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Chapter Three
Numerical Descriptive
Measures
1.
Age
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(1)
Smallest(1)
20.61
0.552
20
19
3.172
10.059
7.659
2.663
15
18
33
680
33
33
18
4.
Study /
Week
(hrs)
13.67
1.165
12.5
10
6.693
44.792
1.160
1.123
28
5
33
451
33
33
5
5.
Auto Cost
($)
21303.45
2971.803
18000
18000
16003.649
256116773.399
1.750
1.320
65000
1000
66000
617800
29
66000
1000
6.
7.
Alch bev / Sodas /
wk (#)
wk (#)
8.83
1.957
6
0
11.071
122.558
5.879
2.242
50
0
50
282.5
32
50
0
3.56
0.798
2
0
4.584
21.012
4.415
1.967
20
0
20
117.5
33
20
0
10.
9.
TV /
8.
No. units
video
Hrs. Paid this sem game /
/ wk (hrs)
(#)
wk (hrs)
9.53
2.140
4.5
0
12.105
146.531
0.969
1.291
40
0
40
305
32
40
0
16.56
0.359
16
16
2.061
4.246
2.604
0.983
10
13
23
546.5
33
23
13
7.41
0.917
5
4
5.269
27.757
0.072
1.020
18
2
20
244.5
33
20
2
11.
Movie
theater /
wk (#)
5.47
0.831
3.5
3
4.775
22.796
3.973
2.046
20
0
20
180.5
33
20
0
12.
$/wk
entertain
($)
46.41
6.335
40
100
35.836
1284.249
0.840
1.091
150
0
150
1485
32
150
0
17.
13.
last
volunteer
14.
16.
semest
19.
/ year
$ in
largest bal on bad class
18.
Int'l trips
(hrs)
wallet ($)
cc ($)
(#)
GPA now
(#)
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(1)
Smallest(1)
16.91
4.333
5
0
24.889
619.460
1.784
1.674
80
0
80
558
33
80
0
26.35
4.161
20
20
23.901
571.258
1.841
1.364
100
0
100
869.5
33
100
0
1143.79
296.888
700
0
1705.491
2908697.922
8.437
2.811
7300
0
7300
37745
33
7300
0
1.09
0.159
1
1
0.914
0.835
1.589
0.858
4
0
4
36
33
4
0
3.21
0.075
3.3
3.3
0.418
0.175
-0.445
-0.164
1.62
2.38
4
99.49
31
4
2.38
5.95
1.441
3
1
8.150
66.425
4.965
2.228
35
0
35
190.5
32
35
0
20.
Gamble
Indian
Casino
(#)
1.34
0.634
0
0
3.589
12.878
8.294
3.006
15
0
15
43
32
15
0
21.
Fly since
9/11
6.28
0.912
5
4
5.157
26.596
1.259
1.113
20
0
20
201
32
20
0
22.
fast car
(mph)
120.71
5.000
120
100
27.837
774.880
-0.912
0.272
110
70
180
3742
31
180
70
Commonly used Descriptive
Measures:
1) Measures of Central Tendency
2) Measures of Variation
3) Measures of Position
4) Measures of Shape
Measures of Central Tendency
Purpose: To
determine
the “centre”
of the data
values.
Measures of Central Tendency
Answer questions
• Where is the
middle of my data?
{Mean, Median,
Midrange}
• Which data value
occurs most often?
{Mode}
The Mean
The sample mean is
denoted by x-bar
The population mean is
denoted by µ (mu)
x = individual data values
X-bar = Σx / n
µ = Σx / N
Example:
The following are
accident data for a 5
month period:
6, 9, 7, 23, & 5
To calculate the average number of
accidents per month:
X-bar = Σx / n
X-bar = (6 + 9 + 7 + 23 + 5) ÷ 5
X- bar = 10.0
Statistic
What is the average person’s monetary
value to society?
The Median
is the centre value in a
data set when the data
are arranged
from
smallest
to
largest.
What do we call this ordering
process?
By arranging the data in an
Ordered Array:
5, 6, 7, 9, & 23
With an even number of observations, the
value that has an equal number of items to
the right and to the left is the Median.
Md = 7
To calculate the median with an
even number of observations,
average the two center values of
the ordered set.
Example: With an ordered array: 5, 6, 7, & 9
Md = ( 6 + 7 ) ÷ 2 = 6.5
If there is an odd number of
observations:
Md = (n + 1 ) ÷ 2
where
n = # of observations
Remember:
Median describes
the centrally
placed location
of a value
relative to the
rest of the data.
Question
Is the mean or median more
sensitive to extreme values
(outliers)?
Explain.
The mean is
affected by every
value.
The median is
unaffected by
extreme values.
The mean is pulled
toward extreme
values.
The median does
not use all data
information
available.
Question:
When dealing with data
that are likely to
contain outliers
(personal income,
ages, or prices of
houses), would the
Mean or Median be
preferred as the
measure of central
tendency? Why?
Think of the Median
as providing a more
“typical” or
“representative”
value of the situation.
The Mode
(Mo)
The value that
occurs most
frequently.
Questions?
1) Can there be more than one mode?
2) Is the mode affected by extreme
values?
3) For continuous variables, is it possible
that a mode does not exist? Explain?
4) Is the mode always a measure of
central tendency?
Give an example of when the mode
may provide more useful
information than the mean or the
median.
Example
From a purchaser’s
standpoint, the most
common hat or
jeans size is what
you would like to
know, not the
average hat or
jeans size.
Measures of Central Tendency
are useful.
Means
Medians
Modes
The use of any single statistic to
describe a complete distribution
fails to reveal important facts.
Dig Deeper!
Measures of Variation
Answers the question:
“How spread out are my data values?”
Consider Two Scenarios
Scenario 1:
Jack buys a car & pays
$1000.
Jill buys a car & pays
$21,000.
Average Price = $11,000
Scenario 2:
Bob buys a car & pays
$10,000.
Mary buys a car &
pays $12,000.
Average Price = $11,000
Based on the data, both
scenarios report the same
“average price.”
What’s the difference?
Quiz
Suppose you are a purchasing agent
for a large manufacturing company.
Your two suppliers fill your orders in
an average of 10 days.
The following histograms plot the
delivery time of the two suppliers.
Do the two suppliers have the
same reliability in terms of making
deliveries on time?
Homogeneity: the degree of
similarity within a set of data values.
The mean of a homogeneous data set
is far more representative of the typical
value than a mean of a heterogeneous
data set.
If all the data values in a sample are
identical, then the mean provides
perfect information, the variation is
zero, and the data are perfectly
homogeneous.
Variation:
the tendency of data values to
scatter about the mean, x-bar.
If all the data values in a sample are
identical, then the mean provides
perfect information, the variation is
zero, and the data are perfectly
homogeneous.
Commonly used Measures of
Variation:
•
•
•
•
Range
Variance
Standard Deviation
Coefficient of Variation (CV)
The Range
Range = H – L
The value of the range is strongly
influenced by an outlier in the sample
data.
Variance & Standard Deviation
During a five week production period, a
small company produced 5,9,16,17,& 18
computers, respectfully.
The average = 13 computers/wk
Describe the variability in these five
weeks of production.
Variance & Standard Deviation
Formulas for Variance & Standard
Deviation
1.
Age
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(1)
Smallest(1)
20.61
0.552
20
19
3.172
10.059
7.659
2.663
15
18
33
680
33
33
18
4.
Study /
Week
(hrs)
13.67
1.165
12.5
10
6.693
44.792
1.160
1.123
28
5
33
451
33
33
5
5.
Auto Cost
($)
21303.45
2971.803
18000
18000
16003.649
256116773.399
1.750
1.320
65000
1000
66000
617800
29
66000
1000
6.
7.
Alch bev / Sodas /
wk (#)
wk (#)
8.83
1.957
6
0
11.071
122.558
5.879
2.242
50
0
50
282.5
32
50
0
3.56
0.798
2
0
4.584
21.012
4.415
1.967
20
0
20
117.5
33
20
0
10.
9.
TV /
8.
No. units
video
Hrs. Paid this sem game /
/ wk (hrs)
(#)
wk (hrs)
9.53
2.140
4.5
0
12.105
146.531
0.969
1.291
40
0
40
305
32
40
0
16.56
0.359
16
16
2.061
4.246
2.604
0.983
10
13
23
546.5
33
23
13
7.41
0.917
5
4
5.269
27.757
0.072
1.020
18
2
20
244.5
33
20
2
11.
Movie
theater /
wk (#)
5.47
0.831
3.5
3
4.775
22.796
3.973
2.046
20
0
20
180.5
33
20
0
12.
$/wk
entertain
($)
46.41
6.335
40
100
35.836
1284.249
0.840
1.091
150
0
150
1485
32
150
0
Empirical Rule
Normally Distributed Data w/
Empirical Rule
Example: Empirical Rule
A company produces a lightweight valve that is
specified to weigh 1365 g.
Unfortunately, because of imperfections in the
manufacturing process not all of the valves
produced weigh exactly 1365 grams.
In fact, the weights of the valves produced are
normally distributed with a mean weight of 1365
grams and a standard deviation of 294 grams.
Question?
1) Within what range of weights would
approximately 95% of the valve weights
fall?
2) Approximately 16% of the weights would
be more than what value?
3) Approximately 0.15% of the weights
would be less than what value?
Answers:
1) 1365 +/- 2σ = 777 to 1953
2) 1365 + 1 σ = 1659
3) 1365 - 3 σ = 483
Example 2: Standard Deviation & the
Empirical Rule
A recent report states that for California the
average statewide price of a gallon of
regular gasoline is $1.52.
Suppose regular gas prices vary across the
state with a standard deviation of $0.08
are normally distributed.
With x-bar = $1.52 & s = $0.08
1) Nearly all gas prices (97.7%) should fall
between what prices?
2) Approximately 16% of the gas prices
should be less than what price?
3) Approximately 2.5% of the gas prices
should be more than what price?
Answers:
1) µ +/- 3σ = $1.28 and $1.76
2) $1.44 (Since 68% of the
prices lie w/in 1σ of the
mean, 32% lie outside this
range: 16% in each tail.
3) $1.68 (Since 95% of the
price lie w/in 2 σ of the
mean, 5% lie outside this
range: 2.5% in each tail.
Coefficient of Variation
Compares the variation between
two data sets with different means
and different standard deviations
and measures the variation in
relative terms.
Coefficient of Variation (CV) formula
CV = σ / µ (100)
CV Example 1
Spot, the dog, weighs 65 pounds. Spot’s
weight fluctuates 5 pounds depending on
Spot’s exercise level.
Sea Biscuit, the horse, weighs 1200 pounds.
Sea Biscuit’s weight fluctuates 125 pounds
depending on the number of rides Sea
Biscuit goes on.
Question?
Relatively
speaking, which
animal’s weight,
Spot or Sea
Biscuit’s, varies
the most?
Coefficient of Variation vs. Standard
Deviation
Some financial investors use
the coefficient of variation or
the standard deviation or
both as measures of risk.
What does the Coefficient of
Variation tell us about the risk of a
stock that the standard deviation
does not?
Relative to the amount invested in
a stock, the coefficient of variation
reveals the risk of a stock in terms
of the size of standard deviation
relative to the size of the mean (in
percentage).
CV Example 2:
SUPPOSE:
Five weeks of average prices for stock A are:
$57, $68, $64, $71, and $62.
While five weeks of average prices for stock B
are:
$12, $17, $8, $15, and $13.
QUESTION:
Relative to the
amount of money
invested in the
stock, which
stock, A or B, is
riskier?
Stock A vs. Stock B in terms of Risk
Stock A
Stock B
µ = 64.40
µ = 13
σ = 4.84
σ = 3.03
CV = σ/ µ (100) = 7.5%
CV = σ/ µ (100) = 23.3%
Measures of Position
Indicate how a particular value fits in with all
the other data values.
Commonly used measures of position are:
Percentiles
Quartiles
Z-scores
TO FIND THE LOCATION OF THE Pth
PERCENTILE:
• Determine n ∙ P /100 and use one of the
following two location rules:
• Location rule 1. If n ∙ P /100 is NOT a counting
number, round up, and the Pth percentile will be
the value in this position of the ordered data.
• Location rule 2. If n ∙ P /100 is a counting
number, the Pth percentile is the average of the
number in this location (of the ordered data) and
the number in the next largest location.
Use the two rules of percentiles and the following
data to determine both the 85th and the 50th
percentile for starting salary.
Starting Salary Data:
3130
2940
2920
2710
2850
2880
2755
3050
2880
3325
2950
2890
Step 1: Arrange the data in ascending
order
2710
2920
2755
2940
2850
2950
2880
3050
2880
3130
2890
3325
Step 2:
Use the formula for percentiles
n ∙ P /100
& Identify the 85th percentile given 12
observations
i = n (p /100) = 12 (85/100) = 10.2
Because i is not an integer, round
up.
The position of the 85th percentile is
the next integer greater than 10.2, the
11th position.
2710
2920
2755
2940
2850
2950
2880
3050
2880
3130
2890
3325
From the data, the 85th percentile is the
value in the 11th position, or $3130.
To calculate the 50th percentile, apply
step 2:
•
n ∙ P /100
•
i = 12 (50/100) = 6
• Because i is an integer, the 50th percentile is the
average of the sixth and seventh values:
(2890 + 2920) /2 = 2905.
2710
2920
2755
2940
2850
2950
2880
3050
2880
3130
2890
3325
Quartiles
• Quartiles are merely particular percentiles that
divide the data into quarters:
• Q1 = 1st quartile = 25th percentile (P25)
• Q2 = 2nd quartile = 50th percentile
(P50)
• Q3 = 3rd quartile = 75th percentile (P75)
• Quartiles are used as benchmarks, much like
the use of A,B,C,D, and F on exam grades.
Z- Scores
A z-score determines the relative position of any
particular data value x, and is expressed in
terms of the number of standard deviations
above or below the mean.
Measures of Shape
Measures of shape address
skewness and kurtosis.
Skewness
• Symmetric data = the sample mean =
sample median
• Right-skewed (positive) = mean >
median
• Left-skewed (negative) = mean <
median
Closing Example:
The number of defects in 10 rolls of
carpets are:
3, 2, 6, 0, 1, 3, 2, 1, 0, 4
•
•
What are the 75th percentile and the
50th percentile?
What are the mean, standard deviation,
and coefficient of variation?