2. 3
Numerical Summary Measures

3. Measures of central tendency Mean = 2.95 liters 75% percentile = 3.38
Three values are larger than 3.38
and nine are smaller.
25% percentile = 2.60
Rank of FEV
2.15, 2.25, 2.30, 2.60, 2.68, 2.75, 2.82, 2.85, 3.00, 3.38, 3.50, 4.02, 4.05
Chapter3 p39

4. Chapter3 p40 0 = female 1 = male Mean = 0.615
61.5% of the study subjects are males
Median
Not as sensitive to the value of each measurement
the 50th percentile of a set of measurements
- N is odd, the median is the middle value, (N+1)/2
- N is even, the median is the average of the middlemost values, the N/2 th and (N/2)+1 th observations
Chapter3 p40

5. Mode
The mode of a set of values is the observation that occurs most frequently
Table 3.1 do not have a unique mode, since each of the value occurs only once.
The mode for the data in Table 3.2 is 1.
Fig. 3.1(a) – symmetric, and unimodal
mean ~ median ~ mode
Fig. 3.1(b) – symmetric and bimodal
mean ~ median, two peaks  two
distinct group
Fig. 3.1(c) – skewed to the right
not symmetric, mean lies to the right of
the median
Fig. 3.1(d) – skewed to the left
not symmetric, not symmetric, mean lies to the left of the median
When the data are not symmetric – median is the best measure of central tendency
median mean
Chapter3 p42

6. mean = median = mode
Need to have some idea about the variation among the measurements.
Chapter3 p39

7. Measures of Dispersion
Range = max. – min.
Its usefulness is limited, because it considers only the extreme values of a data set.
Chapter3 p45

8. Chapter3 p46 In all cases, mean > median  skewed+
Interquartile range
= 75% percentile – 25% percentile
= 3.38 – 2.60 = 0.78 liters
How to find the kth percentile ?
Let n = number of observations (obs.)
Rank the data from the smallest to the largest
If nk/100 = integers
kth percentile  ½ (nk/100)th + (nk/ )th largest obs.
If nk/100 is not an integers
kth percentile  (j+1)th largest measurement where j is the largest integer that is < nk/100
Example - Table 3.2
The 25% percentile
13(25)/100 = 3.25, so the 25% percentile is the = 4th largest measurement = 2.60 liters
The 75% percentile
13(75)/100 = 9.75, so the 75% percentile is the = 10th largest measurement = 3.38 liters
In all cases, mean > median  skewed
(means – median) is smaller for post-AIDS
Rank of FEV
2.15, 2.25, 2.30, 2.60, 2.68, 2.75, 2.82, 2.85, 3.00, 3.38, 3.50, 4.02, 4.05
Chapter3 p46

9. Variance (s2) and standard deviation
Table 3.1
Mean = 2.95 liters
Variance = 0.39 liters2
Standard deviation = 0.62 liters
Coefficient of variation
Chapter3 p46

10. Chapter3 p49 Group mean = ungroup mean Table 3.3
Two/three subjects have 12/11 years of duration
Grouped mean = [3(5)+1(6)+1(8)+3(11)+2(12)]/10=8.6 years
Chapter3 p49

11. Grouped variance
Midpoint
99.5
139.5
Chapter3 p51

12. Symmetric and unimodal (%)
Chebychev’s inequality
For symmetric and unimodal distribution
~67% of the data lie in the interval mean ± 1s
~95% of the data lie in the interval mean ± 2s
~99% of the data lie in the interval mean ± 1s
Not symmetric and unimodal
 Chebychev’s inequality
For any number , at least [1- (1/k)2] of the measurements in the set of data lie within k standard deviation of their mean
Given k = 2  1 – (1/2)2 = ¾, or 75% of the data lie within mean ± 2s
k (std. dev.)
Chebychev (%)
Symmetric and unimodal (%) 1
≧ 0
~ 67 2
≧ 75
~ 95 3
≧ 89
~ 99
Chapter3 p49

13. 9 patients without #7 (beats/min.)
10 patients (beats/min.)
9 patients without #7 (beats/min.)
mean
130.8
140.9
median
143
150
mode
range
127 47
interquartile
25%percentile=120
75%percentile=150
=150 – 120 = 30
Std. deviation
35.5
Chapter3 p54

14. outliner
Chapter3 p56

15. Grouped mean
= grams
Grouped std.dev.
= grams
Chapter3 p57

16. Chapter3 p59

17. Chapter3 p59

18. Using Excel
Statistical analysis – AVERAGE, STDEV, VAR, MAX, MIN, MODE, PERCENTILE, TDIST, TINV, TTEST ….
Graph plotting using Excel

19. Exercises
Chapter3 p61

20. Chapter3 p62