Section 3-2

Measures of Variation

Objectives Compute the range, variance, and

standard deviation

Example: You own a bank and wish to determine which customer waiting line system is best Branch A (Single Waiting Line)

Branch B (Multiple Waiting Lines)

(in minutes)6.5 6.6 6.7 6.8

7.1 7.3 7.4 7.7

7.7 7.7

(in minutes)4.2 5.4 5.8 6.26.7 7.7 7.7 8.5

9.3 10.0Find the measures of central tendency and compare the two customer waiting line systems. Which is best?

Which is best –Branch A or Branch B?

Branch A (Single Wait Line)

Branch B (Multiple Wait Lines)

Mean 7.15 7.15Median 7.2 7.2Mode 7.7 7.7Midrange 7.1 7.1

Does this information help us to decide which is best?

Let’s take a look at the distributions of each branch’s

wait times

Which is best –Branch A or Branch B?

InsightsSince measures of central tendency are

equal, one might conclude that neither customer waiting line system is better.

But, if examined graphically, a somewhat different conclusion might be drawn. The waiting times for customers at Branch B (multiple lines) vary much more than those at Branch A (single line).

Measures of VariationRange VarianceStandard Deviation

RangeRange is the simplest of the three

measuresRange is the highest value (maximum)

minus the lowest value (minimum) Denoted by R

R = maximum – minimum

Not as useful as other two measures since it only depends on maximum and minimum

Example: You own a bank and wish to determine which customer waiting line system is best Branch A (Single Waiting Line)

Branch B (Multiple Waiting Lines)

(in minutes)6.5 6.6 6.7 6.8

7.1 7.3 7.4 7.7

7.7 7.7

(in minutes)4.2 5.4 5.8 6.26.7 7.7 7.7 8.5

9.3 10.0Find the range for each branch

VarianceAllows us to look in more detail at how much

each piece of data differs from the mean (measure of center)----page 115

Variance is an “unbiased estimator” (the variance for a sample tends to target the variance for a population instead of systematically under/over estimating the population variance)

Serious disadvantage: the units of variance are different from the units of the raw data (variance = units squared or (units)2

NotationsPopulation Variance Sample Variance

sizesamplenmeansamplex

podataaisxwhere

nxx

s

int

1)( 2

2

Standard DeviationIs the square root of the variance (gives the

same units as raw data)Provides a measure of how much we might

expect a typical member of the data set to differ from the mean. The greater the standard deviation, the more the

data is “spread out”Standard deviation can NOT ever be negative

Allows us to interpret differences from the mean with a sense of scale (make a judgment of whether a difference is large or small, in a systematic way)

NotationsPopulation Standard Deviation Sample Standard Deviation

sizesamplenmeansamplex

podataaisxwhere

nxx

s

int

1)( 2

NO WORRIES!!!Since the formulas are so involved, we will

use our calculators or MINITAB to determine the variance or standard deviation and focus our attention on the interpretation of the variance or standard deviation

Why did I bother showing you? So you have some sense of what is going on behind the scenes and realize it is not magic, it’s MATH

Uses of the Variance and Standard DeviationVariances and standard deviations are used to

determine the spread of the data. If the variance or standard deviation is large, the

data is more dispersed. This information is useful in comparing two or more data sets to determine which is more (most) variable

The measures of variance and standard deviation are used to determine the consistency of a variableFor example, in manufacturing of fittings, such

as nuts and bolts, the variation in the diameters must be small, or the parts will not fit together

Example: You own a bank and wish to determine which customer waiting line system is best Branch A (Single Waiting Line)

Branch B (Multiple Waiting Lines)

(in minutes)6.5 6.6 6.7 6.8

7.1 7.3 7.4 7.7

7.7 7.7

(in minutes)4.2 5.4 5.8 6.26.7 7.7 7.7 8.5

9.3 10.0Find the standard deviation for each branch

Which is best –Branch A or Branch B?

Branch A (Single Wait Line)

Branch B (Multiple Wait Lines)

Mean 7.15 7.15Median 7.2 7.2Mode 7.7 7.7Midrange 7.1 7.1Standard Deviation 0.48 1.82

Does this information help us to decide which is best?

“Usual” Values

Minimum “usual” value =

Maximum “usual” value = sx 2

sx 2

Empirical (Normal) RuleOnly applies to bell-shaped (normal) symmetric distributionsUsed to estimate the percentage of values within a few

standard deviations of the mean

Chebyshev’s Theorem (p.123)

2

11k

• Specifies the proportions of the spread in terms of the standard deviation

• Applies to ANY distribution

• The proportion of data values from a data set that will fall with k standard deviations of the mean will be AT LEAST

ExampleLengths of Longest 3-point kick for NCAA Division 1-A Football (in yards) 29 31 31 32 32 34 35 36 3737 43 43 45 45 47 54 54 5557 59

1) Construct a frequency distribution of the lengths of 3-point kicks. Use 7 classes with a class width of 5, beginning with a lower class limit of 25.

2) Use MINITAB to create a histogram. Does the histogram appear symmetric, skewed to right, or skewed to left?

3) Use MINITAB to create a dotplot. Does the histogram appear symmetric, skewed to right, or skewed to left?

4) Use MINITAB to find mean, median, maximum, minimum, and standard deviation

5) Use formulas and MINITAB results to find mode, midrange, range, variance, minimum “usual” value, and maximum “usual” value

AssignmentPage 124 #1-15 odd