Why is the study of variability Why is the study of variability important?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 VariabilityMeasures of Variability

• range (max-min)• interquartile range (Q3-Q1)• deviations • variance• standard deviation

xx

2

Lower case Greek letter sigma

Suppose that we have these data values:

24 34 26 30 3716 28 21 35 29

Find the mean.

Find the deviations. xx

What is the sum of the deviations from the mean?

24 34 26 30 3716 28 21 35 29

Square the deviations: 2xx

Find the average of the squared deviations:

n

xx 2

The average of the deviations squared is called the variance.

Population Sample

2 2s

parameter statistic

Calculation of variance Calculation of variance of a sampleof a sample

1

2

2

n

xxs n

df

A standard deviation is a measure of the average deviation from the mean.

Calculation of standard Calculation of standard deviationdeviation

1

2

n

xxs n

Degrees of Freedom (df)Degrees of Freedom (df)

• n deviations contain (n - 1) independent pieces of information about variability

Use calculatorUse calculator

Which measure(s) of variability is/are

resistant?

Linear transformation ruleLinear 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:

xbax

xbax

a

ba

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?

25.61$)25.1(2530

33.8$31

25

Rules for Combining two variablesRules 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:

baba

baba

22baba

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 3.5 0.7

Assembly 21.8 2.4

Tuning 12.3 2.7

minutes6.373.128.215.3 T

minutes680.37.24.27.0 222 T