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3
Numerical Summary Measures
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
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
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
median mean
When the data are not symmetric – median
is the best measure of central tendency
Chapter3 p42
mean = median = mode
Need to have some idea about the
variation among the measurements.
Chapter3 p39
Measures of Dispersion
Range = max. – min.
Its usefulness is limited,
because it considers
only the extreme values
of a data set.
Chapter3 p45
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/100 + 1)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
+
In all cases, mean > median  skewed
(means – median) is smaller for post-AIDS
Example - Table 3.2
The 25% percentile
13(25)/100 = 3.25, so the 25% percentile is the 3
+ 1 = 4th largest measurement = 2.60 liters
The 75% percentile
13(75)/100 = 9.75, so the 75% percentile is the 9
+ 1 = 10th largest measurement = 3.38 liters
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
Variance (s2) and standard deviation
n
1
2
s2 
(
x

x
)
 i
n  1 i 1
Table 3.1
Mean = 2.95 liters
Variance = 0.39 liters2
Standard deviation = 0.62 liters
Coefficient of variation
s
0.62
CV  100% 
100%  21.0%
x
2.95
Chapter3 p46
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
k
x

i 1
mi f i
k 
 f i 
 i 1 
k  number _ of _ int ervals
mi  midpo int_ of _ the _ ith _ int erval
f i  frequency _ associated _ with _ the _ ith _ int erval
Chapter3 p49
Grouped variance
k
s 
2
Midpoint
99.5
139.5
 (m  x )
i 1
i
2
fi
k 
 f i   1
 i 1 
k  number _ of _ int ervals
mi  midpo int_ of _ the _ ith _ int erval
f i  frequency _ associated _ with _ the _ ith _ int erval
Chapter3 p51
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
10 patients
(beats/min.)
9 patients without
#7 (beats/min.)
mean
130.8
140.9
median
143
150
mode
150
150
range
127
47
interquartile
25%percentile=
120
75%percentile=
150
=150 – 120 = 30
Std.
deviation
35.5
Chapter3 p54
outliner
Chapter3 p56
Grouped mean
= 3348.2 grams
Grouped std.dev.
= 616.1 grams
Chapter3 p57
Chapter3 p59
Chapter3 p59
Using Excel
Statistical analysis – AVERAGE, STDEV, VAR, MAX, MIN,
MODE, PERCENTILE, TDIST, TINV, TTEST ….
Graph plotting using Excel
Exercises
Chapter3 p61
Chapter3 p62