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Lecture

Lecture 5

11 Pages
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Department
Quantitative Methods
Course Code
QMS 102
Professor
Clare Chua

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1
Measures Central Tendency
Last lecture we covered:
Measures
Central
Tendency
MODE MEDIANMEAN
OBJECTIVES
1. Which is the preferred measure? Mean or
Median
2. Shape
3
MfPitiPtil
3
.
M
easure o
f
P
os
iti
on <
P
ercen
til
e>
4. Measures of variation
How to compute mean?
Data
Raw Data
Example 3.1
(page 133)
Grouped Data
In the form of
Frequency
Distribution
Example 3.7
(page 136)
Weights
Examples 3.2, 3.3, 3.4,
3.5
(pages 133, 134,135 )
Total
Example 3.6
(page 135)
Median Salary by City - Job: Family Physician /
Doctor (Canada)
Which is the preferred
measure? Mean or Median
Page 139
www.notesolution.com
2
Illustration 1
The following data represents the prices of houses sold last week
in a particular neighbourhood. The data is:
$240,000 245,000 250,000 260,000 265,000
Mean house price = $252,000
You would get the impression as to the size and type of the typical
hithihbhd
h
ouse
i
n
th
e ne
i
g
hb
our
h
oo
d
.
Median house price = $250,000
You would get the same impression as to the size and type of the
typical house in the neighbourhood.
If the mean and median are CLOSE,
then the mean will give the correct impression and is the
preferred measure.
Illustration 2
The following data represents the prices of houses sold
last week in a particular neighbourhood. The data is:
$240,000 245,000 250,000 260,000 465,000
Mean house price = $292,000
You would
g
et the im
p
ression as to the size and t
yp
e of
gp yp
the typical house in the neighbourhood.
Median house price = $250,000
You would get the different impression as to the size and
type of the typical house in the neighbourhood.
If the mean and median are NOT CLOSE,
then the median will give the correct impression
Note: The median didn’t change
Two issues
1. Median is less sensitive to extreme
scores
2. How do we conclude that the mean
and median are close or not close?
Median and Mean
The median is the middle of a distribution:
half the scores are above the median and
half are below the median.
The median is less sensitiveto extreme
scores than the mean and this makes it a
better measure than the mean for highly
skewed distributions. The median income
is usually more informative than the mean
income, for example
How do we conclude that the
mean and median are close or
not close?
not
close?
We need some rules.
Heuristic Rule
How do we conclude that the
mean and median are close or
not close?
not
close?
10% Rule
The "Ten Percent Rule" is a
general rule of thumb
www.notesolution.com
3
What is 10% Rule?page 140
Which is a better measure of tendency, mean or median?
To help you determine if the mean and median are really different, use the
following rule:
1: Calculate the ‘difference' between the mean and median.
Difference = ~mean- median~
2: Calculate ‘10% of the smaller' of the mean or median.
10% smaller = Minimum (10% umean , 10% umedian)
3: Compare ‘Difference’ with ‘10% smaller
The following decision rule is:
If Difference is less than 10% of smaller, you conclude that mean is
approximately equal to median, in which case the mean is the preferred
measure.
If Difference is greater than 10% of smaller, you conclude that mean is
NOTequal to median, in which case the median is the preferred measure.
Example 1
Consider the following:
Mean = 12 and Median = 14
Apply the 10% rule, you have
1: Difference = ~Mean-Median~= ~12 - 14~=2
2: 10% smaller = Minimum ( 10% u12 , 10%
u
14)
= Minimum ( 1.2 , 1.4) = 1.2
3: Since Difference of 2 is greater than 10%
smaller of 1.2, which indicate that mean is NOT
equal to median, in which case the median is the
preferred measure.
Illustration
Mean house price = $252,000
Median house price = $250,000
Difference = Mean Median
Difference= 252,000-250,000
= 2,000
10% smaller
=
10%(250 000)
Mean house price = $292,000
Median house price = $250,000
Difference = Mean Median
Difference= 292,000-250,000
= 42,000
10% smaller
=
10%(250 000)
10%
smaller
10%(250
,
000)
= 25,000
Conclusion: 2,000 < 25,000
If Difference < 10% of
smaller ; mean is equal to
median; mean is the
preferred measure
10%
smaller
10%(250
,
000)
= 25,000
Conclusion: 42,000 > 25,000
If Difference >10% of
smaller ; mean is NOT
equal to median; median
is the preferred measure
3.3 Descriptive Statistics III
Shape
Page 160
Shape
Describe a set of numerical data
Pattern of the distribution of data values
throughout the entire range of all the
3 shapes:
(1) Symmetrical
(2) Left-skewed (Negative Skewed)
(3) Right-skewed (Positive Skewed)
Left-Skewed
Mostof the data are in the upper
Median
XX
Median
Right-Skewed
Most
of
the
data
are
in
the
upper
portion of the distribution
Mean < Median
Most of the data are in the
lower portion of the
distribution
A long tail and distortion to the left
that is caused by some extremely
small values.
These extremely small values pull
the mean downward so that
A long tail and distortion
to the right that is caused by
some extremely large
values.
These extremely large values pull
the mean downward so that
Mean > Median
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Description
OBJECTIVES Last lecture we covered: 1. Which is the preferred measure? Mean or Median 2. Shape MeeasuuessCeentallTendeency 3. Measure off PPosiittiionPercenttiille> 4. Measures of variation MODE MEDIAN MEAN How to compute mean? Data Raw Data Example 3.1 Grouped Data (page 133) In the form of Distribution Examples 3.2, 3.3, 3.4,tal Example 3.7 3.5 Example 3.6 (page 136) (pages 133, 134,135 )age 135) Median Salaryby C ity - Job: Family ian c Doctor (Canada) Which is the preferred measure? Mean or Median Page 139 1 www.notesolution.com Illustration 1 Illustration 2 The following data represents the prices of houses sold laThe following data represents the prices of houses sold in a particular neighbourhood. The data is: last week in a particular neighbourhood. The data is: $240,000 2250,0260,0265,000 $240,000 245,000 250,000 260,000 465,000 Mean house price = $252,000 Mean house price = $292,000 You would get the impression as to the size and type of thYou would get the ipression as to the size ype of house ihe neibourhood.h d the typical house in the neighbourhood. Median house price = $250,000 Median house price = $250,000 Note: The median didnt change You would get the same impression as to the size and type You would get the different impression as to the size and typical house in the neighbourhood. type of the typical house in the neighbourhood. If the mean and median are, then the mean will give the correct impression and is the If the mean and median areOT CLOSE, preferred measure. then the median will give the correct impression Median and Mean Two issues The median is the middle of a distribution: half the scores are above the median and half are below the median. 1. Median is less sensitive to extreme The median is less sensitive to extreme scores scores than the mean and this makes it a 2. How do we conclude that the mean better measure than the mean for highly and median are close or not close? skewed distributions. The median income is usually more informative than the mean income, for example How do we conclude that the How do we conclude that the mean and median are close or mean and median are close or noot closee? nottclosse?? We need some rules. 10% Rule Heuristic Rule The Ten Percent Rule is a general rule of thumb 2 www.notesolution.com
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