frequency distributions and graphing data: levels of measurement frequency distributions
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Frequency distributions and graphing data:
Levels of Measurement
Frequency distributions
Graphing data
Stages in scientific investigation:
Obtain your data:
Usually get data from a sample, taken from a population.
Descriptive statistics:
Reveal the information that's lurking in your data.
Inferential statistics:
Use data from a sample to reveal characteristics of the population from which the sample data were presumably selected.
Levels of measurement:
1. Nominal (categorical or frequency data):
When numbers are used as names.
e.g. street numbers, footballers' numbers.
All you can do with nominal data is count how often each number occurs (i.e. get frequencies of categories).
2. Ordinal:
When numbers are used as ranks.
e.g. order of finishing in a race: the first three finishers are "1", "2" and "3", but the difference between "1" and "2" is unlikely to be the same as between "2" and "3".
Many measurements in psychology are ordinal data - e.g., attitude scales.
3. Interval:
When measurements are made on a scale with equal intervals between points on the scale, but the scale has no true zero point.
e.g. temperature on Celsius scale: 100 is water's boiling point; 0 is an arbitrary zero-point (when water freezes), not a true absence of temperature. Equal intervals represent equal amounts, but ratio statements are meaningless - e.g., 60 deg C is not twice as hot as 30 deg!
Many measurements in psychology are interval data - e.g., IQ scores.
-4 -3 -2 -1 0 1 2 3 4
1 2 3 4 5 6 7 8 9
4. Ratio:
When measurements are made on a scale with equal intervals between points on the scale, and the scale has a true zero point.
e.g. height, weight, time, distance.
Measurements in psychology which are ratio data include reaction times, number correct, error scores.
What kind of measurement is a Likert scale?
Likert scales are often used to measure attitudes and opinions:
Strictly speaking, these are ordinal data – but commonly treated as interval measurements.
1 2 3 4 5 6 7
How attractive is Simon Cowell?
(1 = “highly unattractive”, 7 = “highly attractive”)
1 2 3 4 5 6 7
1 2 3 4 5 6 7
Is this scale interval?
Or ordinal?
Nominal data masquerading as scale measurements:
SPSS uses numbers as codes for nominal data.
Here “1” = “male” and “2” = “female. These are names, not numbers!
Frequency distributions:
50 scores on a statistics exam (max = 100):
84 82 72 70 72
80 62 96 86 68
68 87 89 85 82
87 85 84 88 89
86 86 78 70 81
70 86 88 79 69
79 61 68 75 77
90 86 78 89 81
67 91 82 73 77
80 78 76 86 83
Raw (ungrouped) Frequency Distribution:
Score Freq Score Freq Score Freq Score Freq
96 1 86 6 76 1 66 0
95 0 85 2 75 1 65 0
94 0 84 2 74 0 64 0
93 0 83 1 73 1 63 0
92 0 82 3 72 2 62 1
91 1 81 2 71 0 61 1
90 1 80 2 70 3
89 3 79 2 69 1
88 2 78 3 68 3
87 2 77 2 67 1
Class interval width = 3
Score Frequency
94-96 1
91-93 1
88-90 6
85-87 10
82-84 6
79-81 6
76-78 6
73-75 2
70-72 5
67-69 5
64-66 0
61-63 2
Class interval width = 5
Score Frequency
95-99 1
90-94 2
85-89 15
80-84 10
75-79 9
70-74 6
65-69 5
60-64 2
Grouped Frequency Distributions:
Grouped Frequency Distributions:
Raw Frequency of Scores (Class Interval = 3):
0
2
4
6
8
10
12
94
-96
91
-93
88
-90
85
-87
82
-84
79
-81
76
-78
73
-75
70
-72
67
-69
64
-66
61
-63
Score
Ra
w F
req
ue
nc
y
Raw Frequency of Scores (Class Interval = 5):
0
2
4
6
8
10
12
14
16
95
-99
90
-94
85
-89
80
-84
75
-79
70
-74
65
-69
60
-64
Score
Ra
w F
req
ue
nc
y
Score Raw Freq.
(=total in each cell)
94-96 1
91-93 1
88-90 6
85-87 10
82-84 6
79-81 6
76-78 6
73-75 2
70-72 5
67-69 5
64-66 0
61-63 2
Cumulative Frequency Distributions:
Cumulative freq.
(=each cell total + all preceding cell totals)
50
49
48
42
32
26
20
14 ( = 2+5+5+0+2)
12 ( = 5+5+0+2)
7 ( = 5+0+2)
2 ( = 0+2)
2 ( = 2)
Cumulative freq.
(= cum. freq. as % of total)
100
98
96
84
64
52
40
28 ( = (14/50)*100 )
24 ( = (12/50)*100 )
14 ( = (7/50)*100 )
4 ( = (2/50)*100 )
4 ( = (2/50)*100 )
Cumulative frequency graph
0
10
20
30
40
50
60
70
80
90
100
62 65 68 71 74 77 80 83 86 89 92 95
Score
Fre
qu
en
cy
(%
to
tal)
Relative Frequency Distributions:
Useful for comparing groups with different totals.
Group A: N = 50
Score Raw Freq.
96-100 3
91-95 4
86-90 11
81-85 15
76-80 8
71-75 4
66-70 2
61-65 3
Total: 50
Group B: N = 80
Score Raw Freq.
96-100 3
91-95 4
86-90 18
81-85 24
76-80 11
71-75 9
66-70 5
61-65 6
Total: 80
Rel. Freq.
6 %
8 %
22 %
30 %
16 %
8 %
4 %
6 %
100 %
Rel. Freq.
3.75 %
5.00 %
22.50 %
30.00 %
13.75 %
11.25 %
6.25 %
7.50 %
100 %
Relative frequency = (cell total/overall total) x 100
Raw Frequencies of Scores (N = 50)
0
2
4
6
8
10
12
14
16
96-100 91-95 86-90 81-85 76-80 71-75 66-70 61-65
Score
Raw
fre
qu
ency
Raw Frequency and Relative Frequency Distributions:
Only the scale of the graph changes - not the pattern of frequencies.
Relative Frequencies of Scores (N = 50)
0
5
10
15
20
25
30
35
96-100 91-95 86-90 81-85 76-80 71-75 66-70 61-65
Score
Rel
ativ
e fr
equ
ency
(%
)
Frequency of accidents
0
10
20
30
40
50
volvo mini porsche
Type of car driven
No
. o
f a
cc
ide
nts
pe
r y
ea
r
Effects of aspect ratio and scale on graph appearance:
(a) A graph aimed at giving an accurate impression...
Frequency of accidents
0
10
20
30
40
50
volvo mini porsche
Type of car driven
No
. o
f a
cc
ide
nts
pe
r y
ea
r
(b) A tall thin graph exaggerates apparent differences...
Frequency of accidents
0
10
20
30
40
50
volvo mini porsche
Type of car driven
No
. o
f ac
cid
ents
per
ye
ar
(c) A low wide graph minimises apparent differences...
Frequency of accidents
10
20
30
40
50
volvo mini porsche
Type of car driven
No
. of
ac
cid
en
ts p
er
ye
ar
(d) Starting the scale at a value other than zero can also exaggerate apparent differences.
Graphing averages:
If plotting averages, always include a measure of how scores are spread out around the average.
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