Download 2.2

Measures of variation
(dispersion) [‫]مقاييس التشتت‬
Formula:
1. For ungrouped data population variance:
N

2

 (x
i 1
i
 )2
N
2. For ungrouped data sample variance
n
S 
2
 (x
i 1
i
 x)
n 1
2
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
Standard Deviation (S)
1. It is the square root of variance.
2. Most commonly used measure of
variance.
3. Shows variation about mean.
4. It used to compare between more than
one data set when the means are equal,
the best one is the minimum.
S S
2
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Example


( x  x) ( x  x) 2
Year of
graduation
No. of
Students
2004
4
-2
4
2005
6
0
0
2006
5
-1
1
2007
8
2
4
2008
7
1
1
Total
30
0
10
• Calculate variance and standard deviation
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
n
• Solution: Variance
S 
2
 (x
i 1
i
 x )2
n 1

10
 2.5
4
• Standard Deviation S  S 2  2.5  1.58
• Interpretation: The observations fall 1.58
units from the mean.
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
Variance and standard deviation for grouped
data
•
Formula
n
n
S 
2
fx
i 1
i
2
i

( f x )
i 1
n 1
i
n
i
2
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Example: the table below shows the temperature of a sample of
50 cities taken at the same time on a certain day; determine the
mean and standard deviation of the sample.
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
Temp.
f
Cum
ulativ
e F.
Midp
oint
f .x
x2
f .x 2
10-14
10
10
12
144
120
1440
15-19
12
22
17
289
204
3468
20-24
18
40
22
484
396
8712
25-29
6
46
27
729
162
4374
30-34
4
50
32
1024
128
4096
Total
50
1010
22090
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
f .x 1010

x

 20.2
f
50
n
n
S 
2
fx
i 1
S
i
2
i

( f x )
i 1
i
n
n 1
S
2

i
2
(1010) 2
22090 
1688
50


 34.45
49
49
34.45  5.87
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Coefficient of Variation (C.V) (‫)معامل االختالف‬
1. It is the main important application of the
mean and standard deviation.
2. Measures relative variation and always in
percentage (%).
3. Can be used to compare two or more data
sets measured in different units.
4. Can be used widely in chemistry and
engineering science.
5. The variable with smaller C.V is less
dispersed than others so it is the better.
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Formula:
• Coefficient of Variation
S
C.V   100%
x
• Example: Suppose that technician A
completes 40 analysis daily with standard
deviation of 5, technician B completes 160
analysis per day with standard deviation of
15.
• Which employee shows less variability or
better?
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Sol.
S
5
C.V ( A)   100% 
 100%  12.5%
x
40
S
15
C.V ( B)   100% 
 100%  9.4%
x
160
• Employee B is better than A because
he have the less variation.
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
Interquartile Range [IQR] (‫)المدى الربيعي‬
There are three quartiles Q1, Q2, Q3
1. Q1 is a 25% of sorted data.
2. Q2 is a 50% of sorted data or median.
3. Q3 is a 75% of sorted data.
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
Formulas
Q1  L1
 N


F
1


  4
  C1
f
1






Q2 the same as median formula.
 3N


F
3 
 4
Q3  L3  
  C3
f3




Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Example: you have the frequency table:
Class boundaries
31.5 – 36.5
36.5 – 41.5
41.5 – 46.5
46.5 – 51.5
51.5 – 56.5
65.5 – 61.5
Frequency (f) Cumulative frequency (F)
4
4
7
11
10
21
7
28
18
46
4
50
Calculate Q1, Q2, Q3 and interquartile range.
Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Sol.
Q1  L1
step 2 
 N
 F1

  4
f1





  C1



50
 12.5
4
Step (3) the first quartile class is [41.5-46.5]
step(4) : L1  41.5, F1  11, f1  10, C1  5
N

 50


F

11
1
4
 4

Q1  L1  

C

41
.
5

 1

  5  42.25
 f1 
 10 




Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Sol. Q2
step 2 
50
 25
2
Step (3) the median quartile class is [46.5-51.5]
step(4) : L1  46.5, F1  21, f1  7, C1  5
N


F
2
2
 25  21
Q2  L  

C

46
.
5

 5  49.36
 2


 7 
 f2 


Measures of variation
(dispersion) [‫]مقاييس التشتت‬
• Sol. Q3
50  3
step 2 
 37.5
4
Step (3) the third quartile class is [51.5-56.5]
step(4) : L3  51.5, F3  28, f 3  18, C3  5
 N 3

 3  50


F

28
3


 4

Q3  L3   4

C

51
.
5



  5  54.14
3
f
18
3








Interquartile range = Q3 - Q1 =54.14-42.25=11.89.
Box plot
• A box plot is a descriptive
statistics and it is a convenient
way of graphically depicting
groups of numerical data through
their quartiles.
• Example: Plot a box plot for {7, 4,
3, 5, 6, 8, 10, 1}
Box plot
• Solution:
• Sort data as: {1, 3, 4, 5, 6, 7, 8, 10}.
• Minimum value is 1, maximum value is
10.
• Calculate Q1, Q2, Q3 as:
position of Q1 
k (n  1) 25(9)

 2.25
100
100
3 4
Q1 
 3.5
2
Box plot
• Q2 and Q3
k (n  1) 50(9)
position of Q 2 

 4.5
100
100
56
Q2 
 5 .5
2
k (n  1) 75(9)
position of Q3 

 6.75
100
100
78
Q3 
 7 .5
2
Box plot
Box plot
Note: The above plotting done by computer using
software R as:
> A <- c(7, 4, 3, 5, 6, 8, 10 ,1)
> quantile(A)
0%
1.00
25%
3.75
50%
5.50
75%
7.25
100%
10.00