MA-250 Probability and Statistics Nazar Khan PUCIT 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 error SD(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 rule might not be followed. Solution: remove the outliers and then do the normal approximation.

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.