Understanding Data
There are three basic concepts necessary to understand data
Measures of Central Tendency
Measures of Dispersion
Measures of Association
I.
Measures of Central Tendency
Defines a value about which the data clusters
A.
Mean (Arithmetic average)
Represents the most common value based on a calculation
B.
Median
Middle Value
Rank the data from lowest to highest, pick out the one in the middle. (half
above - half below)
C.
Mode
The most frequently occurring value
Count the occurrence of each value in the range of your data. The value with
the greatest number of occurrences is the mode.
Multi-modal
Ties in the number of occurrences
NOTES
The mean, mode and median values need not be the same.
For symmetric distributions the three are usually close.
The mean is influenced by outlying observations, while the median is not.
If there are values that are far removed from the observations, the median may be a better
measure of central tendency than the mean.
Eg. Yearling Sales Data
Mean Yearling Sale Price
Median Yearling Sale Price
27,295
9,162
What does this difference mean?
Stud fees of weanlings cataloged in Keeneland’s Nov. Sale
Interval
100-200,000
50-100,000
20-50,000
0-20,000
mean = 19,126
Number in Interval
fi
8
39
228
750
Σ=1,025
Midpoint
Xi
150,000
75,000
35,000
10,000
fiXi
1,200,000
2,925,000
7,980,000
7,500,000
Σ=19,605,000
II.
Measures of Dispersion
How much difference (variation) exists in the data.
The greater the variation in the data, the smarter you need to be in interpreting the data.
A.
Range
It is the difference between the minimum and maximum value.
Useful index relative to the mean
Does not account for the distribution of observations between min. and max.
values.
B.
Variance
Sum of squared differences from the mean divided by N-1.
Measures the variation in the data.
In squared terms - so not directly comparable to the mean.
Formula
C.
Standard Deviation
Square root of the variance
Directly comparable to the mean
Rule of Thumb – “If the SD is greater than the mean - your dealing with a lot of
variation in the data”.
Formula
D.
Standard error of the mean
Standard deviation of the distribution of the mean.
If s.e.m. is small, we expect the sample mean to be closer to the true mean of the
population
III.
Measures of Association
A
Scatter plots - graph illustrating relationship between two variables.
Examples
B
Covariance - a measure of how two variables vary relative to each other.
Measures negative or positive relationship
Has no relative (ie comparative) merit
No indication of statistical significance
C.
Correlation coefficient - measures the STRENGTH of association between two variables.
Measures negative or positive relationship
Ranges between 0 and 1
0 = no association
1 = perfect association
A measure of statistical significance can be calculated
Note statistical significance is important because it provides a measure of confidence in the analysis and
recommendations.
Statistical significance is impacted by the variance in the data. The greater the variance, the
harder it is to achieve target measures of significance.
Rules of thumb measures of statistical significance.
0.10 = 10% is the minimum for significance
0.05 = 5% is the desired level of significance
0.01 = 1% is a strong level of significance
D.
Regression Coefficient - Measures the association between one variable and a group of
variables.
Properties of a Normal Distribution
Necessary to understand
hypothesis testing
level of confidence
level of significance
sample vs. population characteristics
Normal distribution approximates many aspects of society and nature.
Often described as the “Bell-Shaped” curve. Each half is symmetric.
What does this tell you about the mean, median and mode?
Confidence Interval - is an interval estimate that provides a range of values in which the “true
value” lies between with an associated level of confidence.
Confidence interval calculation
mean + (confidence factor * standard error of mean)
Confidence factors:
68% level confidence factor = 1.0
90% level confidence factor = 1.64
95% level confidence factor = 1.96
Confidence intervals differ from the calculation that explains the dispersion of observations, i.e.,
data around the mean for a normal distribution.
Assuming you have a normal distribution, the range of values that includes 68% of the
observations is calculated as
mean + 1.0 * standard deviation
The range of values that includes 95% of the observations is calculated as
mean + 2.0 * standard deviation
Example Graph
The standard deviation is a statistic that tells you how tightly all the various examples are clustered
around the mean in a set of data. When the examples are pretty tightly bunched together and the bellshaped curve is steep, the standard deviation is small. When the examples are spread apart and the
bell curve is relatively flat, that tells you you have a relatively large standard deviation.
Computing the value of a standard deviation is complicated. But let me show you graphically what a
standard deviation represents...
One standard deviation away from the mean in either direction on the horizontal axis (the red area on
the above graph) accounts for somewhere around 68 percent of the people in this group. Two
standard deviations away from the mean (the red and green areas) account for roughly 95 percent of
the people. And three standard deviations (the red, green and blue areas) account for about 99
percent of the people.
If this curve were flatter and more spread out, the standard deviation would have to be larger in order
to account for those 68 percent or so of the people. So that's why the standard deviation can tell you
how spread out the examples in a set are from the mean.
ANOVA
Analysis of Variance - is in some ways a misleading name for a collection of statistical tests to
compare means.
Basic purpose is to compare a mean to a value or across groups.
Is the mean sale price > $75,000
Is the mean sale price of fillies < colts
One method would be to calculate means and compare the values.
Problem is there is no accounting for the variation in sale price between groups.
Anova - tests for the variability between groups.
Assumptions needed for ANOVA
Each group is independent
(One group doesn’t dictate the outcome of the second group)
The data on the group is from a normal distribution.
One-way ANOVA only one variable is used to classify data into different groups.
Variable - open
Open = 1
Breed = 0
Regression Analysis
Regression is a method of analysis that applies when research is dealing with an dependent
variable (variable of interest) and one or more independent variables (factors affecting the
dependent variable).
The purposes of regression are to quantify how the dependent variable is related to the
independent variables, and to make predictions on the dependent variable using
knowledge/estimates of independent variables.
Regression analysis is a measure of association between a dependent variable and one or more
independent variables.
Dependent variable = endogenous variable = value estimated by the regression model
Independent variable = exogenous variable = value determined outside the system but influences
the system by affecting the value of the endogenous variable.
Regression analysis uses data on dependent and independent variables to estimate an equation.
In regression, the dependent variable is a linear function of the independent variables.
Y = Intercept + Β1 X 1 + Β2 X 2 + ...+ Βn X n + ε
Where
Y = dependent variable
Intercept = an intercept term of a line (constant) - calculated by computer
Bi = estimated regression coefficient - (calculated by computer)
Xi = independent variable (variable and its value is specified by the modeler using the data)
i = index 1,2 up to n
n = number of independent variables in the model
e = error term
Interpreting Regression Results
1. Bi?or Slope Coefficients
Measures the change in the dependent variable, for a given change in an
independent variable.
Note, the change in the independent variable should be within the
range of the data.
The sign (+/-) on the B coefficient indicates a positive or negative correlation
between the independent and dependent variables. It is important that the
correlation corresponds to "theoretical" rules. If it doesn’t there is problems
with the results.
Standard Error of B, is a measure of dispersion of the estimated coefficient
which is in turn used in calculated the T statistic.
The T value is the calculated T statistic used to test if a coefficient is different
from zero. The T statistic is calculated by dividing an estimated slope
coefficient by its standard error, ( Bi / SEi ). It is used to test the null
hypothesis:
Ho : B i = 0
Ha: Bi > 0 or Ha: Bi < 0
It tells you if your results are statistically significant.
Standardized regression coefficient, labeled Beta in the SPSS output. Is a
dimension less coefficient that allows you to determine the relative
importance of independent variables in your model. The greater the Beta the
greater the effect that variable has on the dependent variable.
R2 measures the proportion of the variation in Y (independent variable) which is
explained by the regression equation.
R2 is often informally used as a goodness of fit statistic and to compare the validity
of regression results under alternative specifications of the independent variables in
the model.
R2 is a proportion, therefor varies from 0 to 100 percent, or 0 to 1 depending
on how its reported.
The addition of independent variables will never lower R2 and may increase
it.
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The adjusted R2 takes into account the number of independent variables in
the model.
The R2 of time series data is usually higher than cross section data.
The F statistic printed out can be used to test the significance of the R2 statistic. The
F statistic indicates if the model is statistically significant. The F statistic tests the
joint null hypothesis
B2 = B3 = ... = Bn = 0
Durbin-Watson tests for correlation in the error terms. If the error terms are
correlated, you can improve the regression by taking this information into account.
Multicollinearity tests for correlation between independent variables. The problem
with correlated variables is that they provide similar information and thus make it
hard to separate the effects of individual variables.
Elasticity
A common analysis procedure for economic and business studies is to calculate and
examine elasticities, which are readily available from regression model results.
An elasticity measures the effect of a 1 percent change in an independent variable on
the dependent variable.
Ei = β i *
AvgerageXi
AverageY
Elasticity at the mean is calculated as:
There are important economic implications if the elasticity is
Elastic Ei > 1.0 the dependent variable is sensitive to changes in Xi
Inelastic Ei < 1.0 the dependent variable is insensitive to changes in Xi
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Dummy Variables
Sometimes called indicator variables
Many variables of interest in business, economics, and social sciences are not
quantitative (continuous), but are qualitative (discrete). Qualitative variables can
be modeled by regression but they must represented as dummy variables.
Dummy variables are defined to take on a value of 0 or 1.
Example
Race
Sex
Income class
Location
It is no problem to have dummy variables as independent variables, but "advanced"
statistical methods are required for models with dummy variables as a dependent
variable.
A qualitative variable with c categories will be represented by c-1 dummy variables,
each taking on the values 0 and 1.
For Example - North, South, East, West
Category
East
West
North
South
X1
1
0
0
0
Dummy Variables
X2
X3
0
0
1
0
0
1
0
0
In this case if you used 4 dummy variables, the solution procedure used to solve the
regression problem will fail.
The coefficients on the dummy variables measures the differential impact between
the indicated category (receiving a value of 1) and the category or dummy which has
been dropped from the regression, in this case south.
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If the result of our regression analyzing differences by regions was:
Y = 250,000 + 50,000 X1 + 80,000 X2 - 6,000 X3
You would make the following interpretation for regional differences
South
Y = 250,000
East
Y = 250,000 + 50,000(1) = 300,000
West
Y = 250,000 + 80,000(1) = 330,000
North
Y = 250,000 - 6,000(1) = 244,000
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Regression
Descriptive Statistics
Mean
SALE_PRC 74349.099
STUD_FEE
17466.65
AGE
8.4730
BTYPPROD
.1526
GDSTPROD 7.151E-02
USRACEE
65741.88
HIP_NUMB 1807.0144
Std. Deviation
Sale Price in $
Stud fee of in vitro foal in $
Age in years
Black type producer = 1
Graded stakes producer = 1
Race earnings in US $
Hip number of mare
N
1664
1664
1664
1664
1664
1664
1664
Model Summary
Model
1
R
R Square
.697a
.486
Adjusted
R Square
.484
Std. Error of
the Estimate
105497.158
a. Predictors: (Constant), HIP_NUMB, AGE, USRACEE,
GDSTPROD, STUD_FEE, BTYPPROD
ANOVAb
Model
1
Regression
Residual
Total
Sum of
Squares
1.74E+13
1.84E+13
3.59E+13
df
6
1657
1663
Mean Square
2.902E+12
1.113E+10
F
260.769
Sig.
.000a
a. Predictors: (Constant), HIP_NUMB, AGE, USRACEE, GDSTPROD, STUD_FEE,
BTYPPROD
b. Dependent Variable: SALE_PRC
Coefficientsa
Model
1
Unstandardized
Coefficients
B
Std. Error
(Constant)
74963.604
8541.390
STUD_FEE
3.174
.118
AGE
-6125.122
778.121
BTYPPROD 18015.341 10646.326
GDSTPROD 39840.094 13501.066
USRACEE
.217
.021
HIP_NUMB
-13.312
2.816
Standardized
Coefficients
Beta
.547
-.177
.044
.070
.195
-.099
t
8.777
26.879
-7.872
1.692
2.951
10.571
-4.727
Sig.
.000
.000
.000
.091
.003
.000
.000
a. Dependent Variable: SALE_PRC
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