Download Descriptive statistics

Research Methods: 2
M.Sc.
Physiotherapy/Podiatry/Pain
Descriptive statistics
Magic Brain Pills
• Research Methods
Assessment
• mark with
• n=15
• Research Methods
Assessment
• mark without
• n=15
• 32 47 76 24 66 43 56
30 43 52 46 28 74 28
65
• 56 29 21 49 56 39 47
34 42 44 42 53 25 41
8
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 Unimodal
Sample 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  x
Interval or Ratio
• 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 = 15.9
Sample 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
• Sample 2: Days to
recovery without Rx
• 12 8 18 22 24 17 15
• 15 12 19 18 25 16 19
14
• Calculate Median and
Interquartile range
• 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
•
•
•
•
•
•
 2
x )
SD= [(Xi –
 (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
Standard Deviation
X3
Half x-mean = -ve
Half x-mean = +ve
X1
X2
X4
Mean
So  (x - mean) always = 0
So Square then sum and take square root of
Sum of Squares/ n-1
Exercise: SD
• Sample 1
• Sample 2
• 28.0, 19.2, 25.0, 20.0,
26.6
• 21.7, 19.3, 21.6, 28.7,
10.3
• Calculate Mean, SD,
and Range
• 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
4/15 failed
• Without
• Mean 39.1
• SD 13.6
6/15 failed