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ThiQar college of Medicine
Family & Community medicine dept.
Biostatistics Lecture
Third stage
by: Dr. Muslim N. Saeed
December 15th ,2016
Mathematical Presentation of Data
Measures of Dispersion
Quintiles, Centiles & Quartiles
 A quintile is a value below which a certain proportion
of observations occurred in the ordered set of data
values.
 A centiles are values, in a series of observations,
arranged in ascending order of magnitude, which
divide the distribution into 100 equal parts (10th
Percentile, 3rd, 97th, and the 50th (median)
percentile).
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Quintiles, Centiles & Quartiles
 Quartiles are the observations in an array that divide
the distribution into four equal parts.
 lower Quartile: the value below which 25% of
observations lie in an ordered array
 2nd quartile = Median = 50th percentile
 Upper Quartile = 75th percentile
 Interquartile Range: is the middle 50% of all
observations
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Distance travelled in Miles
Villages
Distance travelled in miles
Village (1)
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
7.7
7.7
Village (2)
4.2
5.4
5.8
6.2
6.7
7.7
7.7
8.5
9.3
10
Measures of
Central
Tendency
Village (1)
Village (2)
Mean
7.15
7.15
Median
7.2
7.2
Mode
7.7
7.7
Dot plots of Distance Travelled
Village (1)
Village (2)
Even though the measures of center tendency are all the same,
it is obvious from the dot plots of each group of data that there
are some differences in the ‘spread’ (or variation) of the data
Consider these means for weekly candy bar consumption
Mean = {12, 2, 0, 14, 10, 9, 5, 4}
= (12+2+0+14+10+9+5+4)/8
=7
Mean = {7, 8, 6, 7, 7, 6, 8, 7}
= (7+8+6+7+7+6+8+7)/8
=7
Measures of Dispersion
 As well as measures of central tendency we need
measures of how variable the data are.
 Dispersion is a key concept in statistical thinking.
 The basic question being asked is how much do the scores
deviate around the Mean?
Measures of Dispersion; These are
 The range
 The Variance
 Standard Deviation
 Standard Error
 Coefficient of Variation
Measures of Dispersion; The Range
 The range is an important measurement
Range
Highest
Value
Lowest
Value
However, they do not give
much indication of the
spread of observations about
the mean
 Simple to calculate
 Easy to understand
 It neglect all values in the center and depend on the extreme value,
extreme value are dependent on sample size
 It is not based on all observations
 It is not amenable for further mathematic treatment
 should be used in conjunction with other measures of variability
Variance:
The mean sum of squares of the deviation from the mean.
e.g. if the data is: 1,2,3,4,5.
The mean for these data=3
the difference of each value in the set from the mean:
1-3= -2
2-3= -1
3-3= 0
4-3= 1
5-3= 2

The summation of the differences =zero

Summation of square of the differences is not zero
The Variance
Variance can never be a
negative value
All observations
considered
are
The problem with the
Another formula for the variance variance is the squared
unit
The standard deviation is the square root of the
variance
 The standard deviation measured the variability between
observations in the sample or the population from the mean of that
sample or that population.
 The unit is not squared
 SD is the most widely used measure of dispersion
Standard Error of the mean(SE)



It measures the variability or dispersion of the
sample mean from population mean
It is used to estimate the population mean, and to
estimate differences between populations means
SE=SD/√ n
Coefficient of variation (CV):






It expresses the SD as a percentage of the mean
CV= (S /mean) x 100 (mean of the sample)
It has no unit
It is used to compare dispersion in two sets of data
especially when the units are different
It measures relative rather than absolute variation
It takes in consideration all values in the set
Exercise
A sample of 11
patients admitted
to a psychiatric
ward experienced
the
following
lengths of stay,
calculate measures
of dispersion.
No.
Length
No.
length
1
29
7
28
2
14
8
14
3
11
9
18
4
24
10
22
5
14
11
14
6
14
total