INTRODUCTION TO STATISTICS & PROBABILITY

Chapter 2:

Looking at Data–Relationships (Part 1)

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Dr. Nahid Sultana

Chapter 2: Looking at Data–Relationships

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2.1: Scatterplots

2.2: Correlation

2.3: Least-Squares Regression

2.5: Data Analysis for Two-Way Tables

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Objectives

Bivariate data

Explanatory and response variables

Scatterplots

Interpreting scatterplots

Outliers

Categorical variables in scatterplots

2.1: Scatterplots

Bivariate data 4

For each individual studied, we record data on two variables.

We then examine whether there is a relationship between these two variables: Do changes in one variable tend to be associated with specific changes in the other variables?

Student ID

Number of Beers

Blood Alcohol Content

1 5 0.1

2 2 0.03

3 9 0.19

6 7 0.095

7 3 0.07

9 3 0.02

11 4 0.07

13 5 0.085

4 8 0.12

5 3 0.04

8 5 0.06

10 5 0.05

12 6 0.1

14 7 0.09

15 1 0.01

16 4 0.05

Here we have two quantitative variables recorded for each of 16 students:

1. how many beers they drank 2. their resulting blood alcohol content

(BAC)

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Many interesting examples of the use of statistics involve relationships between pairs of variables.

Two variables measured on the same cases are associated if

knowing the value of one of the variables tells you something about the values of the other variable that you would not know without this information.

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Associations Between Variables

A response (dependent) variable measures an outcome of a study.

An explanatory (independent) variable explains changes in the response variable.

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Scatterplot

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The most useful graph for displaying the relationship between two quantitative variables on the same individuals is a scatterplot.

1. Decide which variable should go on which axis.

2. Typically, the explanatory or independent variable is plotted on the x-axis, and the response or dependent variable is plotted on the y-axis.

3. Label and scale your axes.

4. Plot individual data values.

How to Make a Scatterplot

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Scatterplot (Cont…) Example: Make a scatterplot of the relationship between body weight and backpack weight for a group of hikers.

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Body weight (lb) 120 187 109 103 131 165 158 116

Backpack weight (lb) 26 30 26 24 29 35 31 28

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Interpreting Scatterplots

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After plotting two variables on a scatterplot, we describe the overall pattern of the relationship. Specifically, we look for form, direction, and strength .

Form: linear, curved, clusters, no pattern

Direction: positive, negative, no direction

Strength: how closely the points fit the “form”

… and clear deviations from that pattern

Outliers of the relationship, , an individual value that falls outside the overall pattern of the relationship

How to Examine a Scatterplot

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Linear

Nonlinear

No relationship

Interpreting Scatterplots (Cont…) (Form)

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Interpreting Scatterplots (Cont…) (Direction)

Positive association: High values of one variable tend to occur together with high values of the other variable.

Negative association: High values of one variable tend to occur together with low values of the other variable

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Interpreting Scatterplots (Cont…)

No relationship: X and Y vary independently. Knowing X tells you nothing about Y.

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Interpreting Scatterplots (Cont…) (Strength)

The strength of the relationship between the two variables can be seen by how much variation, or scatter, there is around the main form.

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Interpreting Scatterplots (Cont…) (Outliers)

In a scatterplot, outliers are points that fall outside of the overall pattern of the relationship.

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Interpreting Scatterplots (Cont…)

Direction Form Strength

There is one possible outlier―the hiker with the body weight of 187 pounds seems to be carrying relatively less weight than are the other group members.

There is a moderately strong, positive, linear relationship between body weight and backpack weight.

It appears that lighter hikers are carrying lighter backpacks.

How to scale a scatterplot

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Using an inappropriate scale for a scatterplot can give an incorrect impression. Both variables should be given a similar amount of space: • Plot roughly square • Points should occupy all the plot space (no blank space)

Same data in all four plots

Categorical variables in scatterplots 16

What may look like a positive

linear relationship is in fact a

series of negative linear

associations.

Plotting different habitats in

different colors allows us to

make that important distinction.

To add a categorical variable, use a different plot color or symbol for each category.

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Categorical variables in scatterplots (Cont…)

Comparison of men and women racing records over time. Each group shows a very strong negative linear relationship that would not be apparent without the gender categorization.

Relationship between lean body mass and metabolic rate in men and women. Both men and women follow the same positive linear trend, but women show a stronger association.

Categorical explanatory variables

When the explanatory variable is categorical, you cannot make a scatterplot, but you can compare the different categories side by side on the same graph (boxplots, or mean +/− standard deviation).

Comparison of income (quantitative response variable) for different education levels (five categories).

But be careful in your interpretation: This is NOT a positive association, because education is not quantitative.