Research Methods: 1M.Sc.

Physiotherapy/Podiatry/Pain

Descriptive statistics

• Research Methods Assessment

• mark with• n=15

• 32 47 76 24 66 43 56 30 43 52 46 28 74 28 65

• Research Methods Assessment

• mark without• n=15

• 56 29 21 49 56 39 47 34 42 44 42 53 25 41 8

Magic Brain Pills

Descriptive/Summary Statistics

Measures of centrality

Measures of dispersion

Summary Statistics

Measures of centrality

Mode, Median and Mean

The Mode

• Tally observations in set

• Frequency table

• Nominal, Ordinal, Interval or Ratio

• Uses?

Exercise: The Mode

• Sample one 5 5 6 9 8 9 7 4 5 8

• Sample two 16 14 12 18 19 14 12 18

• Sample three 81 79 78 75 81 69 75

Sample one Mode = 5 UnimodalSample two Mode = 14 or 12 or 18 Tri or Multi-Modal Sample three Mode = 75 or 81 Bi-Modal

The Median

• Put set in rank order

• Median lies in middle, half values greater half lower

• If n is even, median = mean of the middle two positions

• Ordinal, Interval or Ratio

• Uses ?

Exercise: Median

• Sample one 5 5 6 9 8 9 7 4 5 8

• Sample two 16 14 12 18 19 20 14 12 18

• Sample three 81 79 78 75 81 69 75 72

Sample one Median = 6.5 (4 5 5 5 6 7 8 8 9 9)Sample two Median = 16 (12 12 14 14 16 18 18 19 20)Sample three Median = 76.5 (69 72 75 75 78 79 81 81)

The Arithmetic Mean

• Add all the values

• Divide by the number of values

• Mean = Xi/n

• Interval or Ratio

x

• Uses ?

Exercise: Mean

• Sample one 5 5 6 9 8 9 7 4 5 8

• Sample two 16 14 12 18 19 20 14 12 18

• Sample three 81 79 78 75 81 69 75

Sample one Mean = 6.6 Sample two Mean = 15.9Sample three Mean = 76.8

Summary Statistics

Measures of dispersion

Range, Inter and Semi Interquartile Range and Standard Deviation

The Range

• Find Minimum value

• Find Maximum value

• Subtract Max-Min

• Uses?

Interquartile Range

• Put set in rank order

• Find the median = Q2

• Q1 = (n+1)/4th position

• Q3 = 3(n+1)/4th position

• Q3-Q1 Interquartile range

• (Q3-Q1)/2 Semi-Interquartile range

Exercise: Interquartile Range

• Sample 1: Days to recovery with Rx

• 12 8 18 22 24 17 15

• Calculate Median and Interquartile range

• Sample 2: Days to recovery without Rx

• 15 12 19 18 25 16 19 14

• Calculate Median and Interquartile range

Answer Sample 1

12 8 18 22 24 17 15

8 12 15 17 18 22 24

Q2 = 17 (4th), n = 7

Q1= (7+1)/4th position = 2nd = 12

Q3 = 3(7+1)/4th position = 6th = 22

IQR = 10

Answer Sample 2

15 12 19 18 25 16 19 14

12 14 15 16 18 19 19 25

Q2 = 17 (4th + 5th / 2), n = 8

Answer Sample 2

12 14 15 16 18 19 19 25

Q1 = (8+1)/4th position = 2¼th;

position 2 = 14 position 3 = 15 Q1 = 14 ¼

Q3 = 3(8+1)/4th position = 6¾th;

position 6 = 19 position 7 = 19 Q3 =19

IQR = 4¾

Standard Deviation

• SD= [(Xi – )2 (n – 1)] (for samples <30)

• Calculate the mean

• Subtract each value from the mean

• Square each answer and sum

• Divide by n-1

• Take square root of that answer

x

Standard DeviationX3

X1

X2

X4

Mean

Half x-mean = +veHalf x-mean = -ve

So (x - mean) always = 0 So Square then sum and take square root of

Sum of Squares/ n-1

Exercise: SD

• Sample 1

• 28.0, 19.2, 25.0, 20.0, 26.6

• Calculate Mean, SD, and Range

• Sample 2

• 21.7, 19.3, 21.6, 28.7, 10.3

• Calculate Mean, SD, and Range

Answer

• Sample 1 Mean 23.76, SD 3.95, Range 8.8

• Sample 2 Mean 20.30, SD 6.62, Range 18.4

• Are they different ?

• Are they different enough ?

Magic Brain Pills

• With• Mean 47.7• SD 17.4

• Without• Mean 39.1• SD 13.6

4/15 failed 6/15 failed