Transcript Chapter 3.2 Variance and Standard Deviation

Chapter 3.2
Variance and Standard Deviation
Comparison of Outdoor Paint
A testing lab wishes to test two experimental brands
of outdoor paint to see how long each will last before fades.
The testing lab makes 6 gallons of each paint to test. Since
different chemical agents are added to each group and only
six cans involved, these two groups constitute two small
populations. The results in months are shown.
Brand A
10
60
50
30
40
20
Brand B
35
45
30
35
40
25
Find the mean for Brand A and Brand B

Recall formula for Mean
Results?

Although the means are the same, we cannot conclude
that both brands of paint last equally well.

Find the range:
Definitions and formulas for variability of a
data set:

The variance is the average of the squares of the distance
of each value is from the mean.
( X   )
 
n
2
2

The standard deviation is the square root of the variance.
( X   )
  
n
2
2

X = individual value, N = population size,
µ = population mean
Steps for finding Variance and
Standard Deviation
Find the mean of the data
2. Subtract the mean from each data value
3. Square each result
4. Find the sum of the squares
5. Divide the sum by N to get the variance
6. Take the square root of the variance to get the standard
deviation
(It might be helpful to organize the data in a table)
1.
Find the Standard Deviation and Variance
of both brands of paint
Brand A
Values, X
X-µ
Brand B
(X - µ)2
Values, X
10
35
60
45
50
30
30
35
40
40
20
25
X-µ
(X - µ)2
Conclusions

Since the standard deviation of brand A is greater than
the standard deviation of brand B, the data are more
variable for brand A.

In summary, when the means are equal, the larger the
variance or standard deviation is, the more variable the
data are.
Formulas for samples

Variance:
n ( X )  ( X )
s 
n(n  1)
2
2
2

Standard Deviation:
n ( X )  ( X )
s
n(n  1)
2
2
Steps for finding the sample variance and
standard deviation:
1.
Find the sum of the values (∑X)
2.
Square each value and find the sum (∑X2)
3.
Substitute in the formulas and solve
Example:

Find the sample variance and standard deviation for the
amount of European auto sales for a sample of 6 years
shown. The data are in millions of dollars.
11.2, 11.9, 12.0, 12.8, 13.4, 14.3
Precipitation and High Temperatures

The normal daily high temperatures (in degrees
Fahrenheit) in January for 10 selected cities are as follows:
50, 37, 29, 54, 30, 61, 47, 38, 34, 61
The normal monthly precipitation (in inches) for these
same 10 cities is listed here:
4.8, 2.6, 1.5, 1.8, 1.8, 3.3, 5.1, 1.1, 1.8, 2.5
Which set is more variable?