Slide 1Lecture 4: Measures of Variation

• Given a stem –and-leaf plot Be able to find

» Mean • (40+42+3*50+51+2*52+64+67)/10=46.7

» Median• (50+51)/2=50.5

» mode • 50

Stem (tens) Leaves (units)

4 0 2

5 0 0 0 1 2 2

6 4 7

# of phones (x)

f fx Cum Freq

4 2 8 2

3 4 12 6

2 5 10 11

1 16 16 27

0 13 0 40=n

Review of Lecture 3: Measures of Center

• Given a regular frequency distribution Be able to find

» Sample size •2+4+5+16+13=40

» Mean •(8+12+10+16+0)/40=1.15

» Median:•average of the two middle values=1

Mediangroup

5th 6th

Slide 22.5 Measures of Variation

Statistics handles variation. Thus this section one of the most important sections in the entire book

Measure of Variation (Measure of Dispersion): A measure helps us to know the spread of a data set.

Candidates: Range Standard Deviation, Variance Coefficient of Variation

Slide 3Definition

The range of a set of data is the difference between the highest value and the lowest value

Range=(Highest value) – (Lowest value)

Example: Range of {1, 3, 14} is 14-1=13.

Slide 4Standard Deviation

The standard deviation of a set of values is a measure of variation of values about the mean

We introduce two standard deviation: • Sample standard deviation• Population standard deviation

Slide 5Sample Standard Deviation Formula

Formula 2-4

(x - x)2

n - 1S =

Sample size

Data value

Slide 6Sample Standard Deviation

(Shortcut Formula)

Formula 2-5

n (n - 1)s =

n (x2) - (x)2

Slide 7Example: Publix check-out waiting times in minutes

Data: 1, 4, 10. Find the sample mean and sample standard deviation.

x1 -4 16

4 -1 1

10 5 25

15 42

xx 2)( xx

min0.53

15x

min6.421

13

42

1

2

n

xxs

1

16

100

117

2x

x2)( xx 2 x

Using the shortcut formula:

min6.421

6

126

6

225351

)13(3

15)117(3

)1(

2

22

nn

xxns

n=3

Slide 8Standard Deviation -

Key Points

The standard deviation is a measure of variation of all values from the mean

The value of the standard deviation s is usually positive and always non-negative.

The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others)

The units of the standard deviation s are the same as the units of the original data values

Slide 9Population Standard

Deviation

2 (x - µ)

N =

This formula is similar to Formula 2-4, but instead the population mean and population size are used

Slide 10

Population variance: Square of the population standard deviation

Variance

The variance of a set of values is a measure of variation equal to the square of the standard deviation.

Sample variance s2: Square of the sample standard deviation s

Slide 11Variance - Notation

standard deviation squared

s

2

2

}Notation

Sample variance

Population variance

Slide 12Round-off Rule

for Measures of Variation

Carry one more decimal place than is present in the original set of data.

Round only the final answer, not values in the middle of a calculation.

Slide 13Definition

The coefficient of variation (or CV) for a set of sample or population data, expressed as a percent, describes the standard deviation relative to the mean

100%s

xCV =

100%CV =

Sample Population

• A measure good at comparing variation between populations• No unit makes comparing apple and pear possible.

Slide 14Example: How to compare the variability

in heights and weights of men?

Sample: 40 males were randomly selected. The summarized statistics are given below.

Sample mean Sample standard deviation

Height 68.34 in 3.02 in

Weight 172.55 lb 26.33 lb

Solution: Use CV to compare the variability

Heights:

Weights: %26.15%10055.172

33.26%100

%42.4%10034.68

02.3%100

x

sCV

x

sCV

Conclusion:

Heights (with CV=4.42%) have considerably less variation than weights (with CV=15.26%)

Slide 15Standard Deviation from a

Frequency Distribution

Use the class midpoints as the x values

Formula 2-6

n (n - 1)S =

n [(f • x 2)] - [(f • x)]2

Slide 16Example: Number of TV sets

Owned by households• A random sample of 80 households was selected

• Number of TV owned is collected given below.

TV sets (x) # of Households (f) fx fx2

0 4 0 0

1 33 33 33

2 28 56 112

3 10 30 90

4 5 20 80

Total 80 139 315

sets 0.1

6320

5879

)180(80

)139()315(80

)1(

)( (b)

sets7.180

139 (a)

222

nn

fxfxns

x

Compute:

(a) the sample mean

(b) the sample standard deviation

Slide 17Estimation of Standard

DeviationRange Rule of Thumb

For estimating a value of the standard deviation s,

Use

Where range = (highest value) – (lowest value)

Range

4s

Slide 18Estimation of Standard

DeviationRange Rule of Thumb

For interpreting a known value of the standard deviation s, find rough estimates of the minimum and maximum “usual” values by using:

Minimum “usual” value (mean) – 2 X (standard deviation)

Maximum “usual” value (mean) + 2 X (standard deviation)

Slide 19Definition

Empirical (68-95-99.7) Rule

For data sets having a distribution that is approximately bell shaped, the following properties apply:

About 68% of all values fall within 1 standard deviation of the mean

About 95% of all values fall within 2 standard deviations of the mean

About 99.7% of all values fall within 3 standard deviations of the mean

Slide 20The Empirical Rule

FIGURE 2-13

Slide 21The Empirical Rule

FIGURE 2-13

Slide 22The Empirical Rule

FIGURE 2-13

Slide 23Recap

In this section we have looked at:

Range

Standard deviation of a sample and population

Variance of a sample and population

Coefficient of Variation (CV)

Standard deviation using a frequency distribution

Range Rule of Thumb

Empirical Distribution

Slide 24Homework Assignment 4

• problems 2.5: 1, 3, 7, 9, 11, 17, 23, 25, 27, 31

• Read: section 2.6: Measures of relative standing.