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D. Statistical Analysis · Two-Variable Data Analysis and Validity

Two-Variable Data Analysis and Validity: Unit Review

D2.1 D2.2 D2.3 D2.4 D2.5 D3.1 D3.2 D3.3About 40 minutes
Learning goals
  • I can describe, measure and model relationships between two variables and check my models.
  • I can explain the different reasons two variables can be related, and spot hidden variables.
  • I can judge statistics and claims in the media and explain how data management is used in careers.

Words to know

correlation coefficient (r)
A number from −1 to 1 measuring the direction and strength of a linear relationship.
least-squares line
The line that minimizes the sum of squared residuals; it passes through (x̄, ȳ).
residual
Actual minus predicted: e = y − ŷ.
common-cause factor
A third variable that causes both X and Y.

Let's review!

In this unit you moved from describing scatter plots to calculating the correlation coefficient and r², organizing categorical data in contingency tables, comparing groups with side-by-side box plots, fitting the least-squares line by hand, and checking models with residuals, outliers and influential points. You made and judged predictions, compared related relationships, and sorted correlations into types of relationships, including hidden variables and Simpson's paradox. Finally, you turned these tools on statistics in the media, learned a checklist for judging validity, and explored careers that use data management.

The last unit of the course, Culminating Data Management Investigation, asks you to use all of these skills on a question of your own. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time with a calculator (and Desmos or a spreadsheet to check, if you have one). Show your sums when you calculate r or a regression line; most errors are in Σx² vs (Σx)². For every "why" question, answer in full sentences using the variables' names. For each question you miss, write down why (formula slip? wrong denominator? forgot to check the graph? confused correlation with cause?) and go back to that lesson. Finish with the 10-question quiz.

1. Graphscatter plot2. Describeform, direction,strength, outliers3. Measurer and r²4. Modelŷ = ax + b5. Checkresidual plot,influence6. Concludecontext, range,cause?
A workflow for any two-variable analysis. Skipping step 1 or step 5 is how most mistakes in this unit happen.
−1−0.500.51rink salesmonthly tempsEnglish vs mathscience vs mathr values from this unit (r ≈ 0.29 hides a strong curve!)
Some r values from this unit. Notice the monthly temperatures: a small r, but a very strong curved relationship.

Two-Variable Data and Scatter Plots

1

Which display would you use for province and whether a household owns a snowblower?

2

Describe a scatter plot whose points rise to the right in a tight straight band, with one point far below.

3

Name the explanatory variable: hours of daylight and electricity used for lighting.

The Correlation Coefficient r: Meaning and Calculation

1

For 5 pairs: nΣxy − ΣxΣy = 60, nΣx² − (Σx)² = 40, nΣy² − (Σy)² = 160. Find r.

2

A point is below and to the right of (x̄, ȳ). What sign is (x − x̄)(y − ȳ)?

3

Find r for (1, 5), (2, 3), (3, 4), (4, 1).

Interpreting r: the Coefficient of Determination and the Limits of r

1

r = −0.8. Find and interpret r² for price and sales.

2

Give two situations in which r misleads.

3

Does converting metres to feet change r?

Two Categorical Variables: Contingency Tables

1

Of 120 Grade 12 students, 75 have a driver's licence. Of those 75, 45 drive to school. What percent of licensed students drive to school?

2

Row totals 70 and 30, column totals 40 and 60, grand total 100. Find the expected count for row 2, column 1.

3

When are two categorical variables associated?

Comparing Groups: Side-by-Side Box Plots

1

Find the five-number summary of 2, 4, 5, 7, 8, 10, 30 and check for outliers.

2

Two box plots barely overlap. What does that say about the relationship?

3

Name one thing a box plot hides.

The Line of Best Fit by Least Squares

1

n = 5, Σx = 20, Σy = 50, Σxy = 230, Σx² = 90. Find the least-squares line.

2

r = 0.6, s_x = 5, s_y = 15. Find the slope.

3

Interpret the slope in ŷ = −0.4x + 30, where x is days since purchase and y is the percent of a cleaning spray left.

Residuals and Residual Plots: How Good Is the Fit?

1

ŷ = 5x + 2. Find the residual for (3, 20).

2

A residual plot is an arch. What should you do?

3

Residuals 2, −1, −2, 1, 0. Find s_e.

Outliers and Influential Points

1

s_e = 3. Is a residual of −7 worth investigating?

2

Why can an influential point have a small residual?

3

Give an acceptable reason to remove a point.

Using Models: Predictions and Comparing Relationships

1

A model was built for x from 5 to 25. Is predicting at x = 40 interpolation or extrapolation?

2

Relationship A: r = 0.9, slope 0.5. Relationship B: r = 0.4, slope 4. Which is stronger?

3

List four things a complete description of a relationship should include.

Correlation and Causation: Types of Relationships

1

Classify: tree age and trunk diameter.

2

Classify: ice cream sales and forest fires.

3

Classify: the consumer price index and the number of known planets.

4

What study design best supports a causal claim?

Hidden Variables, Simpson's Paradox and Misleading Summaries

1

Player A makes 40/50 easy shots and 10/50 hard shots; Player B makes 9/10 easy and 30/90 hard. Who is better at each type and overall?

2

What is the ecological fallacy?

3

How do you protect against a lurking variable?

Statistics in the Media and Advertising

1

Support goes from 40% to 46%. Give the change in percentage points and as a percent.

2

A bar graph's axis starts at 70. Bars show 72 and 78. How many times as tall does the second bar look?

3

Give one misuse of statistics named in the curriculum.

Evaluating the Validity of Statistical Claims

1

An online poll on a sports site gets 30 000 votes. Why is it not a valid measure of public opinion?

2

A random poll of 1000 has a margin of error of ±3 percentage points, 19 times out of 20, and reports 49% support. Can a headline say "Most oppose"?

3

Name the unstated assumption in "Teens who play sports have higher marks, so sports raise marks."

Careers and University Programs in Data Management

1

A policy has a 1% chance of a $12 000 claim per year. Find the expected claim cost.

2

Name the four example careers in the curriculum.

3

Where should you check a university program's admission requirements?

Just for fun

Final review riddle: what do you call a scatter plot with r = 1, no outliers, a random residual plot, and a randomized experiment behind it?

A statistician's dream. Enjoy it while it lasts; real data are messier.

Check your understanding

Quiz

1.

What is r if nΣxy − ΣxΣy = −90, nΣx² − (Σx)² = 100 and nΣy² − (Σy)² = 144?

2.

r = 0.9. What fraction of the variation in y is explained by the linear model?

3.

The least-squares line always passes through:

4.

A residual plot shows a clear U-shape. The best conclusion is:

5.

In a contingency table, row totals are 80 and 120, column totals 50 and 150, grand total 200. The expected count for row 1, column 1 is:

6.

A point far to the right of the other data changes the slope a lot when removed. It is:

7.

Ice cream sales and forest fires are correlated because of hot weather. This is:

8.

Group B has the higher success rate in each of two categories but the lower rate overall. This is:

9.

A rate rises from 10% to 12%. Which statement is correct?

10.

Which occupation uses expected value to set insurance premiums?

For parents and teachers

Use the review to find what to revisit:

  • Wrong display chosen, or vague scatter-plot descriptions → Lesson 1 (Two-Variable Data and Scatter Plots).
  • Errors calculating r, especially Σx² vs (Σx)² → Lesson 2 (The Correlation Coefficient r).
  • Trusting r without a graph, or misreading r² → Lesson 3 (Interpreting r).
  • Dividing by the wrong total in a two-way table → Lesson 4 (Contingency Tables).
  • Quartile or fence mistakes, or comparisons with no numbers → Lesson 5 (Side-by-Side Box Plots).
  • Slope or intercept errors, or using r as the slope → Lesson 6 (The Line of Best Fit by Least Squares).
  • Residual sign errors, or judging fit by r alone → Lesson 7 (Residuals and Residual Plots).
  • Deleting points without reason, or missing influential points → Lesson 8 (Outliers and Influential Points).
  • Extrapolating without warning, or confusing slope with strength → Lesson 9 (Using Models).
  • Claiming cause from correlation, or mixing up the types of relationships → Lesson 10 (Correlation and Causation).
  • Missing a lurking variable or Simpson's paradox → Lesson 11 (Hidden Variables).
  • Percent vs percentage points, relative vs absolute change, truncated axes → Lesson 12 (Statistics in the Media).
  • Accepting claims without checking source, sample and method → Lesson 13 (Evaluating the Validity of Statistical Claims).
  • Unsure about careers or admission requirements → Lesson 14 (Careers and University Programs).

Looking ahead: the Culminating Data Management Investigation asks your teen to pose a question, collect or find data, analyse it with the tools from the whole course, and present conclusions with their limitations. This unit's checklist for validity is a good self-check for that project.