1. Overview; Variables, Constants, Tables & Graphs Dr Gwilym Pryce
Faculty of Social Sciences Induction Block: Maths & Statistics Lecture 1
Overview; Variables, Constants, Tables & Graphs
Dr Gwilym Pryce

2. Aims and Objectives of the Maths & Stats Induction
Aim: to revise basic maths relevant to the course.
Objectives: by the end of the Induction Programme students should be able to:
Understand the meaning and types of variables and constants
Understand how to graph scale and categorical variables
Be familiar with basic algebraic notation
Understand the simple mathematical representation of relationships, both algebraically and graphically
Understand the basic principles and laws of probability
Outline the main issues surrounding sampling.

3. Why do social scientists need to learn about statistics?
Theories have to be verified empirically otherwise they remain conjectures
Need for evidenced based practice & policy:
medicine
public health
economics
informed decisions better than uninformed decisions
information is complex and needs summarising in a way that reflects the underlying data in a meaningful way

4. Why do we need mathematics?
Statistics can be represented in a non-mathematical way, but some understanding and application of maths will help us:
spoken language can be ambiguous varies across countries and cultures

5. Different cultures find different things funny
Different cultures and languages express ideas differently
But mathematical notation is:
unambiguous and concise
common notation is understood across cultures and languages
Research & ideas expressed mathematically can easily reach an international audience

6. Plan of Maths & Stats Induction
Lecture 1: Variables, Constants, Tables & Graphs
Lecture 2: Algebra and Notation
Lecture 3: Precise and Approx Relationships between variables
Lecture 4: Probability
Lecture 5: Inference
Lecture 6: Hypothesis tests
Tutorial: Samples and populations; Validity and Reliability

7. Plan of Maths & Stats Lecture 1: Variables and Constants
1. What is a variable?
2. What is a constant?
3. Types of variables
4. Graphs of single variables
Why summarise?
Tables & graphs of categorical data
Tables & Graphs of Continuous / Quantitative/Scale variables

8. 1. What is a variable?
A measurement or quantity that can take on more than one value:
E.g. size of planet: varies from planet to planet
E.g. weight: varies from person to person
E.g. gender: varies from person to person
E.g. fear of crime: varies from person to person
E.g. income: varies from HH to HH
I.e. values vary across ‘individuals’ = the objects described by our data

9. Individuals = basic units of a data set whom we observe or experiment on in a controlled way
not necessary persons
(could be schools, organisations, countries, groups, policies, or objects such as cars or safety pins)
Variables = information that can vary across the individuals we observe
e.g. age, height, gender, income, exam scores, whether signed Nuclear Test Ban Treaty

10. 2. What is a constant?
A measurement or quantity that has only one value for all the objects described in our data
Also called a ‘scalar’ or ‘intercept’ or ‘parameter’
E.g. speed of light in a vacuum: constant for all light transmissions
E.g. ratio of diameter to circumf.: constant for all circles
E.g. ave. increase in life expectancy: constant at 1 year pa since 1900

11. Often it is a constant that want to estimate:
we employ statistical techniques to estimate ‘parameters’ or ‘constants’ that summarise or link variables.
e.g. mean = ‘typical’ value of a variable = measure of central tendency
e.g. standard deviation = measure of the variability of a variable = measure of spread
e.g. correlation coefficient = measures the correlation between two variables
e.g. slope coefficients = how much y increases when x increases

12. 3. Types of variables:
Numeric = values are numbers that can be used in calculations.
String = Values are not numeric, and hence not used in calculations.
But can often be coded: I.e. transformed into a numerical variable:
e.g. If (country = ‘Argentina’) X = 1.
If (country = ‘Brazil’) X = etc.

13. Scale or quantitative Variables = data values are numeric values on an interval or ratio scale
(e.g., age, income). Scale variables must be numeric.
E.g. dimmer switch: brightness of light can be measured along a continuum from dark to full brightness
Categorical Variables = variables that have values which fall into two or more discrete categories
E.g. conventional light switch: either total darkness or full brightness, on or off.
Male or female, employment category, country of origin

14. Two types of Ordinal variables:
Ordinal variables = Data values represent categories with some intrinsic order
(e.g., low, medium, high; strongly agree, agree, disagree, strongly disagree).
Ordinal variables can be either string (alphanumeric) or numeric values that represent distinct categories (e.g., 1=low, 2=medium, 3=high).

15. Ordinal variables: Values fall within discrete but ordered categories
I.e. the sequence of categories has meaning
e.g. education categories:
1 = primary
2 = secondary
3 = college
4 = university undergraduate
5 = university postgraduate masters
6 = university postgraduate phd
e.g. 1= Very poor, 2= poor, 3=good, 4=very good

16. Nominal variables
Nominal Variables = Data values represent categories with no intrinsic order
sequence of categories is arbitary -- ordering has no meaning in and of itself:
e.g. country of origin: Wales, Scotland, Germany…
e.g. make of car: Ford, Vauxhall
e.g. job category
e.g. company division
Nominal variables can be either string (alphanumeric) or numeric values that represent distinct categories (e.g., 1=Male, 2=Female).

17. 4. Graphs of Variables: Why summarise?
Tables & graphs of categorical data
Tables & Graphs of Continuous / Quantitative/Scale variables

18. Why Summarise? Small data sets can be presented in their entirety
e.g. if only have 10 observations and 3 variables, can list all data
but even then we might want to know what is the typical value of a variable
Large data sets require summary
Lots of information can be confusing, particularly if numerical
most of us need headline figures or stylised facts to be able to absorb information.

19. Graphical summaries: Summary statistics:
allow us to visualise the distribution of data across different values or categories
how many (or what proportion) of cases fall within certain categories or ranges of values?
Summary statistics:
describe the distribution of a single variable

20. Tables of Categorical Data
Categories are listed either in columns or rows (respecting order if ordinal)
Count or % of cases in each category listed
If number of categories is large, may be useful to group categories together:
e.g. Country of origin ---> collapse to continents
Good tables:
give clear messages: tell a story
too much info in a table defeats its purpose
Source always given

23. Graphs of Categorical Data
Pie Charts
If all the categories sum to a meaningful total, then you can use a pie chart
Pie charts emphasise the differences in proportions between categories
OK for a single snapshot, but not very good for showing trends
would need to have a separate pie chart for each year

24. What’s missing?

25. Bar Charts can show either % or count
not very good for showing trends in more than one category

30. Beware of scaling...

33. Beware of small print...

34. Tabulating and Graphing Scale Data
Scale or quantitative data: usually a measurement of size or quantity
not meaningful to report % or count unless break into categories (& then it becomes categorical data!)
e.g. income
Tables of raw data not much use unless only a few values...

35. How tabulate 129,000 observations?

36. What are we interested in when describing the income data?
Is income evenly spread?
Or are most people rich?
Or are most people poor?
Or are most reasonably well off?
This are all questions about the variable’s Distribution
We can represent the whole data set with one picture...