Chapter 3.2

Variance and Standard Deviation

Comparison of Outdoor PaintA 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 Brand B10 3560 4550 3030 3540 4020 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.

The standard deviation is the square root of the variance.

X = individual value, N = population size, µ = population mean

n

X 22 )(

n

X 22 )(

Steps for finding Variance and Standard Deviation1. Find the mean of the data2. Subtract the mean from each data value3. Square each result4. Find the sum of the squares5. Divide the sum by N to get the variance6. Take the square root of the variance to get

the standard deviation(It might be helpful to organize the data in a table)

Find the Standard Deviation and Variance of both brands of paint

Brand A Brand B

Values, X X - µ (X - µ)2 Values, X X - µ (X - µ)2

10     35    

60     45    

50     30    

30     35    

40     40    

20     25    

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:

Standard Deviation:

)1(

)()( 222

nn

XXns

)1(

)()( 22

nn

XXns

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?