DESCRIPTIVE STATISTICS
© LOUIS COHEN, LAWRENCE
MANION & KEITH MORRISON
STRUCTURE OF THE CHAPTER
• Frequencies, percentages and
crosstabulations
• Measures of central tendency and dispersal
• Ta...
FREQUENCIES AND PERCENTAGES
• Graphical forms of data presentation:
– Frequency and percentage tables;
– Bar charts (for n...
FREQUENCIES AND PERCENTAGES
• Bar charts for presenting categorical and discrete
data, highest and lowest;
• Avoid using a...
FREQUENCIES AND PERCENTAGES
• Pie charts and bar charts for showing proportions;
• Interdependence can be shown through cr...
• A crosstabulation is a presentational device.
– Rows for nominal data, columns for ordinal
data.
– Independent variables...
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...
TRIVARIATE CROSSTABULATION
Acceptability of formal, written public examinations
Traditionalist Progressivist/
child-centre...
MEASURES OF CENTRAL
TENDENCY AND DISPERSAL
• The mode (the score obtained by the greatest
number of people);
– For categor...
MEASURES OF CENTRAL
TENDENCY AND DISPERSAL
• The median (the score obtained by the middle
person in a ranked group of peop...
MEASURES OF CENTRAL
TENDENCY AND DISPERSAL
• Standard deviation (the average distance of
each score from the mean, the ave...
STANDARD DEVIATION
• The standard deviation is calculated, in its most
simplified form as:
or
• d2
= the deviation of the ...
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 4 20
Mean = 6
High...
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 11
Mean = 6
Moder...
9
8
Mean
7 |
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 7
Mean = 6
Low standar...
THE RANGE AND INTERQUARTILE
RANGE
• The range:
– The difference between the minimum and
maximum score.
– A measure of disp...
CORRELATION
• Measure of association between two
variables.
• Note the direction of the correlation:
– Positive: As one va...
CORRELATION
• Note the magnitude of the correlation
coefficient:
− 0.20 to 0.35: slight association
− 0.35 to 0.65: suffic...
CURVILINEAR RELATIONSHIP
0
10
20
30
40
50
0
10
20
30
40
50
60
70
80
90
Age
Muscularstrength
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 rat...
BIVARIATE CORRELATIONS
• Correlations
– Spearman correlation for nominal and
ordinal data
– Pearson correlation for interv...
MULTIPLE AND PARTIAL
CORRELATIONS
• Multiple correlation:
– The degree of association between three
or more variables simu...
RELIABILITY
• Split-half reliability (correlation between one
half of a test and the other matched half)
• The alpha coeff...
SPLIT-HALF RELIABILITY
(Spearman-Brown)
Reliability =
r = the actual correlation between the two halves of
the instrument ...
CRONBACH ALPHA
• Reliability as internal consistency: Cronbach’s
alpha (the alpha coefficient of reliability).
• A coeffic...
INTERPRETING THE RELIABILITY
COEFFICIENT
Maximum is +1
>.90 very highly reliable
.80-.90 highly reliable
.70-.79 reliable
...
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Chapter35

  1. 1. DESCRIPTIVE STATISTICS © LOUIS COHEN, LAWRENCE MANION & KEITH MORRISON
  2. 2. STRUCTURE OF THE CHAPTER • Frequencies, percentages and crosstabulations • Measures of central tendency and dispersal • Taking stock • Correlations and measures of association • Partial correlations • Reliability
  3. 3. 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).
  4. 4. 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;
  5. 5. 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.
  6. 6. • 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
  7. 7. 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 at all very little a little quite a lot a very great deal the course was too hard Total
  8. 8. 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%
  9. 9. 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
  10. 10. 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
  11. 11. 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
  12. 12. 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 d DS         = ∑ N d DS 2 ..
  13. 13. 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 4 20 Mean = 6 High standard deviation
  14. 14. 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 11 Mean = 6 Moderately high standard deviation
  15. 15. 9 8 Mean 7 | 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 7 Mean = 6 Low standard deviation
  16. 16. 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.
  17. 17. 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.
  18. 18. 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)
  19. 19. CURVILINEAR RELATIONSHIP 0 10 20 30 40 50 0 10 20 30 40 50 60 70 80 90 Age Muscularstrength
  20. 20. CORRELATION Foot size Hand size 1 1 2 2 3 3 4 4 5 5 Perfect positive correlation: + 1
  21. 21. CORRELATION Foot size Hand size 1 5 2 4 3 3 4 2 5 1 Perfect negative correlation: + 1
  22. 22. CORRELATION Hand size Foot size 1 2 2 1 3 4 4 3 5 5 Positive correlation: <+1
  23. 23. 0 1 2 3 4 5 6 7 Line 1 PERFECT POSITIVE CORRELATION
  24. 24. 0 1 2 3 4 5 6 7 Line 1 PERFECT NEGATIVE CORRELATION
  25. 25. 0 2 4 6 8 10 Line 1 MIXED CORRELATION
  26. 26. CORRELATIONS • Correlations – Spearman correlation for nominal and ordinal data – Pearson correlation for interval and ratio data
  27. 27. BIVARIATE CORRELATIONS • Correlations – Spearman correlation for nominal and ordinal data – Pearson correlation for interval and ratio data
  28. 28. 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.
  29. 29. RELIABILITY • Split-half reliability (correlation between one half of a test and the other matched half) • The alpha coefficient
  30. 30. 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) r r +1 2 85.01 )85.0(2 + 185 70.1
  31. 31. 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.
  32. 32. 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

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