Download Lecture 3

STATISTICS FOR MANAGERS
LECTURE 3:
LOOKING AT DATA
AND MAKING
INFERENCES
1. LOOKING AT DATA
Central part of statistics: describing/
summarizing data
 Take into account that data come in
different types
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Sales
 Security rating
 Sector
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1.1 TYPES OF DATA
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Qualitative/categorical
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Attribute (nominal) data
Ranked (ordinal) data
Quantitative/numerical
Different types of data require different
treatment
One can use:
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Graphical summaries
Numerical summaries
1.2 QUALITATIVE DATA
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Graphical summaries
Pie chart
 Bar chart
 Ordered bar chart
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Numerical summaries
Frequency tables
 Percentage tables
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1.3 QUANTITATIVE DATA
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Graphical summaries
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Run chart: Example: stock prices
Histogram. Example: tick data
Box plot
Numerical summaries
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Arithmetic mean
Median
Standard deviation
Quartiles
1.3.1 RUN CHART
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For data collected over time (time series)
X-axis: date or number of data point
Y-axis: numerical value of data point
Things to look for
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Trends
Seasonality
Cycles
Outliers
1.3.1 RUN CHART (cont.)
Figure 2. Ratio of survey income to NAS consumption per capita.
P rovincial averages over time.
1.2
1.18
1.16
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
1988 1989
1990 1991 1992
1993 1994
1995 1996
Weighted by population
1997 1998 1999
Unw eighted
2000 2001
1.3.1 RUN CHART (cont.)
FIGURE 6: DAY 27/02/97
TRANSACTIONS-CLOCK TIME RELATIONSHIP
30000
20000
15000
10000
5000
Transactions
4281
4067
3853
3639
3425
3211
2997
2783
2569
2355
2141
1927
1713
1499
1285
1071
857
643
429
215
0
1
Seconds after 9 a.m.
25000
1.3.2 HISTOGRAM
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Determine the range of data
Decompose into bins of equal width
Count how many data points fall within each bin
Construct a bar chart based on these counts
Only problem: have to choose the width of the
bin
Allows to judge
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Center/location
Spread/variation
Symmetry
Outliers
1.3.2 HISTOGRAM (cont.)
F IGU R E 12 : D A Y 2 2 / 0 2 / 9 7
F R EQU EN C Y OF PR IC E C HA N GES ( T IC KS)
3500
3000
2500
2000
1500
1000
500
0
-4
-3
-2
-1
0
1
2
3
4
1.3.3 BOX PLOT
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Pack a lot of information in a single plot
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Box that extend from Q1 to Q3
A line inside the box indicates the median
Whiskers extend to bottom and top
Outliers are denoted by asterisks
Can compare data sets by lining up their
box plots.
1.3.3 BOX PLOT (cont.)
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1
1.2
1.4
1.6
1.8
Figure 3. Box plot of the ratio over time
1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
1.3.4 LOCATION
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Mean: sum up all the data and divide by
the number of points
Median
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Sort all the data from smallest to largest
Take the middle one (for odd number of data)
Take the average of the middle two (for even
number of data)
1.3.4 LOCATION (cont.)
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Mean versus median
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The median is more robust than the mean.
This means that it is less affected by extreme
observations
As a function of symmetry of the data
• Skewed to the left: mean<median
• Symmetric: mean approximately equal to median
• Skewed to the right: mean>median
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For skewed data the median is a more typical
observation
1.3.5 SPREAD
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Standard deviation
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Measures a typical deviation from the mean
Do not bother to do it yourself. Let EXCEL or
any other program do it for you.
Inter-quartile range
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Q1 is median of the bottom half of data
Q3 is median of the top half of data
IQR=Q3-Q1
1.3.6 OUTLIER DETECTION
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Graphically
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Use histogram
Look for points away from the rest
Numerically
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Points more than 3 standard deviations away
from the mean
Points more than 1.5*IQR away from Q1 and
Q3.
2. SAMPLING
All statistical information is based on
data
 The process of collecting data is
called sampling
 It is important to do it right
 Not everybody seems to understand
this importance
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2. GENERAL SITUATION
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We study a population
Can be a population in the strict sense
but it could also be an experiment
We are interested in certain
characteristics of the population
(parameter)
Want to learn as much as possible about
the parameter
2. EXAMPLES
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Population of Beijing
What is the average income?
 What percentage speak Cantonese?
 What percentage has Internet?
 What is the average price of the square
meter? (300.000 euros buy only 174
squares meters).
 What is the percentage of people that
have a DVD?
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2. BASIC PROBLEM
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Most populations are very large, or even
infinite
Hence it is typically impossible to exactly
determine a parameter (sometime
unfeasible from a cost perspective)
But it is possible to learn something
about a parameter
By collecting a sample from the
population we can obtain information
But the quality of information cna only be
as good as the quality of the sample
2. GOOD SAMPLE, IS THIS HARD?
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The sample has to be representative of
the population
In collecting data, we must not favor (or
disfavor) any particular segment of the
population
If we do we get biased samples
Biased samples yield biased estimates.
Example of biased samples: Internet.
2. NO VOLUNTEERS PLEASE
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A sample into which people have entered
at their own choice is called voluntary
response sample or self-selected.
This typically happens when polls are
posted on the internet, the TV,..
The scheme favors people with strong
opinions.
The resulting sample is rarely
representative of the population
As so often you get what you pay for!
(although something they pay for!)
2. HOW TO DO IT RIGHT
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Analogy
Have one ball per member of the
population
 Put all the balls in a big urn
 Mix well
 Take out n balls
 The result is called simple random
sample
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2. DO IT RIGHT...
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There are other ways to get
representative samples:
Stratified sampling
 Systematic sampling
 Cluster sampling (multistage)
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2. ... BUT AN ESTIMATE IS JUST THAT
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We can estimate a parameter from the
sample (a mean or a proportion)
... but an estimate is not equal to the
parameter!
... Because a sample is not equal to a
population
We must be aware of sampling error
Many people are not!
They sell us estimates as if they were
parameters. Shame on them. Will do it
right.
3. BASIC ESTIMATION
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General estimation
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We are interested in a population parameter
We collect a random sample
In a first step we estimate the parameter. This
is usually straighforward.
In a second step, we deal with the sampling
error.
This requires more work but it is worhwhile.
3.1. ESTIMATING A PROPORTION
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We are interested in a population proportion p
We collect a random sample size n
We compute the sample proportion p̂
This is a natural estimator for p,
But due to the sampling error is not equal to the
true parameter p
Goal: quantify the sample uncertainty contained
in the estimator of p
Intuition: the larger n the smaller the
uncertainty.
3.1. ESTIMATING A PROPORTION
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From probability theory we know that the
central limit theorem applies, under some
assumptions
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For n large, then with a probability 95% the
population proportion p will be in between
pˆ (1  pˆ )
n
For the interval to be trusted we require
pˆ  1.96
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npˆ  10 and
n(1  pˆ )  10
3.1. ESTIMATING A PROPORTION
A confidence interval has the following general
form
CI=estimator ±constant x std error (SE)
=estimator ± margin of error (ME)
For a proportion
pˆ (1  pˆ )
SE=
n
The SE does not depend on the confidence level
but the ME does because of the constant,
which is often abbreviated as z
3.1. ESTIMATING A PROPORTION
How is the ME affected by its various inputs?
ME= z
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3.
pˆ (1  pˆ )
n
As the confidence level increases the ME goes up.
As the estimator moves towards 0.5 the ME goes up
As n increases the ME goes down
We control de confidence level and n, but not the
estimator of p
3.1. ESTIMATING A PROPORTION
Want a CI with a specified level and a specif ME? How
large a sample size n is needed?
Use ME and solve for n
pˆ (1  pˆ )
ME= z
n
2
Solution: n   z pˆ (1  pˆ ) 
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ME
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Catch 22: we have not collected the sample yet, and
therefore the estimate for p is not available yet.
Solutions:
1. Worst case scenario estimator=0.5
2. Use a guess based on previous information
WHAT CONFIDENCE LEVEL?
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You may want a confidence level other than
95%.
Most common: 90%, 95% and 99%.
The formula for the CI is equal
You only change the constant 1.96
Conf. Level
90% 95% 99%
Constant z
1.64 1.96 2.57
Higher confidence level give a wider interval
3.2. ESTIMATING A MEAN
We are interested in a population mean and use as
estimator the sample mean.
CI=estimator ±constant x std error (SE)
=estimator ± margin of error (ME)
For a mean
SE=
s
n
CI= X  z
s
n
Rule of thumb: need more than 50 obs. To trust this
interval
3.2. ESTIMATING A MEAN
How is the margin of error (ME) affected by its
various inputs?
ME= z
s
n
As confidence level increases, ME goes up
2.
As s increases the ME goes up
3. As n increases the ME goes down
We control n and conf. level but not s.
1.
3.2. ESTIMATING A MEAN
Want a CI with a specified level and a specif ME? How
large a sample size n is needed?
Use ME and solve for n
s
ME= z
n
Solution: n   zs 
 ME 
2
Catch 22: we have not collected the sample yet, and
therefore the estimate for p is not available yet. In
this case there is not worst case scenario. Use a
guess based on previous information
3.3. HYPOTHESIS TESTING
If you care about wether a
parameter is equal to a certain
prespecified value, there is an
alternative to hypothesis testing
 Just check whether the prespecified
value is contained in the confidence
interval
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3.3. HYPOTHESIS TESTING
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If the prespecified value is contained in
the CI
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It is one of the (many) plausible values
So we can only make a weak positive
statement
If the prespecified values is not contained
in the CI
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It is not one of the plausible values
We can make a strong negative statement
3.3. CI PERFECT SUBSTITUTE
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We wonder if a parameter is equal to a
prespecified value?
The technique of hypothesis testing give a “yesor-no” answer (at a certain level of significance)
We can get the same from the level of
confidence
... But in addition we get the range of all
plausible values! This is valuable info.
Moral: a confidence interval tends to be safer
and more informative than hypothesis testing
3.4. CAVEAT
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Our confidence intervals are simple, yet
powerful.
But you can’t use them blindly!
Two conditions to trust them
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We need a large sample
We need a random sample
Data that is collected over time is usually NOT a
random sample: the data point of today is
usually related to the data point yesterday
Stock returns are an exception to this rule.
Small sample and time series are for the pros!
3.5. WHAT ABOUT OTHER PARAMETERS
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We have covered confidence intervals for
 Population proportions
 Population means
Both are based on the CLT
There are other interesting parameters
 Population median
 Population standard deviation
 etc
Unfortunately they cannot be handle by the CLT
CI can be constructed but the corresponding techniques
are more difficult.