Chapter 3: Describing Relationships

The Practice of Statistics | Starnes and Tabor

What is univariate data?

What is univariate data?

a one-variable data set

What is bivariate data?

a data set that describes the relationship between 2 variables

What is a response variable?

the variable that measures the outcome of a study

What is an explanatory variable?

the variable that may help predict or explain changes in a response variable

What does a scatterplot show?

the relationship (association) between two quantitative variables measured on the same individuals

Why do we study relationships between 2 variables?

to help us explain how one variable affects another and why something happens

How do you make a scatterplot?

1. Label the axes

2. Scale the axes

3. Plot individual data values

How do you describe a scatterplot?

using DUFS – direction, unusual features, form and strength

What is the easiest way to lose points when making a scatterplot?

not labeling the axes

How do you know which variable to put on what axis?

the explanatory always goes on the x-axis, and the response variable goes on the y-axis; if there is no explanatory variable, either variable can go on the x-axis

Where do you start each axis of a scatterplot?

at a number smaller than the smallest value of that variable

When do 2 variables have a positive association?

when above average values of one variable tend to accompany above average values of the other variable

When do 2 variables have a negative association?

when above average values of one variable tend to accompany below average values of the other variable

When do 2 variables have no association?

if knowing one variable does not help us predict the value of the other variable

How do you describe direction of a scatterplot?

positive association, negative association, or no association

How do you describe form of a scatterplot?

linear or nonlinear

How do you describe strength of a scatterplot?

weak, moderate, or strong

How do you describe unusual features of a scatterplot?

outliers or clusters

What does correlation r measure?

the direction and strength of the association of the linear relationship between two quantitative variables

What interval does r always fall between?

-1 to 1

What does r > 0 indicate?

a positive association

What does r < 0 indicate?

a negative association

What only occurs in the case of a perfect liner relationship?

the extreme values r = -1 and r = 1

What do values of r close to -1 or 1 indicate?

a very strong relationship

What do values of r close to 0 indicate?

a very weak relationship

What does correlation not imply?

causation

What type of relationships should correlation only be used on?

linear

True or false: Correlation is not a resistant measure of strength

true

True or false: You can determine the form of a relationship using only correlation

false

How do you find r on a calculator?

after entering the values in the lists, press STAT → CALC → 8: LinReg(a+bx) → Calculate

What does correlation require of both variables?

that they be quantitative

What does correlation make no distinction between?

explanatory and response variables

True or false: r does not change when change the units of measurement of x, y, or both

true

What is a regression line?

a line that describes how a response variable (y) changes as an explanatory variable (x) changes

What form are regression lines expressed in?

ŷ = a + bx

What is the regression line used to predict?

the value of y for a given value of x

What is extrapolation?

the use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line

Why is extrapolation dangerous?

there is no guarantee the linear pattern we see will continue beyond the given data

True or false: The regression line will pass exactly through all the points in a scatterplot

false

What is a residual?

the difference between an observed value of the response variable and the value of y predicted by the regression line

What is the equation to find a residual?

y - ŷ

How do you interpret a residual?

give the size and direction of the residual

The actual value of [response variable] is [residual value] more/less than the value predicted by the regression line with x = [explanatory variable]

What does a represent in the regression line equation ŷ = a + bx?

the y-intercept, the predicted value of y when x = 0

What does b represent in the regression line equation ŷ = a + bx?

the slope, the amount by which the predicted value of y changes when x increases by 1 unit

How do you interpret slope?

The predicted value of [response variable] goes up/down by b for each [unit of x].

How do you interpret the y-intercept?

The predicted value of a [individual] that has 0 [unit of x] is [y-intercept]

What regression line do we want?

the one that minimizes the sum of the squared residuals

What is the least-squares regression line?

the line that makes the sum of the squared residuals as small as possible

What point is always guaranteed to be on the least squares regression line?

(x̄, ȳ)

How do outliers affect the least squares regression line?

they strongly influence the line

What is a residual plot?

a scatterplot of the residuals on the vertical axis and the explanatory variable on the horizontal axis

How do you find the residual plot on a calculator?

2nd → y= → Plot 1 → Enter → Adjust settings → Zoom → 9: ZoomStat → Enter

How does a residual plot work?

it magnifies the deviations of the points from the line, making it easier to see unusual observations and patterns

What is the purpose of a residual plot?

to assess linearity with a tool other than the actual scatterplot

What do you look for in a residual plot?

random scattered points above and below the regression line

How can you tell if a linear model is appropriate?

if there are no obvious patterns

What does the standard deviation of the residuals?

s, which gives the typical size of a prediction error (residual)

How do you calculate the standard deviation of the residuals?

2nd → STAT -> Math → 7: stdDev( → 2nd → STAT → RESID

How do you interpret the standard deviation of the residuals?

Using the LRSL that predicts [y] using [x], we will typically be off by about “s” units in our predictions

What is the coefficient of determination?

r², which measures the percent reduction in the sum of squared residuals when using the least-squares regression line to make predictions, rather than the mean value of y

How do you calculate r²?

STAT → CALC → 8: LinReg(a+bx)

How do you interpret r2?

[r² as a percentage]% of variation in [y variable] is accounted for by the least squares regression line with x = [x variable]

What are points with high leverage in regression?

points that have much larger or much smaller x-values than the other points in the data set

What is an outlier in regression?

a point that does not follow the pattern of the data and has a large residual

What is an influential point in regression?

any point, that if removed, substantially changes the slope, y-intercept, correlation, coefficient of determination, or standard deviation of the residuals