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Why is the study of variability
important?
• Allows us to distinguish between
usual & unusual values
• In some situations, want more/less
variability
– scores on standardized tests
– time bombs
– medicine
Measures of Variability
• range (max-min)
• interquartile range (Q3-Q1)
• deviations  x  x  Lower case
Greek letter
2
sigma
• variance  
• standard deviation  
Suppose that we have these data values:
24
16
34
28
26
21
30
35
37
29
Find the mean.
Find the deviations.
x  x 
What is the sum of the deviations from the mean?
24
16
34
28
26
21
30
35
37
29
x  x 
2
Square the deviations:
Find the average of the squared deviations:
 x  x 
2
n
The average of the deviations
squared is called the variance.
Population parameter

2
Sample
s
2
statistic
Calculation of variance
of a sample
  xn  x 
s 
n 1
2
2
df
A standard deviation is a
measure of the average
deviation from the mean.
Calculation of standard
deviation

xn  x 

s
2
n 1
Degrees of Freedom (df)
• n deviations contain (n - 1)
independent pieces of
information about
variability
Use calculator
Which measure(s) of
variability is/are
resistant?
Linear transformation rule
• When multiplying or adding a constant to a
random variable, the mean changes by both.
• When multiplying or adding a constant to a
random variable, the standard deviation
changes only by multiplication.
• Formulas:
ax b  ax  b
 ax b  a x
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?
  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:
 a  b   a  b
a b  a  b
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
T  3.5  21.8  12.3  37.6 minutes
T  0.7 2  2.42  2.7 2  3.680 minutes