Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Calculate covariance and distinguish its units from correlation.
- Justify the conclusion "Covariance=(2+0+2)/3=4/3 and correlation=1" using the stated assumptions.
Before you start
Means and products of deviations.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For paired data (1,2),(2,4),(3,6), compute population-form covariance.
Why this math matters
Calculate covariance and distinguish its units from correlation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The divisor is n=3, explicitly using population-form covariance.
- The pairs are preserved.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Read shared movement through centered products
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For paired data (1,2),(2,4),(3,6), compute population-form covariance.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Means are x̄=2,ȳ=4
Center both variables before multiplying.
Work through the mathematics
Centered products are 2,0,2
Matching positive or negative deviations produce positive products.
Check the conclusion
Covariance=(2+0+2)/3=4/3 and correlation=1
The exact positive linear relation gives perfect correlation, while covariance retains product units.
The result
Covariance=(2+0+2)/3=4/3 and correlation=1
The exact positive linear relation gives perfect correlation, while covariance retains product units.
Common mistakes to catch
- Sample covariance uses n−1 instead.
- Correlation is dimensionless, but covariance is not.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What happens to covariance if all y values are multiplied by ten?
Show a hint
Each centered y deviation is multiplied by ten.
Reveal answer and explanation
It multiplies by ten
Correlation remains one because its standard-deviation normalization also scales.
Practice 2
Does zero covariance imply independence in every model?
Show a hint
Linear association is only one kind.
Reveal answer and explanation
No
Nonlinear dependence can have zero centered-product average.
Take the idea with you
Explain why covariance magnitudes cannot be compared across arbitrary unit changes without normalization.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Find variables that are pairwise independent but not jointly independent
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.