Grade by Grade
D. Data · Data, Statistics and Mathematical Modelling

Data, Statistics and Mathematical Modelling: Unit Review

A1 A2 D1.1 D1.2 D1.3 D2.1 D2.2 D2.3 D2.4 D2.5About 40 minutes
Learning goals
  • I can describe implications of collecting, storing, representing and using large data sets.
  • I can represent and analyse one-variable data, including with quartiles and box plots.
  • I can create scatter plots, find correlation and test regression models with technology.
  • I can carry out the modelling process: question, plan, model and report.

Words to know

interquartile range (IQR)
Q3 − Q1: the spread of the middle half of the data.
box plot
A graph of the five-number summary.
correlation coefficient (r)
A number from −1 to 1 measuring the direction and strength of a linear relationship.
R²
A number from 0 to 1 measuring how well a model fits the data.
extrapolation
Predicting outside the range of the data.
bias
An error that pushes results consistently in one direction.
mathematical model
A mathematical description of a real situation used to understand, predict or decide.

Let's review!

In this unit you looked at how large data sets are collected, stored, represented and used, and what that means for privacy, security and fairness. You analysed one-variable data with the mean, median, mode, range, quartiles, IQR, box plots, histograms and stem-and-leaf plots. You analysed two-variable data with scatter plots, the correlation coefficient and regression models (linear, quadratic and exponential). Then you put it all together in the mathematical modelling process: asking a question, planning and collecting data, creating a model and reporting it.

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 rather than all at once. Write your answers before checking. For every question you get wrong, write down why (careless arithmetic? forgot a step? did not know the idea?) and go back to that lesson's worked examples. Keep Desmos or a spreadsheet handy for the regression questions. Finish with the 10-question quiz.

Two Grade 9 classesClass A1218222634Class B1520243240101418222630343842minutes to solve a logic puzzle
Box plots of puzzle-solving times for two classes, used in the review questions below.
Training and extra push-ups012345678024681012141618weeks of trainingpush-ups in one minute (extra)y = 2.11x + 0.93
Data for the line-of-best-fit review questions: weeks of training and the extra push-ups a student could do.

Big Data All Around Us: Collecting, Storing and Using Data

1

Name the four stages of the data life cycle and one question to ask at each.

2

A ride-sharing app keeps every trip's start and end location for 10 years. Describe one benefit and one risk.

3

What is data sovereignty, and what do the OCAP® principles stand for?

Describing One-Variable Data: Mean, Median, Mode and Range

1

Find the mean, median, mode and range: 3, 5, 6, 8, 9, 10, 12, 13, 15, 30.

2

For the data in the last question, which measure of centre is better, and why?

3

Five quiz marks have a mean of 8. Four of them are 6, 7, 9 and 10. Find the fifth.

Quartiles, the Five-Number Summary and the Interquartile Range

1

Find the five-number summary and IQR of 3, 5, 6, 8, 9, 10, 12, 13, 15, 30.

2

Is 30 an outlier in that data?

3

A student is at the 80th percentile for height. What does that mean?

Box Plots: Drawing, Reading and Comparing

1

Using the box plots above, which class has the higher median? By how much?

2

Which class is more consistent? Use the IQR.

3

About what percent of Class B took more than 32 minutes?

4

About what percent of Class A took between 12 and 22 minutes?

Histograms, Stem-and-Leaf Plots and the Shape of Data

1

A stem-and-leaf plot shows 4 | 1 5 8; 5 | 0 2 2 7; 6 | 3 (key 4 | 1 = 41). Find the median.

2

A distribution has mean 18 and median 24. Name the likely shape.

3

Why would a histogram be better than a dot plot for 2 000 heights?

Scatter Plots and the Correlation Coefficient

1

Describe r = −0.72.

2

Which shows a stronger relationship, r = 0.55 or r = −0.80?

3

Sales of snow shovels and sales of winter boots are strongly correlated. Does buying shovels cause boot sales? Explain.

Lines of Best Fit, Linear Regression and Predictions

1

Using the graph above (y = 2.11x + 0.93), predict the extra push-ups after 4.5 weeks.

2

For the point (6, 14), find the residual.

3

Is a prediction for 20 weeks of training reliable? Explain.

Testing Regression Models: Linear, Quadratic and Exponential

1

Which model would you test first for the value of a car that loses 15% of its value each year?

2

Three models give R² = 0.71, 0.96 and 0.98. The 0.98 model predicts a negative population next year. Which would you choose?

3

Use y = 300(1.05)ˣ to find y when x = 2.

Just for fun

Final riddle of the unit: Why did the statistician bring a ladder to the review?

To reach the upper quartile.

What Is Mathematical Modelling?

1

Name the steps of the modelling process.

2

Give an example of a model used to make a decision in Canada.

3

What is an assumption? Give one for a model of how long a bag of milk lasts.

Asking a Question of Interest

1

Improve the question "Is the cafeteria good?"

2

In "How does the number of hours a lamp is on affect the temperature of the bulb's surface?", identify the variables.

3

Name two things to keep the same when testing how ramp height affects how far a toy car rolls.

Planning and Collecting Data: Sources, Samples, Bias and Assumptions

1

A school has 400 students in Grade 9 and 600 in Grade 10. For a stratified sample of 40, how many from each grade?

2

Identify the bias: a survey about sleep handed out only at a 7 a.m. swim practice.

3

Why does averaging repeated trials help, and what can it not fix?

Creating a Mathematical Model from Data

1

What display and model suit: daily high temperature and number of people at a beach?

2

With y = 0.513x + 1.61, find the drop height for a 75 cm bounce.

3

What should a residual plot look like for a good model?

Reporting a Model: Answers, Fit, Limitations and Predictions

1

List the seven parts of a modelling report.

2

A model predicts 52 and a new test value is 50. Find the percent error relative to the actual value.

3

Rewrite with appropriate uncertainty: "Our town will need exactly 172.5 t of salt next winter."

Check your understanding

Quiz

1.

Removing names from a data set...

2.

What is the median of 4, 9, 2, 7, 5, 8?

3.

For 1, 3, 4, 6, 8, 9, 11, what is Q3?

4.

On a box plot, about what percent of the data lies in the right whisker?

5.

A histogram has a long tail to the right. Which is most likely?

6.

Which r shows the weakest linear relationship?

7.

Data covers x = 2 to x = 12. A prediction at x = 7 is...

8.

Which model best fits a quantity that doubles every hour?

9.

Choosing every 10th person on a list after a random start is...

10.

Which belongs in the limitations section of a report?

For parents and teachers

How to use the review results:

  • Trouble with privacy, consent or data sovereignty: revisit Lesson 1 (Big Data All Around Us).
  • Errors with the mean, median or choosing a measure: Lesson 2.
  • Wrong quartiles (often from not ordering the data, or including the median in the halves when n is odd): Lesson 3.
  • Misreading box plots or comparing groups without numbers: Lesson 4.
  • Mixing up skewed left and skewed right, or histogram errors: Lesson 5.
  • Confusing the sign and the strength of r, or correlation with causation: Lesson 6.
  • Slope/intercept interpretation, residual sign errors, or trusting extrapolations: Lesson 7.
  • Choosing a model by R² alone, or exponent arithmetic errors: Lesson 8.
  • Vague answers about what modelling is: Lesson 9.
  • Questions that are opinions or too broad: Lesson 10.
  • Sampling methods, bias and planning: Lesson 11.
  • Choosing displays and models, or solving for x: Lesson 12.
  • Reports with no evidence, no limitations or overconfident predictions: Lesson 13.

The skills in this unit connect to the linear and non-linear relations unit of this course and to Grade 10 mathematics, where students work further with linear and quadratic relations. Statistics and data management are studied in more depth in a Grade 12 data management course.