex. gender, country, age

  • Random selection in each subgroup

• Cluster sampling

  • Divide population in many subgroups

  • Randomly select clusters

  • Study all members of selected clusters

Sampling methods (image)

Designing an experiment

• It should be possible to pick any member of a population

• Every member should be equally likely to be picked (as much as possible)

• In many modern surveys of people (for politics for example) random sampling has been partially abandoned because it is too hard to achieve.

• Experiments that have known biases can still be valid if the biases are taken into account in the analysis.

Probabilistic Survey (Gallup)

• eg. random telephone #'s sample

  • landlines

  • cellphones

==> UNCERTAINTIES

Non-probabilistic survey

• eg. pool of 400 000 people

  • choose within those

==> NO UNCERTANTIES (but is not [completely] random)