Bivariant Data
Bivariate data pair two variables, and a linear model such as \(V=1800-320t\) describes how one changes as the other grows. In NSW Year 10 Advanced maths you interpret the gradient and intercept in context, use models to predict, and decide when a model or a conclusion from the data is not reasonable (NESA outcome MA5‑DAT‑C‑02).
You will tell univariate from bivariate data, read line graphs of rents and sales over time, and use models for car prices, dam levels, sprint times, zoo visitors and fuel use. You will explain why a model should not be used outside its valid range, why a straight line does not fit doubling data, and why association is not the same as cause. Harder questions find model constants and compare two cafes. It is part of Year 10 Advanced Single Variable and Bivariant Statistics.
Every question with a fully worked solution.
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Theory
Bivariate data record two variables for each individual, or one variable over time. They are shown on scatter plots and line graphs and are often summarised by a linear model such as \(y = mx + c\). This Year 10 Advanced maths page covers choosing displays, reading trends, using and building linear models, knowing where a model is valid, and the difference between an association and a cause.
Univariate data have one variable (e.g. heights of students): use a histogram, dot plot or box plot. Bivariate data pair two variables for each individual (heights and masses): use a scatter plot. A variable recorded over time (a time series) is shown with a line graph.
A linear model \(y = mx + c\) describes a linear trend. The gradient \(m\) is the change in \(y\) for each increase of \(1\) in \(x\) (with units, e.g. dollars per year); \(c\) is the value of \(y\) when \(x = 0\).
A model has a valid range: the values of \(x\) it was built from. Outside it (extrapolation) it may give unrealistic answers, such as negative times or masses.
An association between two variables does not show that one causes the other. A third (lurking) variable may affect both.
| Model | Gradient means | Intercept means |
|---|---|---|
| \(C = 0.12d + 1.5\) (fuel, L, for \(d\) km) | \(0.12\) L per km, i.e. \(12\) L per \(100\) km | \(1.5\) L used even for a tiny trip |
| \(P = 950 - 12d\) (rent, \(d\) km from the city) | rent falls \(\$12\) for each extra km | rent at the city centre (if \(d = 0\) is in range) |
Choosing and reading a display
- Count the variables: one (univariate) or two per individual (bivariate)? Is one of them time?
- Line graphs: read values at points, find changes between consecutive points (the steepest segment is the largest change), and describe the overall trend.
- Two lines on one graph: compare where one is above the other; if one rises while the other falls, they have a negative association.
Using a linear model
- Interpret the gradient and intercept with units, and say whether the intercept is sensible.
- Predict by substituting, or solve for the input that gives a target output.
- Build a model from two given facts: two equations in the unknown constants, solved simultaneously.
- Check validity: is the input inside the valid range? Is the output realistic?
- Cause or association? Unless the data come from a controlled experiment, name a possible third variable instead of claiming a cause.
(i) Two variables, year and visitors: bivariate (a time series), so a line graph suits it.
(ii) Year-to-year changes (thousand): \(+3,\ -34,\ +9,\ +18,\ +12,\ +3\). The largest increase, \(18\) thousand, was from 2021 to 2022.
(iii)
| \(\dfrac{63 - 21}{21} \times 100\%\) | \(=\) | \(\mathbf{200\%}\) |
(iv) Visitors were steady at about \(52\) to \(55\) thousand, dropped sharply in 2020, then rose every year, passing the earlier level by 2023.
(i) What do \(1800\) and \(320\) mean?
(ii) Predict the value after \(2.5\) years.
(iii) When does the model give \(\$520\)?
(iv) Explain what goes wrong at \(t = 6\).
(i) \(1800\): the value when new (\(t = 0\)), \(\$1800\). \(320\): the value falls by \(\$320\) each year.
(ii)
| \(V\) | \(=\) | \(1800 - 320(2.5) = \mathbf{\$1000}\) |
(iii)
| \(1800 - 320t\) | \(=\) | \(520\) |
| \(320t\) | \(=\) | \(1280\) |
| \(t\) | \(=\) | \(\mathbf{4}~\text{years}\) |
(iv) \(V = 1800 - 320(6) = -120\). A laptop cannot have a negative value. \(t = 6\) is outside \(0 \le t \le 4\), so the model should not be used there (extrapolation).
(i) Find \(a\) and \(b\), and interpret them.
(ii) For what times does the model give between \(300\) L and \(500\) L?
(i) Two facts give two equations.
| \(a - 4b\) | \(=\) | \(620 \quad (1)\) |
| \(a - 10b\) | \(=\) | \(380 \quad (2)\) |
| \((1) - (2):\quad 6b\) | \(=\) | \(240\) |
| \(b\) | \(=\) | \(\mathbf{40},\quad a = 620 + 160 = \mathbf{780}\) |
The tank starts with \(780\) L and loses \(40\) L per hour.
(ii) Solve \(300 \le 780 - 40t \le 500\). Subtract \(780\), then divide by \(-40\), which reverses the inequality signs.
| \(-480 \le -40t\) | \(\le\) | \(-280\) |
| \(12 \ge t\) | \(\ge\) | \(7\) |
| \(7 \le t\) | \(\le\) | \(12\) |
From \(\mathbf{7}\) to \(\mathbf{12}\) hours after opening, inside the valid range.
(i) Use this trend to predict the number in 2024.
(ii) The actual 2024 number was \(21\) thousand. Find the percentage error of the prediction, compared with the actual value, to \(1\) decimal place.
(iii) Give a reason the prediction was off.
(i) The number rises \(3\) thousand every \(2\) years, that is \(1.5\) thousand per year.
| \(2024\) | \(=\) | \(18 + 4 \times 1.5 = \mathbf{24}~\text{thousand}\) |
(ii)
| \(\text{error}\) | \(=\) | \(24 - 21 = 3~\text{thousand}\) |
| \(\dfrac{3}{21} \times 100\%\) | \(=\) | \(14.28\ldots\% \approx \mathbf{14.3\%}\) |
(iii) 2024 is outside the data (extrapolation). Growth often slows once most people already have the service, so a straight-line trend overestimates.
Common pitfalls
Frequently asked questions
What is bivariate data?
Data that record two variables for each individual, such as the height and mass of each student, or one variable measured over time. It is shown on a scatter plot or a line graph.
What is the difference between univariate and bivariate data?
Univariate data have one variable, displayed with histograms, dot plots or box plots. Bivariate data pair two variables, displayed with scatter plots or line graphs.
What does the gradient of a linear model mean?
It is the change in the output for each increase of 1 in the input, with units, such as dollars per year or litres per kilometre.
Why can a linear model give a wrong answer?
A model is only built from data over a certain range. Outside that range the trend may change, so extrapolation can give unrealistic answers such as negative values.
Does a strong association mean one variable causes the other?
No. It may be a coincidence, or a third variable may affect both. Only a controlled experiment can show cause and effect.