Relationships BetweenMeasurement Variables Statistics lecture 4
Thought Question 1 There is a positive correlation between SAT score and GPA. For used cars, there is a negative correlation between age of the car and selling price. What does that mean?
Thought Question 2 If you had a scatter plot comparing the heights of a number of fathers and their adult sons, how could you use it to predict the adult height of a child?
Thought Question 3 Would these pairs of variables have a positive correlation, a negative correlation, or no correlation? • Calories eaten per day and weight • Calories eaten per day and IQ • Vinho consumed and driving ability • Number of priests and amount of liquor sold in Portugal cities. • Height of husbands and heights of wives
Goals for this lecture • Get the idea of a statistical relationship and statistical significance • Understand the meaning of correlation between two measurement variables • Learn how to use the linear relationship between two variables to predict one value, given the other
Relationships • Deterministic: You can predict one variable exactly given another (example: distance at a constant speed given time) • Statistical: You can describe a relationship between variables, but it isn’t precise because of natural variability (example: the average relationship between height and weight.)
Statistical Significance Often we must use a sample to tell us about a population. We want to know if any relationships observed in the sample are “real” and not just chance.
Rule of Thumb A statistical relationship is considered significant if it is stronger than 95% of the relationships we’d expect to see by chance.
Be aware of sample size Statistical significance is affected by sample size: • It’s easy to rule out chance if you have lots of observations (but the relationship still may not be strong or useful.) • On the other hand, even a strong relationship may not achieve statistical significance if the sample is small.
Strength of Relationship? Correlation (also called the correlation coefficient or Pearson’s r) is the measure of strength of the linear relationship between two variables. Think of strength as how closely the data points come to falling on a line drawn through the data.
Features of Correlation • Correlation can range from +1 to -1 • Positive correlation: As one variable increases, the other increases • Negative correlation: As one variable increases, the other decreases • Zero correlation means the best line through the data is horizontal • Correlation isn’t affected by the units of measurement
r = +.4 r = +1 r = +.8 Positive Correlations r = +.1
Negative Correlations r = -.4 r = -.1 r = -.8 r = -1
Zero correlation r = 0 r = 0
Number of PointsDoesn’t Matter r = .8 r = .8
Important! Correlation does not imply causation.
Linear Regression In addition to figuring the strength of the relationship, we can create a simple equation that describes the best-fit line (also called the “least-squares” line) through the data. This equation will help us predict one variable, given the other.
Remember 9th Grade Algebra? x = horizontal axis y = vertical axis Equation for a line: y = slope*x + intercept or as it often is stated: y = mx + b
Don’t panic! You won’t have to calculate the least-squares line equation yourself. Instead, you can use functions built into common computer programs like Microsoft Excel or even many pocket calculators. (But you do need to know how to use the regression line equation.)
Using the Regression Equationto Predict Y for a Given X • b: intercept = -123 • m: coefficient of height (x) = 4 y = mx + b weight = (4 * height) + -123 “Predicted” weight for 68 inches: weight = (4 * 68) - 123 = 149 pounds
Predict Weight for a Given Height weight = (4 * height) - 123 • 62 inches (4 * height) - 123= 125 lbs. • 75 inches (4 * height) - 123= 177 Lbs. • 70 inches (4 * height) - 123= 157 lbs.
What’s the point? • Regression shows what a dependent (y) variable is “predicted” to be, given a value for the independent (x)variable. • Definition: The residual is the amount an actual dependent (y) value differs from the “predicted” value • Definition: R-squared is the percentage of variance from the mean that is explained by the independent (x) variable
Regression in CAR • School test scores • Cheating in school test scores • Tenure of white vs. black coaches in NBA • Racial profiling in traffic stops • Miami criminal justice
Extrapolation? Beware! Don’t use your regression equation very far outside the boundaries of your data because the relationship may not hold. • Words vs. age (r = .993 for ages 2-6) Words = 562 * Age - 764 Age 1: 562 * 1 -764 = -202 words???
Negative Weight? Data area
Mark Twain and the length of the Mississippi River • From “Life on the Mississippi” (1884) • In 176 years, the river was shortened by 403 kilometers, or about 2.3 kilometers per year • A million years ago, the Mississippi must have been 2.2 million kilometers long • In 742 years, it will be 2.9 kilometers long, joining Cairo, Illinois, and New Orleans • Twain: “There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”