Transcript Measures of Variability

* 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
*variance  
Greek letter
*standard deviation sigma
2
 
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
*
*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. ax b  ax  b
*Formulas:
 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