Scatterplots and Correlation

Transcription

1 Scatterplots and Correlation Chapter 14 April 30, 2012 Relationships Among Variables Scatterplots Eamining Scatterplots Numerical Summaries Eamples

2 1.0 Relationships Among Variables Two quantitative variables. Data displays: scatterplots. Numerical summaries: correlation concept, calculation, interpretation, cautions

3 1.1 Human Development Report Observational units: 39 nations G.D.P. = gross domestic product in $1,0000 U.S. Internet = % adult residents who use the Internet Literacy = adult literacy rate Fertility = mean # of children per adult woman

4 2.0 Scatterplots Definition A scatterplot shows the association between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal ais, and the values of the other variable appear on the vertical ais. Each individual in the data appears as the point in the plot fied by the values of both variables for that individual. Response variable Eplanatory variable

5 2.1 Association Definitions If there is a strong association between two variables, then knowing one helps a lot in predicting the other. But when there is a weak association, information about one variable does not help much in guessing the other. Volume and weight of water perfect Weight and fuel economy of cars medium to strong Heights of fathers and sons medium Caution: Be careful when interpreting empirical associations. Think carefully about whether it makes sense.

6 3.0 Eamining Scatterplots You can describe the overall pattern of a scatterplot by the direction, form and strength of the relationship. An important kind of deviation is an outlier, an individual value that falls outside the overall pattern of the relationship.

7 3.1 Detecting Outliers Moscow Individuals: 26 cities. X-ais: % of population with a higher education. Y-ais: Entrepreneurial environment score. Outlier: Moscow.

8 3.2 Eamining Direction Definition Two variables are positively associated when above-average values of one tend to accompany above-average values of the other and below-average values also tend to occur together. Two variables are negatively associated when above-average values of one tend to accompany below-average values of the other, and vice versa. Positive Association Negative Association Mean Of Y Mean Of Y Mean Of X Mean Of X

9 3.2 Eamining Direction: 26 Cities Avg. Avg. We see values mainly in the top right and bottom left quadrants. This signifies a positive association between having a higher % of educated citizens and the score for entrepreneurial environment.

10 3.3 Eamining Form Life epectancy rises as G.D.P. increases, but then it levels off. non-linear form

11 3.4 Eamining Strength The strength of a relationship is determined by how closely the points follow a clear form. If there is a strong relationship (or association), then knowing one helps a lot in predicting the other. Data cloud Data cloud Strong positive linear form Strong negative linear form Data cloud Data cloud Weak positive linear form Weak negative linear form

12 3.4 Eamining Strength: 26 Cities There is a weak positive linear relationship between % population with a higher education and entrepreneurial environment.

13 4.0 Numerical Summaries Calculate arithmetic mean of and arithmetic mean of y. Plot the point of averages. Calculate S.D. of, S.D. of y. This gives the horizontal spread and vertical spread of the data cloud. Avg. of y 2 S.D Of Y. Avg. 2 S.D Of Y. Avg. of 2 S.D.. Avg. 2 S.D. of Of

14 4.0 Numerical Summaries Find the correlation coefficient r. It measures the clustering of the data points about a line.

15 4.0 Numerical Summaries

16 4.1 Computing the Correlation Table: Length of femur and humerus in 5 Archaeoptery fossils Femur (): Humerus (y):

17 4.1 Computing the Correlation Step 1: Calculate the mean and S.D. for and y. Femur: Humerus: Avg. = c.m., S.D. = c.m. Avg. = 66.0 c.m., S.D. = c.m. Step 2: Calculate standard scores for and y. Standard Score y Standard Score 38 ( )/13.20 = ( )/15.89 = ( )/13.20 = ( )/15.89 = ( )/13.20 = ( )/15.89 = ( )/13.20 = ( )/15.89 = ( )/13.20 = ( )/15.89 = 1.133

18 4.1 Computing the Correlation Step 3: Average the products of these standard scores. Divide by n 1 instead of n. r = 1 [( 1.530) ( 1.573) + ( 0.167) ( 0.189) 4 + (0.061) (0.252) + (0.439) (0.378) + (1.197) (1.133)] r = 1 [( )] 4 = Why does r work as a measure of association?

19 4.1 Computing the Correlation Definition Convert each variable into standard scores. The average of the product gives the correlation coefficient. r= average of (X in standard scores) (Y in standard scores).

20 4.2 Facts About r The larger the value of r, the more tightly clustered the points are around a line. Positive values of r indicate a positive association, and vice versa. The correlation r always falls between -1 and +1. Values of r near zero indicate a weak straight-line relationship. An r of +1 implies that when X increases by one S.D. of X, that Y increases by one S.D. of Y. An r of -1 implies that when X increases by one S.D. of X, that Y decreases by one S.D. of Y. The correlation coefficient for X versus Y is the same as for Y versus X. The correlation coefficient has no units. The correlation does not change when we change the units of measurement of, y or both.

21 4.3 Some Cases When r Fails As A Measure of Association

22 5.0 Eamples Two weathermen compute the correlation between daily maimum temperatures for Washington D.C. and Boston. One does it for June, the other does it for the whole year. Who gets the bigger correlation? A class of fifteen students happens to include 5 basketball players. True or false, and eplain: the relationship between heights and weights for this class should be summarized using r.

23 5.0 Eamples The correlation between height and weight among men age in the U.S. is about Say whether each of these conclusions follows. Taller men tend to be heavier. The correlation between weight and height for men age is also about 0.4. Heavier men tend to be taller. If someone eats more and puts on 10 pounds, he is likely to get somewhat taller.

24 5.0 Eamples For women age 25 and over in the U.S. in 2005, the relationship between age and education level (years of schooling completed) can be summarized as follows. average age 50 years, S.D. 16 years, average ed. level 13.2 years, S.D. 3.0 years, r True or false: as you get older, you become less educated.