Research Methods: 2M.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 Frequency Relative
Frequency012345Total
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 RelativeFrequency
p
0 2 2/25 0.08
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
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 Boundaries Interval Midpoint
56.0 - 58.9
59.0 - 61.9
62.0 - 64.9
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
77.4574.4571.4568.4565.4562.4559.4556.4553.45
10
5
0
Weight
Freq
uenc
yFrequency Distribution for Weights of 50 males.
Histograms; ProbabilityProbability Distribution for weights of 50
males.
0
0.05
0.1
0.15
0.2
0.25
0.3
53.45 56.45 59.45 62.45 65.45 68.45 71.45 74.45 77.45
Weights
Pro
babi
lity
Frequency/Probability Polygons
0
2
4
6
8
10
12
14
Fre
qu
ency
50.45 53.45 56.45 59.45 62.45 65.45 68.45 71.45 74.45 77.45 80.45
Weight (Kg)
Weights of 18 year old males
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
xOr FrequencyProbability
SD or
68.2%± 1SD95.4%± 2SD99.7%± 3SD
p = 0.682± 1SD
p = 0.954± 2SD
p = 0.997± 3SD
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;
16
4 SD 10, x
z = (16 - 10) /4z = 1.5
p = 0.9332 p = 1 - 0.9332 p = 0.067
95% of datap = 0.95p < 0.05