slide 1 lecture 4: measures of variation given a stem –and-leaf plot be able to find »mean...
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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.