DESCRIPTIVE STATISTICS

© LOUIS COHEN, LAWRENCE MANION & KEITH MORRISON

STRUCTURE OF THE CHAPTER

• Frequencies, percentages and crosstabulations

• Measures of central tendency and dispersal• Taking stock• Correlations and measures of association• Partial correlations• Reliability

FREQUENCIES AND PERCENTAGES • Graphical forms of data presentation:

– Frequency and percentage tables;

– Bar charts (for nominal and ordinal data);

– Histograms (for continuous – interval and ratio – data);

– Line graphs;

– Pie charts;

– High and low charts;

– Scatterplots;

– Stem and leaf displays;

– Boxplots (box and whisker plots).

FREQUENCIES AND PERCENTAGES • Bar charts for presenting categorical and discrete

data, highest and lowest;• Avoid using a third dimension (e.g. depth) in a

graph when it is unnecessary; a third dimension to a graph must provide additional information;

• Histograms for presenting continuous data;• Line graphs for showing trends, particularly in

continuous data, for one or more variables at a time;

• Multiple line graphs for showing trends in continuous data on several variables in the same graph;

FREQUENCIES AND PERCENTAGES

• Pie charts and bar charts for showing proportions;

• Interdependence can be shown through cross-tabulations;

• Boxplots for showing the distribution of values for several variables in a single chart, together with their range and medians;

• Stacked bar charts for showing the frequencies of different groups within a specific variable for two or more variables in the same chart;

• Scatterplots for showing the relationship between two variables or several sets of two or more variables on the same chart.

• A crosstabulation is a presentational device. – Rows for nominal data, columns for ordinal

data.– Independent variables as row data,

dependent variables as column data.

CROSSTABULATIONS

BIVARIATE CROSSTABULATION

sex * The course was too hard: crosstabulation

7 11 25 4 3 50

3.7% 5.8% 13.1% 2.1% 1.6% 26.2%

17 38 73 12 1 141

8.9% 19.9% 38.2% 6.3% .5% 73.8%

24 49 98 16 4 191

12.6% 25.7% 51.3% 8.4% 2.1% 100.0%

Count

% of Total

Count

% of Total

Count

% of Total

male

female

Total

not atall

verylittle a little

quite alot

a verygreat deal

the course was too hard

Total

TRIVARIATE CROSSTABULATION

Acceptability of formal, written public examinations

Traditionalist Progressivist/child-centred

Formal, written public exams

Socially advantaged

Socially disadvantaged

Socially advantaged

Socially disadvantaged

In favour 65% 70% 35% 20%

Against 35% 30% 65% 80%

Total per cent

100% 100% 100% 100%

MEASURES OF CENTRAL TENDENCY AND DISPERSAL

• The mode (the score obtained by the greatest number of people);– For categorical (nominal) and ordinal data

• The mean (the average score);– For continuous data– Used if the data are not skewed– Used if there are no outliers

MEASURES OF CENTRAL TENDENCY AND DISPERSAL

• The median (the score obtained by the middle person in a ranked group of people, i.e. it has an equal number of scores above it and below it);– For continuous data– Used of the data are skewed– Used if there are outliers

MEASURES OF CENTRAL TENDENCY AND DISPERSAL

• Standard deviation (the average distance of each score from the mean, the average difference between each score and the mean, and how much, the scores, as a group, deviate from the mean. – A standardized measure of dispersal. – For interval and ratio data

STANDARD DEVIATION

• The standard deviation is calculated, in its most simplified form as:

or

• d2 = the deviation of the score from the mean (average), squared

= the sum of• N = the number of cases• A low standard deviation indicates that the

scores cluster together, whilst a high standard deviation indicates that the scores are widely dispersed.

1..

2

N

dDS

N

dDS

2

..

9

8 Mean

7 |

6 |

5 |

4 |

3 |

2 |

1 X X X X | X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 2 3 420 Mean = 6

High standard deviation

9

8 Mean

7 |

6 |

5 |

4 |

3 |

2 |

1 X X X X X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 2 6 10 11Mean = 6

Moderately high standard deviation

9

8 Mean7 |

6 |

5 |

4 |

3 X

2 X

1 X X X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

5 6 6 6 7Mean = 6

Low standard deviation

THE RANGE AND INTERQUARTILE RANGE

• The range:– The difference between the minimum and

maximum score.– A measure of dispersal.– Outliers exert a disproportionate effect.

• The interquartile range:– The difference between the first and the third

quartile, the difference between the 25th and the 75th percentile, i.e. the middle 50 per cent of scores (the second and third quartiles).

– Overcomes problems of outliers/extreme scores.

CORRELATION

• Measure of association between two variables.

• Note the direction of the correlation:– Positive: As one variable increases, the

other variables increases– Negative: As one variable increases, the

other variable decreases– The strongest positive correlation

coefficient is +1.– The strongest negative correlation

coefficient is -1.

CORRELATION

• Note the magnitude of the correlation coefficient:− 0.20 to 0.35: slight association− 0.35 to 0.65: sufficient for crude prediction− 0.65 to 0.85: sufficient for accurate prediction− >0.85: strong correlation

• Ensure that the relationships are linear and not curvilinear (i.e. the line reaches an inflection point)

CURVILINEAR RELATIONSHIP

0

10

20

30

40

50

Age

Mu

scu

lar

stre

ng

th

CORRELATION

Foot size Hand size

1 1

2 2

3 3

4 4

5 5

Perfect positive correlation: + 1

CORRELATION

Foot size Hand size

1 5

2 4

3 3

4 2

5 1

Perfect negative correlation: + 1

CORRELATION

Hand size Foot size 1 2 2 1 3 4 4 3 5 5

Positive correlation: <+1

0

1

2

3

4

5

6

7

Line 1

PERFECT POSITIVE CORRELATION

0

1

2

3

4

5

6

7

Line 1

PERFECT NEGATIVE CORRELATION

0

2

4

6

8

10

Line 1

MIXED CORRELATION

CORRELATIONS

• Correlations– Spearman correlation for nominal and

ordinal data– Pearson correlation for interval and ratio

data

BIVARIATE CORRELATIONS

• Correlations– Spearman correlation for nominal and

ordinal data– Pearson correlation for interval and ratio

data

MULTIPLE AND PARTIAL CORRELATIONS

• Multiple correlation:– The degree of association between three

or more variables simultaneously.• Partial correlation:

– The degree of association between two variables after the influence of a third has been controlled or partialled out.

– controlling for the effects of a third variable means holding it constant whilst manipulating the other two variables.

RELIABILITY

• Split-half reliability (correlation between one half of a test and the other matched half)

• The alpha coefficient

SPLIT-HALF RELIABILITY(Spearman-Brown)

Reliability =

r = the actual correlation between the two halves of the instrument (e.g. 0.85);

Reliability = = = 0.919 (very high)

rr

12

85.01)85.0(2

18570.1

CRONBACH ALPHA

• Reliability as internal consistency: Cronbach’s alpha (the alpha coefficient of reliability).

• A coefficient of inter-item correlations. • It calculates the average of all possible split

half reliability coefficients.

INTERPRETING THE RELIABILITY COEFFICIENT

Maximum is +1

>.90 very highly reliable

.80-.90 highly reliable

.70-.79 reliable

.60-.69 marginally/minimally reliable

<.60 unacceptably low reliability