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AGSC 320
Statistical Methods
Numerical descriptive measures
Data representation
1. •
Measures of central tendency
e.g., mean, mode, median, midrange
2. Measures of dispersion
e.g., range, variance, standard deviation
3. Measures of distribution shape
e.g., normal, skewed, uniform, random
4. Measures of position
e.g., percentiles, quartiles, standard scores
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Data organization
Height of 20 trees:
50, 45, 32, 48, 56, 38, 42, 48, 55, 36,
41, 51, 30, 59, 53, 47, 57, 51, 46, 44
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6
5
4
3
2
1
0
30
35
40
45
Height
50
55
60
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Measures of central tendency
1. mean or arithmetic average
Definition: sum of values divided by the total
number of observations
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Measures of central tendency
2. Median:
Definition: the midpoint / middle value in a
group of data
The point that separate the data in two set
with the same number of observations
Steps:
• arrange the data in order
• find the midpoint
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Measures of central tendency
3. Mode
Definition: the most frequently occurring
value / observation
Notes:
• not always unique
• can also be bimodal, multimodal
4. Midrange
Definition: sum of the lowest and highest
values divided by 2
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Measures of central tendency
Summary
Statistics
Mean
Value
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Relationship among mean,median,mode
• Depending on the shape of the histogram /
frequency distribution the mean can be located
differently in respect with median or mode
Mean=Median=Mode
Mode<Median<Mean
Mean<Median<Mode
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Measures of dispersion
1. Range:
Definition: the difference between the largest and
smallest observation
Range = xmax - xmin
where
xmax – largest observation
xmin – smallest observation
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Measures of dispersion
2. Variance:
• Definition: sum of the squared differences
between each observation and the mean,
divided by the number of observations.
Population
Sample
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Measures of dispersion
Working formulas for Variance and
Standard deviation
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Measures of dispersion
3. Standard deviation
Definition: the square root of the variance
• A measure of the spread of the
observations in the original units
Population
Sample
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Measures of dispersion
Variance and Standard deviation
Using definition:
Using working formulas:
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Measures of dispersion
Range rule of thumb
A rough estimate of the standard deviation is a
quarter of range
s
range
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Example using tree data
s
.......
 ...........
.......
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Measures of dispersion
4. Coefficient of variation
Ratio between standard deviation and mean
sample
CV 
s
100
x
population
CV 

100

Example using tree data
CV 
.......
 100  ........... [%]
.......
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Measures of central tendency
grouped data
Mean or Arithmetic average
Definition: sum of values divided by the total
number of observations
c
sample data
x
f
j 1
c
c
j
f
j 1
xj

Population data
f 
j
j 1
c
f
j
j 1
j
j
x j ,   value of the j class midpo int
th
c  number of classes
f j  frequency of the j th class
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Measures of dispersion
grouped data
Variance and Standard deviation for
frequency distribution
c
c
sample data
s2 
f
j 1
j
( x j  x )2
population data
c
f
j 1
j
 
2
f
j 1
1
j
( j   )2
c
f
j 1
j
x j ,   value of the j th class midpo int
c  number of classes
f j  frequency of the j th class
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Measures of dispersion
grouped data
Example: Daily commuting times, in minutes
Calculate mean, variance, standard deviation, CV
Daily commuting time Number of employees
Less than 10 min
4
10 – 20 min
9
20 – 30 min
6
30 – 40 min
4
40 – 50 min
2
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Measures of dispersion
grouped data
• Remember: in a class all individuals are assumed to
have the mid-value of the respective class
• Mid-value of the class = class mean
Commuting time
< 10 min
10 – 20 min
20 – 30 min
30 – 40 min
40 – 50 min
Total
# employees
4
9
6
4
2
Class mean
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fj x μj
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Measures of dispersion
grouped data
• Mean commuting time:
• Variance:
c
 
2
f
j 1
j
( j   )2

c
f
j 1
4(5  ....) 2  9(15  ...) 2  ....

(4  9  6  4  2)
j
• Standard deviation:
σ=……
• Coefficient of variation:
CV=σ/μ x100= ……/…….=
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Use of standard deviation
• Connect mean with standard deviation
• Chebyshev’s Theorem:
For any k>1, at least 1-1/k2 of
the data lie within k standard
deviation from the mean
• Example: if k=2 →1-1/k2=1-1/4=0.75 or 75%
This means that 75% of data values are within
two standard deviation from the mean
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Measures of Distribution Shape
• Skeweness: a measure of the asymmetry
of the frequency distribution

 n

  ( xi  x )3  /( n  1)
 i 1


3/ 2
n


2
   ( xi  x )  /( n  1) 

  i 1

• Kurtosis: measure of the "peakeness" of
the frequency distribution
 n

  ( xi  x ) 4  /( n  1)

3 
2
 n


2
   ( xi  x )  /( n  1) 

  i 1

   i 1
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Measure of position
• Locate the relative position of an observation
/data within dataset
PERCENTILES – divide the data set into 100
groups with equal number of observations
• indicate the position of an individual in a group
– Education
– Health related industry
– Life sciences
percentile 
(# observations less than x)  0.5
100
total # observations
[%]
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Percentiles charts
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Standard scores
• Compare the relative position of observations
within their defining dataset
• Standard score or z-score
z
observation' s value  mean x  x

standard deviation
s
• Allows comparison of different datasets or
different type of data
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Standard score
Example:
Student received 92% Statistics and 75% English
Was the overall student’s performance bad?
Additional info:
• Mean grade for Statistics was 85 and for English was 70
• Variance for Statistics was 36 and for English was 9.
Compute the z-scores:
zStatistics 
x  x ................
 ............ 
s
zEnglish 
x  x ................
 ............ 
s
Conclusion:
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Population vs. statistics
• Various numerical measures can be computed
for the population as well as for a sample
– Mean , median, variance, coefficient of variation
• When the measure is computed for the entire
population then the measure is called
population parameter or simply PARAMATER
• When the measure is computed for a portion of
the population (namely sample), then the
measure is called sample statistics, or simply
STATISTIC.
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