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DESCRIPTIVE STATISTICS
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DESCRIPTIVE STATISTICS
Descriptive statistics are summary measures which
define some important characteristics of data.
1. Measures of location
i. Measures of central tendency
ii. Measures of position
2. Measures of dispersion (variation)
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Measures of Location
i. Measures of central tendency
Measures of central tendency are numerical
values that tend to locate in some sense the
middle of a set of data.
•Arithmetic Mean
•Geometric Mean
•Median
•Harmonic Mean
•Mode
•Proportion
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Measures of central tendency
Arithmetic Mean (Mean)
The arithmetic mean is the most common
measure of the central tendency and is
commonly used for symetrical distributions.
It is used to summarize quantitative data.
n
Sum of all the observations (  x ) divided
by the number of the observations (n).
i 1
i
n
X 
x
i 1
i
n
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Example
Age distribution of seven children attending to
a children clinic is given below
{1,3,6,7,2,3,5}
n
X 
7
x x
i 1
n
i

i 1
7
i
1  3  6  7  2  3  5 27


7
7
X  3,9 years
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Median
The median is the middle value of the set of
data when the data are ranked in order
according to magnitude.
When the data are put in order 50 % of the
observations are less than or equal to the
median, the rest is greater than the median .
Median value is
 n 1


 2 
th
observation.
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Example
n is odd: 5, 28, 8, 10, 9
Ordered data
5, 8, 9, 10, 28
i =(5+1)/2=3
Median is 3rd value which is 9.
n is even: 19, 20, 17, 27, 6, 21
Ordered data 6, 17, 19, 20, 21, 27
i=(6+1)/2=3.5
Median is halfway between the 3rd and 4th
values, which is 19.5.
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Mode
The mode is the value of x that occurs most
frequently.
Data
• Mode : 3
{1,3,7,3,2,3,6,7}
Data
{1,3,7,3,2,3,6,7,1,1}
• Mode : 1 and 3
Data
{1,3,7,0,2,-3, 6,5,-1}
• Mode : No mode
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Example
Suppose the age in years of the first 10 subjects enrolled in
your study are:
34, 24, 56, 52, 21, 44, 64, 44, 42, 46
Then the mean age of this group is 42.7 years
To find the median, first order the data:
21, 24, 34, 42, 44, 44, 46, 52, 56, 64
The median is (44+44)/2 = 44 years
The mode is 44 years.
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Suppose the next patient enrolls and her age is
97 years.
How does the mean, median and mode change?
Ordered data:
21, 24, 34, 42, 44, 44, 46, 52, 56, 64, 97
Mean is 47,6
42,7
Median is 44
44
Mode is 44
44
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Comparison of Mean and Median
• Mean is sensitive to “outliers” (a few very
large or small values), so sometimes mean does
not reflect the quantity desired.
x  31,38
20, 21, 22, 23, 24, 25, 26, 90
87.5% of observations
• Median is “resistant” to outliers
Median = 23.5
•Mean is attractive mathematically.
•50% of sample is above the median, 50% of
sample is below the median.
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Geometric mean is a summary statistic useful
when the measurement scale is not linear, it is
computed as
G  n x1  x2   xn
log( x )

or log( G ) 
i
n
For example, in the area of psychometrics it is well
known that the rated intensity of a stimulus (e.g.,
brightness of a light) is often a logarithmic
function of the actual intensity of the stimulus
(brightness measured in units of Lux). In this
instance, the geometric mean is a better "summary"
of ratings than the simple mean.
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For example, suppose you have an investment
which earns 10% the first year, 50% the second
year, and 30% the third year. What is its average
rate of return? It is not the arithmetic mean,
because what these numbers signify is that on the
first year your investment was multiplied (not
added to) by 1.10, on the second year it was
multiplied by 1.50, and the third year it was
multiplied by 1.30. The relevant quantity is the
geometric mean of these three numbers, which is
about 1.28966 or about 29% annual interest.
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For example, if a stock rose 10% in the first year,
20% in the second year and fell 15% in the third year,
then we compute the geometric mean of the factors
1.10, 1.20 and 0.85 as (1.10 × 1.20 × 0.85)1/3 =
1.0391... and we conclude that the stock rose 3.91
percent per year, on average.
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Harmonic Mean is a "summary" statistic used in
analyses of frequency data. The harmonic mean
is sometimes used to average values that change
in time.
If a variable contains a zero (0) as a valid score,
then the harmonic mean cannot be calculated
(since it implies division by zero).
HM 
n
1
x
i
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In certain situations, the harmonic mean provides the
correct notion of "average". For instance, if for half
the distance of a trip you travel at 40 miles per hour
and for the other half of the distance you travel at
60 miles per hour, then your average speed for the
trip is given by the harmonic mean of 40 and 60,
which is 48; that is, the total amount of time for the
trip is the same as if you traveled the entire trip at
48 miles per hour. (Note however that if you had
traveled for half the time at one speed and the
other half at another, the arithmetic mean, 50 miles
per hour, would provide the correct notion of
"average".)
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Proportion is a fraction in which the numerator is
included within the denominator. A proportion is
often expressed as a percentage.
For example, we might describe the overall
amount of smokers in a population as the
proportion of people in the population who smoke.
Suppose out of 125 people 15 smoke cigarette,
then the proportion of people who smoke is
P=
15
 0.12
125
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ii. Measures of position
Measures of position are used to decribe the location
of a specific piece of data in relation to the rest of the
sample. Quartiles and percentiles are two most popular
measures of position.
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Quartiles
Quartiles are numeric values of variable that divide
the ordered data into quarters; each set of data has
three quartiles.
The first quartile, Q1, is a number such that at
most one-fourth of the data are smaller in value
than Q1 and at most three-fourths are larger.
n 1
Q1 
th
4
The second quartile, Q2, is the median.
2(n  1) n  1
Q2 

th
4
2
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The third quartile, Q3, is a number such that at
most three-fourths of the data are smaller in value
than Q3 and at most one-fourth are larger.
3(n  1)
Q3 
th
4
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Example: Birthweights of 24 infants are as follows:
Obs. Weight Obs. Weight Obs. Weight Obs. Weight
1
2850
7
3150
13
3250
19
3700
2
2900
8
3200
14
3400
20
3800
3
2930
9
3200
15
3450
21
3900
4
2980
10
3200
16
3500
22
4100
5
3000
11
3250
17
3500
23
4400
6
3100
12
3250
18
3600
24
4500
21
25
 6.25 th obs. is the fist
First quartile=Q1=
4
quartile.
6th obs.=3100gr
Q1=3112.5gr.
7th obs.=3150gr
50x0.25=12.5gr
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 12.5 th obs. is
Second quartile=Median=Q2=
the second quartile or median. 2
12th obs.=3250gr
13th obs.=3250gr
Q2=3250gr.
3x 25
 18.75 th obs. is the third
Third quartile=Q3=
4
quartile
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18th obs.=3600gr
19th obs.=3700gr
100x0.75=75gr.
Q3=3675gr.
25% of the infants have birthweights less than
3112,5gr.
Half of the infants have birthweights less than
3250gr.
75%of the infants have birthweights less than
3675 gr.
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Percentiles are numerical values of the variable
that divide a set of ordered data into 100 equal
parts; each set of data has 99 percentiles.
The procedure for determining the value of any kth
percentile involve three basic steps.
1. The data must be ordered
2. The position number i for the percentile in
question must be determined. It is found by first
calculating the value of k (n  1) .
100
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Example: Birth weights of 24 infants.
i.
Find the 30th percentile.
When the data are ordered
30(24  1)
 7.5 th observation, which is the
100
half way between 7th (3150gr) and 8th (3200gr) observations, is the
30th percentile. The 30th percentile is therefore 3175gr, which
means, 30% of the observations lie below 3175gr.
ii. Find the 50th percentile (median).
50(24  1)
 12.5
100
th observation, which is the half way between 12th
(3250gr) and 13th observations (3250gr) is the 50th percentile.
3250gr is the 50th percentile. Half of the infants have birth weights
less than 3250gr.
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2. Measures of dispersion
Range
The range is the simplest measure of dispersion. It
is the difference between the highest valued (H) and
the lowest valued (L) of the observations.
Range= H-L
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Measures of dispersion
Standard deviation is the average distance of
observations to arithmetic mean.
s
 (x
i 1
 x 
x  n
2
n
i
 x)
2
n 1
or
s
i
2
i
n 1
Variance is square of standard deviation.
s2 
2
(
x

x
)
 i
i 1
n 1
 x 
x  n
2
n
or
s2 
i
2
i
n 1
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Step 1 Step 3 Step 4
x
(x  x) (x  x)2
6
1
1
3
-2
4
8
3
9
5
0
0
3
-2
4
25
0
18
x
(x  x)
1
3
5
6
10
25
-4
-2
0
1
5
0
(x  x)2
16
4
0
1
25
18
x 25

x

5
Step 2
Step 5
n
s2 
5
2
(
x

x
)

n 1

18
 4.5
4
s  s 2  4.5  2.12
x 25

x

5
n
s2
5
(x  x)


n 1
2

46
 11.5
4
2
s

s
 11.5  3.39
n
NOTE: The sum of the deviation,  ( xi  x ) , is always zero.
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i 1
The last set of data is more dispersed than the
previous set, and therefore its variance is larger.
First sample
Second sample
1
3
3
5 6
3
5 6
s2=4.5
8
10
s2=11.5
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Coefficient of variation (CV) shows the deviation
as a percentage of arithmetic mean.
s
CV   100
x
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Example: Heights of ten subjects are measured in cm
and m. Data are listed below
x
s
cv
160
180
165
174
190
182
155
165
171
160
170.2
11.23
0.066
1.60
1.80
1.65
1.74
1.90
1.82
1.55
1.65
1.71
1.60
1.702
0.1123
0.066
Although standard
deviations are different,
coefficient of variations are
the same.
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Interquartile Range is a measure of dispersion for
non-symmetric data
IQR = Q3 – Q1
Semi-Interquartile Range is used instead of
standart deviation when distribution of data is
non-symmetric.
SIQR= (Q3 – Q1)/2
Ranked data, increasing order
25% 25% 25%
L
Q1
Q2
25%
Q3
H
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