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MEASURES OF DISPERSION
(VARIABILITY)
•
•
•
•
Range
Variance and Standard Deviation
Coefficient of Variation
Non-central Locations: Inter-fractile
Ranges
Range
Ungrouped data: range = max data value – minimum data value
grouped data: range = upper limit of largest class – lower limit of smallest class
Standard Deviation
Statistical measure which expresses the
average deviation (spread) about the mean.
The sample standard deviation is given by:
s
s
2
2
x

(
x
)
  /n
n 1
2
2
fx

(
fx
)

 /n
n 1
(ungrouped data)
(grouped data)
COEFFICIENT OF VARIATION:
 Relative measure of dispersion
 Ratio of the standard deviation to the arithmetic
mean, expressed as a percentage:
s
CV  (100%)
x
The higher the value of CV, the greater the variability
Example: A sample of movie theaters in a large metropolitan area
tallied the total number of movies showing in a certain week.
Class interval
1–3
4–6
7–9
10 – 12
13 – 15
Frequency (f)
1
4
3
2
1
Σ = 11
Compute each of the following quantities:
(a) range
(b) standard deviation
(c) coefficient of variation.
3- 5
(a)
Range = 15-1 = 14
(b)
s
 fx
2
 ( fx) / n
2
n 1
Movies frequency class
f
showing
midpoint
x
1–3
1
2
fx
fx2
2
4
4–6
4
5
20
100
7–9
3
8
24
192
10- 12
2
11
22
242
13 - 15
1
14
14
196
82
734
Total
11
734  82  82 / 11
s
 3.5
10
(c)
s
CV  (100%)
x
s  3.5, x  7.46
s
3.5
CV  
100%  46.9%
x 7.46
3- 9
Empirical Rule: For any symmetrical, bell-shaped
distribution:

About 68% of the observations will lie within one
standard deviation of the mean

About 95% of the observations will lie within two
standard deviations of the mean

Virtually all the observations will be within three
standard deviations of the mean
3- 10
Bell-shaped curve showing relationship
between m and σ
68%
95%
99.7%
m3s
m2s ms
m
m+s
m+2s m+ 3s
Symmetric Distribution
Zero skewness → :Mean =Median = Mode
Mean
Median
Mode
The Relative Positions of the Mean, Median, and Mode:
3- 11
Positively skewed
(right skewed)
Mean>Median>Mode
Mode
Mean
Median
The Relative Positions of the Mean, Median and Mode.
3- 12
3- 13
Negatively Skewed
(left skewed)
Mean<Median<Mode
Mean
Mode
Median
The Relative Positions of the Mean, Median and Mode
Non-Central Location Measures
(Fractiles or Quantiles)
Fractiles (Quantiles)
Divide size-ordered data sets into subsets of equal
frequency:
• Quartiles
• Sextiles
• Octiles
• Deciles
• percentiles
Most commonly used are quartiles and percentiles.
Quartiles:
Divide size-ordered data sets into four equalfrequency subsets, i.e they define boundaries
of datasets split into 4 equal-frequency
classes.
Lower (first) quartile Q1: Identifies the value below
which the lower 25% of the ordered data set lies. (Q1is
also the 25th percentile)
Middle (second) quartile Q2: Identifies the value that
separates the lower 50% of the data set from the upper
50% of the data set.
Note: Q2 is also the median or 50th percentile)
Upper (third) quartile Q3: Identifies the value
above which the top 25% of the ordered data set
lies.
(Q3 is also the 75th percentile)
Percentiles divide an ordered data set into 100
equal-frequency parts.
A Percentile is a data value below which a
specified percentage of data values in an
ordered data set fall.
CALCULATING QUARTILES FOR
GROUPED DATA
The jth quartile for grouped data is given by:
 jn


F

c
4

Qj  L + 
fQ j
n = sample size
L = lower limit of jth quartile class
F = < cumulative frequency of immediately preceding class.
fQj = frequency of jth quartile class.
Measures of Dispersion using Fractiles
Interquartile Range (IQR):
Difference between Q3 and Q1.
Includes the middle 50% of the observations.
IQR = Q3 - Q1
Middle 80% range = P90 – P10
Middle 40% range = P70 –P30
Quartile deviation QD= (Q3-Q1)/2
Quartile Deviation (QD):
 Measure of spread about the median.
 Equals half the difference between Q3 and Q1.
Q3  Q1
IQR
QD 

2
2
CALCULATING PERCENTILES
GROUPED DATA
The jth percentile for grouped data is given by:
 jn

 F c

100

Pj  L + 
fP j
n = sample size.
L = lower limit of jth percentile class.
F = < cumulative frequency of immediately preceding class.
fPj = frequency of jth percentile class.
3- 23
Example: A sample of movie theaters in a large
metropolitan area tallied the total number of movies
showing in a certain week. Find the 70th percentile
and the IQR.
Class interval
Frequency (f)
1–3
1
4–6
4
7–9
3
10 – 12
2
13 – 15
1
Σ = 11
Movies showing
frequency
f
< cum freq (F)
1–3
1
1
4–6
4
5
7–9
3
8
10- 12
2
10
13 - 15
1
11
Pos P70 = 70x11/100 = 7.7
Movies freq f
1–3
4–6
7–9
10 – 12
13 – 15
1
4
3
2
1
< cum
freq (F)
1
5
8
10
11
(7.7  5)
P70  7 +
 3  7 + 2.7  9.7
3
Pos Q1 = 1x11/4 = 2.75
Pos Q3 = 3 x 2.75 = 8.25
Movies
freq f
1–3
4–6
7–9
10 – 12
13 – 15
1
4
3
2
1
(2.75  1)
Q1  4 +
 3  5.3
4
< cum freq
(F)
1
5
8
10
11
(8.25  8)
Q3  10 +
 3  10.4
2
IQR = Q3 - Q1=10.4 - 5.3 = 5.1