What is a Variable?

• any entity that can take on different values

• not always 'quantitative' or numerical, but we can assign numerical values

• attribute = a specific value of a variableExamples:

• gender: 1=female; 2=male• attitudes: 1 = strongly disagree; 2 = disagree; 3 =

neutral; 4 = agree; 5 = strongly agree

Coding in a data matrix

Gender: Male = 1; Female=2

Political Orientation: Traditionalist=1; Moderate=2; Progressive=3

Social Class: Working=1; Upper working=2; Lower middle=3; Middle=4; Upper middle=5

Coding in a data matrix

Levels of Measurement

• different kinds of variables

(1) Nominal

(2) Ordinal

(3) Interval and Ratio

Nominal Variable

• used to classify things• represents equivalence (=)• adding, subtracting, multiplying or dividing

nominal numbers is meaningless • tells you how many categories there are in

the scheme

Ordinal Variable

• ordering or ranking of the variable• the relationship between numbered items• ‘higher’, ‘lower’, ‘easier’, ‘faster’, ‘more

often’• equivalence (=) and relative size (greater

than) and < (less than)

Interval (and Ratio) Variable

• All arithmetical operations are allowed• intervals between each step are of equal

size• Examples:

- length, weight, elapsed time, speed, temperature

Women’s Shoe Sizes

British 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

European 34 35 35.5 36 37 37.5 38 38.5 39 39.5 40 41 42

American 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5

Japanese (cm)

21.5 22 22.5 23 23 23.5 24 24 24.5 25 25.5 26 26.5

Levels of measurement

Level are names

have an inherent order 

from more to less  or higher to lower

are numbers with equal intervals between them

Nominal  level

Ordinal  level

Interval  level

 

Frequency distributions

• count number of occurrences that fall into each category of each variable

• allow you to compare information between groups of individuals

• also allow you to see what are the highest and lowest values and the value at which most scores cluster

• variables of any level of measurement can be displayed in a frequency table

Frequency table

Percentages• number of cases belonging to particular category divided

by the total number of cases and multiplied by 100.• the total of percentages in any particular group equals 100

per cent.

100% N

f

Graphical presentation

• Pie charts• Barcharts• Line graphs• Histograms

Pie chart

• illustrates the frequency (or percentage) of each individual category of a variable relative to the total.

• Pie charts are not appropriate for displaying

quantitative data. Gender of Sociology Students

81%

19%

Female

Male

SociologyStudents

Female 26Male 6

15

Barcharts

• the height of the bar is proportional to the category of the variable - easy to compare

• used for Nominal or Ordinal level variables (or discrete interval/ratio level variables with relatively few categories)

Marital Status

140

60

8575

30 35

75

0

20

40

60

80

100

120

140

160

Married Living asmarried

Single Divorced Separated Widow ed Missing

Multiple barchart

Marital status

0

20

40

60

80

100

120

140

160

Married Living asmarried

Single Divorced Separated Widow ed

Fre

qu

enci

es

1995

2000

Compound or Component barchart

Sociology Students

41 42

2639

0

20

40

60

80

100

2001 2002Year

Freq

uenc

y Male

Female

Line graphs

• interval/ratio level variables that are discrete• need to arrange the values in order

YEAR

20012000199919981997199619951994199319921991

Va

lue

PR

OD

UC

T

170

160

150

140

130

120

110

Histograms

• represents continuous quantitative data• The height of the bars corresponds to the

frequency or percentage of cases in the class.• The width of the bars represents the size of the

intervals of the variable• The horizontal axis is marked out using the mid

points of class intervals

Example: Histogram

MARKS

85.0

80.0

75.0

70.0

65.0

60.0

55.0

50.0

45.0

40.0

35.0

30.0

25.0

30

20

10

0

Std. Dev = 11.83

Mean = 55.4

N = 100.00

Graphs have the capacity to distort

YEAR

20012000199919981997199619951994199319921991

Va

lue

PR

OD

UC

T

170

160

150

140

130

120

110

YEAR

200120001999199819971996199519941993199219911990V

alu

e P

RO

DU

CT

200

100

0

Measures of Central Tendency

• describe sets of numbers briefly, yet accurately • describe groups of numbers by means of other,

but fewer numbers• Three main measures:

• mean• median• mode

The Mean

• most common type of average that is computed.

When to use the Mean

• When values in a particular group cluster closely around a central value, the mean is a good way of indicating the ‘typical’ score, i.e. it is truly representative of the numbers.

• If the values are very widely spread, are very unevenly distributed, or clustered around extreme values, than the mean can be misleading, and other measures of central tendency should be used instead.

The Median

• Also an average, but of different kind.• It is defined as the midpoint in a set of scores. It

is the point at which one-half, or 50% of the scores fall above and one-half, or 50%, fell below.

• Computing the Median:(1) List the scores in order, either from highest

to lowest or lowest to highest.

(2) Find the middle score. That’s the median.

The Median: Pros and Cons

• time-consuming• if one of the numbers near the middle of the distribution

moves even slightly, than the median would alter, unlike the mean, which is relatively unaffected by a change in one of the central numbers

• if one of the extreme values changes, than the median remains unaltered.

- 2, 80, 100, 120, 130, 140, 160, 200, 3150• single scores which are quite clearly ‘deviant’ when

compared with others, are known as outliers – 2 and 3150

The Mode

• the value in any set of scores that occurs most often

• example 1: – 5, 6, 7, 8, 8, 8, 9, 10, 10, 12 – the mode = 8

• example 2: – 5, 6, 7, 8, 8, 8, 9, 10, 10, 10, 12 –two modes: 8 and

10 – bimodal

• very unstable figure – 1,1,6,7,8,10 – mode = 1– 1,6,7,8,10,10 – mode = 10

When to Use What?

• depends on the type of data that you are describing

– for nominal data - only the mode– for ordinal data - mode and median– for interval data - all of them

• but, for extreme scores - use the median

Measure of dispersion (spread)

• better impression of a distribution’s shape• measures indicate how widely scattered

the numbers are• how different scores are from one

particular score – the mean• variability - a measure of how much each

score in a group of scores differs from the mean

The range

• tells us over how many numbers altogether a distribution is spread

lhr • where

• r is the range• h is the highest score in the data set• l is the lowest score in the data set.

12 13 12 1114 13 12 10 10 11

55

0

10

20

30

40

50

60

0 2 4 6 8 10 12

r = biggest value - smallest value = 55-10 = 45

The mean deviation

• number which indicates how much, on average, the scores in a distribution differ from a central point, the mean.

Mean deviation =

160

580

2

738

6

734

51

689

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10

Mean=370

-210

210

-368

368

-364

364

-319

319

X - mean= (-210)+210+(-368)+368+(-364)+364+(-319)+319 = 0

X - mean= 210+210+368+368+364+364+319+319 = 2522

mean deviation = 2522/8 = 315.25

The standard deviation (SD)

• represents the average amount of variability• It is the average distance from the mean

N

XXs

2

• s the standard deviation• find the sum of what follows• X each individual score• the mean of all the scores• N the sample sizeX

Standard deviations

Shape of Normal Distribution

Mean MedianMode

Symmetrical

Asymptotic tail

The area under the curve

• A normal distribution always has the same relative proportions of scores falling between particular values of the numbers involved.

• Areas under the curve = proportion of scores lying in the various parts of the complete distribution

SS2008N - Surveys

Median

Median

50%50%

SS2008N - Surveys

Quartiles

MedianQuartile 1 Quartile 3

25%

25%

25%

25%

Standard Deviation