Unit 3Section 3-3 – Day 1

3-3: Measures of Variation Range– the highest value minus the lowest

value. The symbol R is used for range.

Variance– the average of the squares of the distance each value is from the mean.

The symbol is σ2

Standard Deviation– the square root for the variance.

The symbol is σ

The Empirical Rule (Normal Rule)

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Applies when the distribution is bell-shaped (or what is called normal).

Approximately 68% of the data falls within 1 standard deviation of the mean.

Approximately 95% of the data falls within 2 standard deviation of the mean.

Approximately 99.7% of the data falls within 3 standard deviation of the mean.

Empirical Rule

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Finding the Variance (Population)

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The formula for finding the variance of a population is:

X = individual valueμ = population meanN = population size

Finding the Variance (Sample)

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The formula for finding the variance of a population is:

X = individual valuen = sample size

Steps for Finding Variance

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Find the mean. Find the difference between each data

value and the mean. Square each difference. Find the sum of their squares. Divide the sum by the number of data

entries.

Finding the Variance

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Find the variance for the population below:

Value X-μ (X-μ)2

10

20

30

40

50

60

Finding the Variance

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Find the variance for the amount of European auto sales for a sample of 6 years as shown.

Value X-μ (X-μ)2

11.2

11.9

12.0

12.8

13.4

14.3

Steps for Finding Standard Deviation

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First, determine the variance of the data set.

Then, take the square root of the variance.

Finding the Standard Deviation

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Find the standard deviation for both of our previous examples.

Finding the Standard Deviation for Grouped Data

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Formula:

Finding the Variance and Standard Deviation for Grouped Data

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Make a table as shown

Find the midpoints of each class and place them in column C.

Multiply the frequency by the midpoint for each class, and place the product in column D.

Multiply the frequency by the square of the midpoint for each class, and place the product in column E.

Find the sums of column B, D, and E.

A B C D E

Class Frequencyf

MidpointXm

f*Xm f*Xm2

To find the Variance: Take the Sum of E, subtract away the

quantity of the Sum of D squared divided by the Sum of B.

Then, divide your value by the Sum of B minus one.

To find the Standard Deviation, take the square root of the variance.

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Finding the Variance and Standard Deviation: Grouped Data

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Find the variance and standard deviation for the grouped data below:

Class Frequency

fMidpoint

Xm

f*Xm f*Xm2

5.5 – 10.5 1

10.5 – 15.5

2

15.5 – 20.5

3

20.5 – 25.5

5

25.5 – 30.5

4

30.5 – 35.5

3

35.5 – 40.5

2

Uses of the Variance and Standard Deviation Variances and standard deviations can be used to

determine the spread of data. If they are large, the data is more dispersed Useful in comparing two or more data sets to determine which is

more variable. Variances and standard deviations are used to determine

the consistency of a variable. Example: manufacturing nuts and bolts, the variation in

diameters must be low so the parts fir together. Variances and standard deviations are used to determine

the number of data values that fall within a specific interval in a distribution.

Variance and standard deviations are used quite often in inferential statistics.

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Coefficient of Variation – the standard deviation divided by the mean.

Notation: CVar The result is expressed as a percentage.

Example: The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variations of the two.

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Range Rule of Thumb A rough estimate of the standard deviation. Standard deviation is approximately the range

divided by four.

Chebyshev’s theorem – the proportion of values from a data set that will fall within k standard deviations of the mean will be at least 1-1/k2, where k is a number greater than 1.

This can be applied to any data set regardless of its distribution or shape

Also states that three-fourths (or 75%) of the data values will fall within 2 standard deviations of the mean.

Chebyshev’s theorem

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A survey of local companies found that the mean amount of travel allowance for executives was $0.25 per mile. The standard deviation was $0.02. Using Chebyshev’s theorem, find the minimum percentage of the data values that will fall between $0.20 and $0.30.

Step 1: Subtract the mean from the larger value

Step 2: Divide the difference by the standard deviation (find k)Step 3: Use Chebyshev’s theorem to find the %

Homework Pg 130-131: 18, 21, 31, 32, 35

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