Research Methods: 2
M.Sc.
Physiotherapy/Podiatry/Pain
Frequency/Probability Polygons,
and the Normal Distribution
Part one: Frequency Tables
Un-grouped
•
•
•
•
Tally observations
Frequency table
Histogram
Polygon
Grouped
•
•
•
•
•
Set class limits
Tally number in class
Frequency table
Histogram
Polygon
Ungrouped Frequency Tables;
Data from n = 25, rating 1-5 of RM2 teaching
0
2
0
2
3
1
2
1
3
4
5
4
4
2
1
2
3
3
1
2
3
1
2
3
2
Ungrouped Frequency Tables;
Frequency Table
Rating
0
1
2
3
4
5
Total
Frequency Relative
Frequency
Ungrouped Frequency Tables;
Data from n = 25, rating 1-5 of RM2 teaching
0
2
0
2
3
1
2
1
3
4
5
4
4
2
1
2
3
3
1
2
3
1
2
3
2
Rating Frequency
0
2
Relative
Frequency
2/25
p
1
5
5/25
0.2
2
8
8/25
0.32
3
6
6/25
0.24
4
3
3/25
0.12
5
1
1/25
0.04
Total
25
25/25
1.00
0.08
Grouped Frequency Tables;
Data of weights (kg) n = 12
56.3
66.4
63.5
71.2
56.4
75.8
68.5
65.9
73.6
58.7
61.7
59.9
Grouped Frequency Tables;
Setting class limits
•
•
•
•
Find range
Choose number of classes (5 < >20)
Classes equal size (Outliers?)
Choose limits at level of measurement
precision
• Tally
Grouped Frequency Tables
• Class boundaries
Half way between classes
One more decimal place than limits
• Class intervals
Distance between boundaries
• Midpoints
Half way between boundaries
Mid point of interval
Grouped Frequency Tables
Limits
56.0 - 58.9
59.0 - 61.9
62.0 - 64.9
Boundaries
Interval Midpoint
Grouped Frequency Tables
Limits
Boundaries
Interval Midpoint
56.0 - 58.9 55.95 - 58.95 3.0
57.45
59.0 - 61.9 58.95 - 61.95 3.0
60.45
62.0 - 64.9 61.95 - 64.95 3.0
63.45
Histograms
•
•
•
•
•
Present information from Frequency tables
Show distribution of the data set
Columns start and end at class boundaries
Midpoints are marked
Join midpoints = Frequency/Probability
Polygon
• Area represent frequency/ probability; total
area under curve; p = 1.00
Histograms; Frequency
F r e q u e n c y D is t r ib u t io n f o r W e ig h t s o f 5 0 m a le s .
Frequency
10
5
0
5 3 .4 5 5 6 .4 5 5 9 .4 5 6 2 .4 5 6 5 .4 5 6 8 .4 5 7 1 .4 5 7 4 .4 5 7 7 .4 5
W e i ght
Histograms; Probability
Probability Distribution for weights of 50
males.
0.3
Probability
0.25
0.2
0.15
0.1
0.05
0
53.45 56.45 59.45 62.45 65.45 68.45 71.45 74.45 77.45
Weights
Frequency/Probability Polygons
Weights of 18 year old males
14
12
Frequency
10
8
6
4
2
0
50.45
53.45
56.45
59.45
62.45
65.45
68.45
Weight (Kg)
71.45
74.45
77.45
80.45
Part two: The Normal
Distribution
•
•
•
•
A type of (family) of distributions
Most important of all known distributions
Natural parameters in populations
Symmetrical bell shaped curve
Normal Distribution
SD or 
Frequency
Probability
x Or 
68.2%
95.4%
99.7%
±±±1SD
2SD
3SD
ppp === 0.682
0.997
0.954
±±± 1SD
3SD
2SD
p if not exact multiple of
SD away from mean ?
Z scores
•
•
•
•
Data point of interest = x
Mean = 
Standard deviation = 
Z score is number of multiples of SD the
data point is away from mean ;
z= x-

Z scores
• Look up the Z score in Tables to find;
Probability associated with values below x
and vice versa.
Why ???
Graph of number of visits to Physiotherapist for
Sports rehabilitation; x  10, SD  4
z = (16 - 10) /4
z = 1.5
 p = 0.9332
 p = 1 - 0.9332
 p = 0.067
16
95% of data
p
<
0.05
p = 0.95