Chapter 2- Looking at Data- Relationships

-associated: term used to describe the relationship between two variables ex. breed and

life span

-examining relationships:

What individuals or cases do the data describe?

What variables are present? How are they measured?

Which variables are quantitative and which are categorical?

Ex. on page 85

-response variable: measures an outcome of a study

-explanatory variable: explains or causes changes in the response variables. Ex. many of

these do not involve direct causation. Ex. sat scores of high school students help predict

future college grades but high sat scores don’t CAUSE high college grades.

-independent variables: called explanatory variables

-dependent variables: called response variables

Response variables rely on explanatory variables

2.1- Scatterplots

-scatterplots: for showing relationship between two quantitative variables measured on

the same individuals.

-explanatory variable(s) on x axis called x. (if no explanatory variable, then any of the

variables can on either axis)

Response variable on y axis called y.

-Interpreting scatterplots:

Look for overall pattern and deviations from pattern ex. outliers- falls outside the pattern of

the relationship.

Describe overall pattern by the form, direction, and strength of the relationship.

-form: ex. clusters

Pg. 87 fig. 2.1 has two clusters

Clusters: groups of points on the graph. They suggest that the data describe several

distinct kinds of individuals.

-positive associated: when two variables are above average values of one tend to

accompany above average values of the other and below average values also tend to occur

together

-negatively associated: when two variables are above average values of one accompany

below average values of the other and vice versa.

-linear relationship: points roughly follow a straight line

Strength of relationship: determined by how closely the points follow a clear form

-to add a categorical variable to a scatterplot, use a different plot colour or symbol for each

category

-smoothing: systematic methods of extracting the overall pattern are helpful. They use

resistant calculations so they are not affected by outliers in the plot.

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-to display a relationship between a categorical explanatory variable and a quantitative

response variable, make a side by side comparison of the distributions of the response for

each category.

2.2- Correlation

-measure used for data analysis by using a numerical measure to supplement the graph.

(since our eyes are not good judges of how strong a relationship is)

-correlation r: helps us see that r is positive when there is a positive association between

the variables. Ex. height and weight have a positive correlation.

Correlation: measures the direction and strength of the linear relationship between two

quantitative variables. It is usually written as r.

Ex. suppose data on variable x and y for n individuals. The means and standard deviations

of the two variables are and for the x values and and for the y values. The

correlation r between x and y is

means : add these terns for all the individuals

This formula helps us see what correlation is but is not convenient for actually calculating r.

the beginning of this formula starts by standardizing the observations.

-ex.

is the standardized height of the ith person. The standardized height says how many SD

above or below the mean a person’s height lies. Standardized values have no units, they

have no longer measured in centimeters. The correlation r is an average of the products of

the standardized height and the standardized weight for the n people.

-properties of correlation:

Correlation for the following:

•Doesn’t make a difference what you make the x or y variable when calculating the

correlation

•Requires that both variables be quantitative, so that it makes sense to do the

arithmetic indicated by the formula for r. ex. city cant be calculated bc its

categorical.

•Because r uses the standardized values of the observations, r does not change when

we change the units of measurement of x, y, or both. Ex. using weight and height.

Cm -> inches or kg -> lbs. doesn’t change the correlation between weight and height.

Correlation r has no unit of measurement

•Positive r indicates positive association between the variables and negative r

indicates negative association

•Correlation r is always a number between -1 and 1. Values of r near 0 means a very

weak linear relationship. Strength of relationship increases as r moves away from 0

toward either -1 or 1. Values of r close -1 or 1 means that the points lie close to a

straight line. The extreme values

r=-1 and r= 1 occur only when the points in a scatterplot lie exactly along a straight line.

•Measures the strength of only the linear relationship between two variables.

Correlation does not describe curved relationships between variables, no matter how

strong they are.

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