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MA-250 Probability and Statistics
Nazar KhanPUCIT
Lecture 5
Measurement Error
• In an ideal world, if the same thing is measured several times, the same result would be obtained each time.
• In reality, there are differences.– Each result is thrown off by chance error.
Individual measurement = exact value + chance error
Measurement Error
• No matter how carefully it is made, a measurement could have been different than it is.
• If repeated, it will be different.• But how much different?– Simple answer:• Repeat the measurements.• Consider the SD
Measurement Error
• Variability in measurements reflects the variability in the chance errors
Individual measurement = exact value + chance errorSD(Measurements) = exact value + SD(chance error)
Measurement Error
• An outlier can affect the – Mean– Standard Deviation
• What if the majority data follows a normal curve?– The outliers will affect the mean and SD such that
the 68-95-99 rule might not be followed.• Solution: remove the outliers and then do
the normal approximation.
Outliers
Outliers
1SD is covering ~86% of the data, so the normal approximation cannot be used.
Outliers
1SD is covering ~68% of the data, so the normal approximation can be used now.
Outliers Removed
Bias
• Chance error changes from measurement to measurement – sometimes positive and sometimes negative.
• Bias affects all measurements in the same way.
Individual measurement = exact value + chance error + bias
below.
Dealing with bi-variate data
• So far, we have dealt with uni-variate data– One variable only– Age, Height, Income, Family Size, etc.
• How can we study relationships between 2 variables?– Relationship between height of father and height
of son– Relationship between income and education
• Answer: scatter diagrams
Can we summarize the scatter diagram?
Summarizing a Scatter Diagram
• Mean• Horizontal SD• Vertical SD
• But these statistics do not measure the strength of the association between the 2 variables.
• How can we summarize the strength of association?
Same mean and horizontal and vertical SDs but the left figure shows more association between the 2 variables.
Correlation
• Correlation measures the strength of association between 2 variables– As one increases, what happens to the other?
• Denoted by r• r=average(x in standard units* y in standard units)
Average = 0.4
How does r measure association strength?
• r=average(x in standard units* y in standard units)• When both x and y are simultaneously above or below their
means, their product in standard units is +ve.• When +ve products dominate, the average of products is +ve
(i.e., correlation r is +ve).• Similarly for –ve products.
Correlation
• r is always between 1 and -1.• r=0 implies no association between x and y.• |r|=1 implies strong linear association.– r=1 implies perfectly linear, positive association.– r=-1 implies perfectly linear, negative association.
Very hard to predict y from x
Easy to predict y from x
Negative association between x and y
Some Properties of the Correlation Coefficient
• r has no units. (Why?)– The correlation between June temperatures for
Lahore and Karachi will be the same in Celcius and Fahrenheit.
• r(x,y)=r(y,x) (Why?)
Exceptions!
Strong linear association without outlier but outlier brings r down to almost 0
r measures linear association only, not all kinds of association.
Association is not Causation!
• Correlation measures association but association is not causation.– In kids, shoe-size and reading skills have a strong
positive linear association. Does a larger foot improve your reading skills?
Summary
• Measurement Errors– Chance Error– Bias
• SD(chance errors) = SD(measurements)• Let’s us determine if an error is by chance or not.
• Correlation measures strength of linear association between 2 variables.– Between -1 and 1
• Not useful for summarizing scatter diagrams with – Outliers, or– Non-linear association.
• Association is not causation.
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