Transcript Variance and Standard Deviation

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  
Range:
•
Single number – not an interval
•
Sensitive to outliers
•
Midrange – average of the max and
min values - VERY
sensitive to outliers
Interquartile Range (IQR):
.
Quartiles:
IQR  Q3  Q1
The first quartile (Q1) is the value for which 25% of the
observations are less than. It is the Median of the first
half of the set of observations. (the 25th percentile)
The third quartile (Q3) is the value for which 75% of the
observations are less than. It is the Median of the second half
of the set of observations. (the 75th percentile)
IQR is insensitive to outliers.
The average of the deviations
squared is called the variance.
Population parameter

2
Sample
s
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
Find the mean
( )
Find the deviations.
30
35
37
29
x  
What is the sum of the deviations from the mean?
24
16
34
28
26
21
Square the deviations:
30
35
37
29
x  
2
Find the average of the squared deviations:

2
x  


n
2
Calculation of variance
of a sample
  xn  x 
s 
n 1
2
2
df
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
When to use which??????
Variance and Standard Deviation are used to
measure spread when the mean is used to
describe center.
IQR is typically used to describe spread when
median is used to describe center.
When the distribution is approximately
symmetric, the mean and standard deviation
are used to summarize the distribution.
If the distribution is skewed, a five number
summary is generally used.
Which measure(s) of
variability is/are
resistant?
The REBEL STAT Company
We have a company with 14 employees that
earn the following monthly salaries:
1200
1900
1400
2100
1800
1000
1300
1300
1700
2300
1200
1400
1100
3500
$1657.14
 = _______________
$634.39
 = _______________
Business has been good, so every
employee receives a raise of $500.
What will happen to the mean and
standard deviation?


$2157.14
= _______________
$634.39
= _______________
Business has NOT been good.
Suppose that we have to cut
everyone’s pay by $500.
What will happen to the mean and
standard deviation?

$1157.14
= _______________
$634.39
 = _______________
What happens to the mean and
standard deviation if a number is
added to each data value?
Business has been good. Suppose
that we give everyone a 30% raise.
What will happen to the mean and
standard deviation?

$2154.29
= _______________

$824.71
= _______________
What happens to the mean and
standard deviation if a number is
multiplied to each data value?
Rules:
a bx  a  bx
 a bx  b  x
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?
  30  25(1.25)  $61.25
1
  25   $8.33
3
Stat Land Activity
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