You can fit a straight line to a scatterplot, predict with it, and judge how well it fits.
Weeks 13-14
Probability & Statistics course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Data displays
Weeks 1-2
You can build and read dot plots, histograms, box plots and two-way tables. You can also describe the shape of a distribution and say what that shape means.
Build a box plot from a five-number summary.
Call a histogram skewed right, skewed left or roughly symmetric.
Read a two-way table to find what percent of one group answered yes.
Worked example
Find the five-number summary of 4, 7, 8, 11, 12, 15, 20
Smallest 4, largest 20
Seven values, so the median is the 4th one: 11
Answer4, 7, 11, 15, 20
Most common mistakeCounting the median into both halves when finding the quartiles here, which gives Q1 = 7.5 and Q3 = 13.5 instead of 7 and 15.
You can find mean, median, range and standard deviation, and say which one describes a data set honestly. You can also flag outliers with a rule instead of a hunch.
Compute both the mean and the median and see which one an outlier drags.
Use the 1.5 times IQR rule to flag an outlier.
Read a standard deviation as a typical distance from the mean.
Worked example
Find the mean and median of 3, 4, 4, 5, 84
Mean = 100 / 5
Median is the middle value once sorted
AnswerMean 20, median 4
Most common mistakeReporting the mean 20 as the typical value: it sits above four of the five numbers because the single value 84 hauls it upward.
You can find the chance of combined events using addition, multiplication and complements. You can also tell independent events from ones that change each other.
Use P(A or B) = P(A) + P(B) - P(A and B) when the events overlap.
Multiply chances for two draws without replacement, changing the second fraction.
Find the chance of at least one by subtracting from 1.
Worked example
A bag holds 5 red and 3 blue marbles. Draw two without putting the first back. Find P(both red).
First draw: 5/8
Only 4 reds are left out of 7: 4/7
Answer(5/8)(4/7) = 20/56, about 0.357
Most common mistakeUsing 5/8 twice: reporting 25/64, about 0.391, which quietly assumes the first marble went back in the bag.
You can build a probability table for a numeric outcome and find its expected value and spread. You can also use the binomial rule for a fixed number of yes-or-no trials.
Find the expected value of a game from its listed payouts and chances.
Recognise a binomial setting: fixed trials, two outcomes, the same chance each time.
Find the mean of a binomial with 50 trials and a 0.2 success rate.
Worked example
A game pays 10 with probability 0.2 and pays 0 with probability 0.8. Find the expected payout.
Weight each payout by its chance: 10(0.2) + 0(0.8)
2 + 0
Answer2 per play
Most common mistakeAveraging the payouts instead of weighting them: reporting (10 + 0)/2 = 5 per play, which would make a losing game look profitable.
You can pick a sample that stands in for a whole group and name the bias when it does not. You can also tell a survey, an observational study and an experiment apart.
Design a stratified sample by splitting a school into grade levels first.
Name the bias in a voluntary online poll.
Explain why only a randomised experiment supports a cause-and-effect claim.
Worked example
A school has 800 students, 200 in each grade. Take a stratified sample of 80.
80 out of 800 is 10% of the school
Take 10% of each grade: 0.10 x 200
Answer20 students from each of the four grades
Most common mistakeTaking all 80 students from one grade and calling it stratified, which measures a single grade rather than the school.
You can build a range that probably holds the true value and say what that range does and does not mean. You can also show how sample size changes the width.
Compute the margin of error for a proportion with 400 people and 55% saying yes.
State the interval as a sentence about the population, not the sample.
Explain why four times the sample size halves the margin of error.
Worked example
400 people, 220 say yes. Build a 95% interval for the true proportion.
Sample proportion = 220/400 = 0.55
Margin = 1.96 x root(0.55 x 0.45 / 400) = 1.96 x 0.0249
Answer0.501 to 0.599
Most common mistakeSaying there is a 95% chance the true value sits in this particular interval: the 95% describes how often the method works across many samples, not this one range.
You can fit a straight line to a scatterplot, predict with it, and judge how well it fits. You can also explain why a strong link is not proof that one thing causes another.
Read a slope of 0.8 as a change per one unit of x.
Use r squared to say how much of the variation the line explains.
Point out a curved residual pattern that means a line is the wrong model.
Worked example
A fitted line is y = 2.5x + 12. Predict y when x = 8.
y = 2.5(8) + 12
20 + 12
Answer32
Most common mistakeTreating the slope as the prediction: answering 2.5 when x = 8, which throws away the starting value of 12 and the multiplication entirely.
Before timing yourself, check whether you can explain Data displays from a blank page. Then connect it to Center and spread. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Statistics calculator can test calculations and representations used in Probability & Statistics. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Probability & Statistics practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Questions about the Probability & Statistics course
Where should I start?
Start with Data displays if you are following the full sequence. If that unit feels automatic, open the Probability & Statistics practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Probability & Statistics test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Statistics calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
