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