1 Faculty of Social Sciences Induction Block: Maths & Statistics Lecture 1 Overview; Variables,...

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Faculty of Social Sciences Induction Block: Maths & Statistics Lecture 1

Overview; Variables, Constants, Tables & Graphs

Dr Gwilym Pryce

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 = 2. etc.

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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

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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).

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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

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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).

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4. Graphs of Variables:

Why summarise? Tables & graphs of categorical data Tables & Graphs of Continuous /

Quantitative/Scale variables

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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.

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Graphical summaries:– 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

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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

Tables of Categorical Data

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Income Support claimants with housing costs by statistical group:May 1999

Total(All

Claimantswith

MortgageInterest)

Aged 60 orover

LoneParents

Disabled Other

000s 000s % 000s % 000s % 000s %1999 263 93 35 67 25 88 33 15 6

DSS Quarterly Statistical Enquiry

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Income Support claimants with housing costs by statistical group:May 1993 to May 1999

Total(All

Claimantswith

MortgageInterest)

Aged 60 orover

LoneParents

Disabled Other

000s 000s % 000s % 000s % 000s %1993 284 88 31 107 38 63 22 27 101994 310 96 31 115 37 74 24 25 81995 329 102 31 115 35 87 26 25 81996 322 103 32 104 32 89 28 25 81997 301 98 33 92 31 89 30 22 71998 281 96 34 78 28 90 32 17 61999 263 93 35 67 25 88 33 15 6


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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

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Income Support claimants with housing costs by statistical group in May 1999

Aged 60 or over36%

Lone Parents25%

Disabled33%

Other6%

What’s missing?

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Bar Charts– can show either % or count– not very good for showing trends in more

than one category

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0

10

20

30

40

50

60

70

80

90

100

Aged 60 or over Lone Parents Disabled Other

Category of Claimant

00

0's


27


0

10

20

30

40

50

60

70

80

90

100

Aged 60 or over Lone Parents Disabled Other

Category of Claimant

00

0's


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1993 19941995

19961997

19981999

Other 000s

Disabled 000s

Lone Parents 000s

Aged 60 or over 000s0

20

40

60

80

100

120

000's

Year

Category

Income Support claimants with housing costs by statistical group: May 1993 to May 1999


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0

20

40

60

80

100

120

140

1993 1994 1995 1996 1997 1998 1999Year

000's

Other 000s

Disabled 000s

Lone Parents 000s

Aged 60 or over 000s

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Beware of scaling...

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60

70

80

90

100

110

120

1993 1994 1995 1996 1997 1998 1999Year

000'sLone Parents 000s

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0

20

40

60

80

100

120

140

160

180

200

1993 1994 1995 1996 1997 1998 1999

Year

000'

s

Lone Parents 000s

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Beware of small print...Notes:1. Numbers are based on a 1 per cent sample up to May 1994, and a 5 per cent sample thereafter.Numbers are subject to a degree of sampling error.2. Children are defined as aged 0 to 15.3. Single with children includes all single people with a child aged between 0 and 15 years of age, andwill include claimants regardless of client group i.e. may include claimants normally classified as"disabled" or "pensioners". Numbers will exclude some single people who onlyhave dependants aged 16 to 19.4. Income Support for the unemployed was replaced by Income-based Jobseeker's Allowance inOctober 1996. To provide a consistent time-series, the unemployed have been removed from thefigures quoted in these tables from 1993 to August 1996.Source: Income Support Statistics Quarterly Enquiries, May 1993 to May 2000.

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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...

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How tabulate 129,000 observations?CM SML 1988 CM SML 1988 CM SML 1988 CM SML 1988

Borrower Total Income Borrower Total IncomeBorrower Total IncomeBorrower Total Income1 . 21 10800 41 . 61 .2 . 22 . 42 7216 62 .3 . 23 19072 43 . 63 .4 . 24 . 44 12000 64 .5 . 25 . 45 9758 65 .6 . 26 . 46 6084 66 .7 . 27 . 47 . 67 .8 . 28 . 48 . 68 .9 . 29 . 49 . 69 18336

10 . 30 . 50 9345 70 1509611 . 31 . 51 9810 71 .12 . 32 . 52 14406 72 1259713 . 33 . 53 9190 73 970014 . 34 . 54 . 74 .15 18720 35 . 55 . 75 .16 16000 36 . 56 . 76 .17 16455 37 . 57 . 77 .18 . 38 11500 58 . 78 529519 7020 39 2912 59 . 79 453920 4576 40 11745 60 . 80 .

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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...

37TOTAL INCOME OF BORROWER(S)5

70

00

.0 -

58

50

0.0

54

00

0.0

- 5

55

00

.05

10

00

.0 -

52

50

0.0

48

00

0.0

- 4

95

00

.04

50

00

.0 -

46

50

0.0

42

00

0.0

- 4

35

00

.03

90

00

.0 -

40

50

0.0

36

00

0.0

- 3

75

00

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30

00

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34

50

0.0

30

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0.0

- 3

15

00

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70

00

.0 -

28

50

0.0

24

00

0.0

- 2

55

00

.02

10

00

.0 -

22

50

0.0

18

00

0.0

- 1

95

00

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50

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16

50

0.0

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00

0.0

- 1

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00

.09

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0.0

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50

0.0

30

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45

00

.00

.0 -

15

00

.0

12000

10000

8000

6000

4000

2000

0

Std. Dev = 12830.02

Mean = 17993.3

N = 125541.00

38LTV16

.75

- 17

.25

15.7

5 -

16.2

514

.75

- 15

.25

13.7

5 -

14.2

512

.75

- 13

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11.7

5 -

12.2

510

.75

- 11

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9.75

- 1

0.25

8.75

- 9

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7.75

- 8

.25

6.75

- 7

.25

5.75

- 6

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4.75

- 5

.25

3.75

- 4

.25

2.75

- 3

.25

1.75

- 2

.25

.75

- 1.

25-.

25 -

.25

LTV Frequency Distribution

All HHs in Low Price Areas

(1995-1998 CML SML Data)

Fre

quen

cy

60000

50000

40000

30000

20000

10000

0

Std. Dev = .25

Mean = .80

N = 74736.00

39LTV

1.45

- 1

.50

1.40

- 1

.45

1.35

- 1

.40

1.30

- 1

.35

1.25

- 1

.30

1.20

- 1

.25

1.15

- 1

.20

1.10

- 1

.15

1.05

- 1

.10

1.00

- 1

.05

.95

- 1.

00.9

0 -

.95

.85

- .9

0.8

0 -

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.75

- .8

0.7

0 -

.75

.65

- .7

0.6

0 -

.65

.55

- .6

0.5

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.55

.45

- .5

0.4

0 -

.45

.35

- .4

0.3

0 -

.35

.25

- .3

0.2

0 -

.25

.15

- .2

0.1

0 -

.15

.05

- .1

00.

00 -

.05




Fre

quen

cy

30000

20000

10000

0

Std. Dev = .22

Mean = .80

N = 74552.00

40LTV

.95

- 1.

00.9

0 -

.95

.85

- .9

0.8

0 -

.85

.75

- .8

0.7

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0.6

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- .6

0.5

0 -

.55

.45

- .5

0.4

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.45

.35

- .4

0.3

0 -

.35

.25

- .3

0.2

0 -

.25

.15

- .2

0.1

0 -

.15

.05

- .1

00.

00 -

.05




Fre

qu

en

cy

30000

20000

10000

0

Std. Dev = .22

Mean = .78

N = 70545.00

41LTV1.

00 -

1.5

0

.50

- 1.

00

0.00

- .

50




Fre

quen

cy

70000

60000

50000

40000

30000

20000

10000

0

Std. Dev = .22

Mean = .80

N = 74552.00

1 Faculty of Social Sciences Induction Block: Maths & Statistics Lecture 1 Overview; Variables,...

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