Transcript Variability Day 1

Why is the study of variability
important?
• Allows us to distinguish between
usual & unusual values
• In some situations, want more/less
variability
– medicine
– scores on standardized tests
– time bombs
Measures of Variability
• range (max-min)
• interquartile range (Q3-Q1)
• deviations  x  x  Lower case
Greek letter
sigma
• variance  
• standard deviation  
2
The average of the deviations
squared is called the variance.
Population

Sample
2
s
parameter
2
statistic
A standard deviation is a
measure of the average
deviation from the mean.
Population

Sample
s
Suppose that we have this population:
24
16
34
28
26
21
30
35
37
29
Find the mean (m).
Find the deviations.
x  m 
What is the sum of the deviations from the mean?
24
16
34
28
26
21
Square the deviations:
16
36
4
4
81
30
35
37
29
x  m 
2
144 0
49
49
1
Find the average of the squared deviations:
x  m 


2

2
n

Calculation of variance
of a sample
  xn  x 
s 
n 1
2
2
Degrees of Freedom (df)
• n deviations contain (n - 1)
independent pieces of
information about
variability
Calculation of standard
deviation of a sample

xn  x 

s
2
n 1
Which measure(s) of
variability is/are
resistant?
Linear transformation rule
• When adding a constant to a random
variable, the mean changes but not the
standard deviation.
• When multiplying a constant to a
random variable, the mean and the
standard deviation changes.
An appliance repair shop charges a $30 service call
to go to a home for a repair. It also charges $25 per
hour for labor. From past history, the average length
of repairs is 1 hour 15 minutes (1.25 hours) with
standard deviation of 20 minutes (1/3 hour).
Including the charge for the service call, what is the
mean and standard deviation for the charges for
labor?
m  30  25(1.25)  $61.25
1
  25   $8.33
3
Rules for Combining two variables
• To find the mean for the sum (or difference), add
(or subtract) the two means
• To find the standard deviation of the sum (or
differences), ALWAYS add the variances, then
take the square root.
• Formulas:
m a b  m a  mb
ma b  ma  mb
2
a
 a b    
2
b
If variables are independent
Bicycles arrive at a bike shop in boxes. Before they can be
sold, they must be unpacked, assembled, and tuned
(lubricated, adjusted, etc.). Based on past experience, the
times for each setup phase are independent with the
following means & standard deviations (in minutes). What
are the mean and standard deviation for the total bicycle
setup times?
Phase
Mean
SD
Unpacking
Assembly
Tuning
3.5
21.8
12.3
0.7
2.4
2.7
mT  3.5  21.8  12.3  37.6 minutes
T  0.7 2  2.42  2.7 2  3.680 minutes