Bivariate data involves two related variables, often visualised with a scatter plot to reveal whether a linear relationship exists. When a linear pattern appears, a line of best fit (often found using least-squares regression) models the trend, expressed in the form y = mx + c. This line can be used to make predictions, though predicting well beyond the range of the original data (extrapolation) is less reliable than predicting within it (interpolation).
Example
A scatter plot of study hours versus test scores might reveal a positive linear trend, with a regression line like score = 5(hours) + 40 — useful for predicting a score for 6 hours of study (interpolation, since it's within the data range), but far less reliable for predicting a score for 40 hours of study (extrapolation, well beyond anything observed).
Key terms
Bivariate data:
Data involving two related variables.
Line of best fit:
A line modelling the trend in a scatter plot of bivariate data.
Extrapolation:
Predicting values beyond the range of the original data.
Questions
1. Bivariate data involves:
Two related variables
Only a single variable
No variables at all
Three or more unrelated variables
2. A line of best fit models:
The trend in a scatter plot of bivariate data
A single unrelated data point
The exact colour of a graph
Nothing measurable in the data
3. Extrapolation means predicting values:
Beyond the range of the original data
Only within the range of the original data
With no connection to any data at all
Using no mathematical model whatsoever
4. Interpolation means predicting values:
Within the range of the original data
Only far beyond the original data
With no connection to any data at all
Using no mathematical model whatsoever
5. A line of best fit is typically expressed in the form:
y = mx + c
y = x² only
A form with no variables at all
A single fixed number
6. A scatter plot is used to visualise:
A relationship between two variables
Only a single value with no relationship
A completely unrelated type of data
Only categorical data with no numbers
7. Extrapolation is generally considered:
Less reliable than interpolation
Always more reliable than interpolation
Exactly as reliable as interpolation in every case
Completely unrelated to reliability
8. Using the regression line score = 5(hours) + 40, the predicted score for 6 hours of study is:
70
46
30
75
9. Using score = 5(hours) + 40, the predicted score for 4 hours of study is:
60
45
50
20
10. Predicting a test score for 40 hours of study, when the original data only ranged from 0-10 hours, is an example of:
Extrapolation
Interpolation
A completely reliable prediction
A prediction with no connection to the regression line
11. Why is predicting within the range of the original data (interpolation) generally more reliable than extrapolation?
The relationship has actually been observed within that range, unlike assumptions made about data beyond it
Extrapolation is always exactly as reliable as interpolation
The original data range has no bearing on how reliable a prediction is
Interpolation is always completely unreliable regardless of the data
12. Why might a scatter plot be examined before fitting a line of best fit, rather than assuming a linear relationship exists?
The data might show a non-linear pattern, in which case a straight line would be a poor model
Every scatter plot always shows a perfectly linear relationship regardless of the data
A line of best fit is always appropriate no matter what pattern the data shows
Examining the scatter plot first has no bearing on choosing an appropriate model
13. Why might a strong linear relationship in a scatter plot not necessarily imply causation between the two variables?
Correlation shows a pattern, but doesn't by itself prove one variable causes changes in the other
A strong linear relationship always definitively proves causation
Correlation and causation are always exactly the same thing
Scatter plots can never reveal any kind of relationship between variables
14. Why might the least-squares method be used specifically to find a line of best fit, rather than simply drawing a line by eye?
It provides a consistent, mathematical method for minimising the overall distance between the line and the data points
Drawing a line by eye always produces a more mathematically accurate result
The least-squares method has no defined mathematical basis or purpose
A line of best fit found by eye is always identical to one found using least-squares
15. Why might extrapolating far beyond a data set risk producing a nonsensical prediction (like a test score over 100%)?
The linear trend observed within the data range may not continue to hold true indefinitely beyond it
Extrapolated predictions are always guaranteed to be sensible and accurate
A regression line always remains valid no matter how far it is extended
Extrapolation never produces predictions that fall outside a realistic range
16. Why might researchers report a correlation coefficient alongside a line of best fit?
It indicates how strongly the data actually follows a linear pattern, showing how much to trust the model
A correlation coefficient has no relevance to evaluating a line of best fit
The line of best fit is always equally reliable regardless of correlation strength
Correlation coefficients are only ever used for non-linear data
17. A scatter plot showing points scattered with no visible pattern suggests:
Little to no linear relationship between the variables
A perfectly strong linear relationship
That a line of best fit will always be highly reliable
That extrapolation is always safe for this data
18. Why might an outlier in a bivariate data set significantly affect the position of a least-squares line of best fit?
The least-squares method is sensitive to points far from the overall pattern, which can pull the line toward them
Outliers never have any effect on a least-squares regression line
A line of best fit always ignores outliers automatically
Outliers only ever affect scatter plots, never the calculated regression line
19. Why might it be misleading to use a regression line built from one population (e.g. adults) to make predictions about a different population (e.g. children)?
The relationship between the variables may not hold the same way across a group the model wasn't built from
Regression lines are always equally valid for any population regardless of the original data source
Population differences never have any effect on the validity of a regression model
A regression line built from one group is always the most accurate model for every other group
20. Why might a weak but statistically real correlation still not be very useful for making individual predictions?
A weak correlation means a wide range of possible outcomes for any given input, limiting predictive precision
Weak correlations always still produce highly precise, useful predictions
Correlation strength has no bearing on how useful a model is for prediction
Any correlation, however weak, always guarantees an accurate individual prediction
21. Why might presenting a scatter plot alongside a line of best fit (rather than the line alone) give a more honest picture of a data set?
It shows the actual spread and variability in the data, not just the summarised trend
A line of best fit always tells the complete story with no need for the underlying data
Showing the scatter plot alongside the line always makes a data set harder to interpret
The spread of data points around a trend line is never relevant information
Answer key (parent copy)
1. Two related variables
2. The trend in a scatter plot of bivariate data
3. Beyond the range of the original data
4. Within the range of the original data
5. y = mx + c
6. A relationship between two variables
7. Less reliable than interpolation
8. 70
9. 60
10. Extrapolation
11. The relationship has actually been observed within that range, unlike assumptions made about data beyond it
12. The data might show a non-linear pattern, in which case a straight line would be a poor model
13. Correlation shows a pattern, but doesn't by itself prove one variable causes changes in the other
14. It provides a consistent, mathematical method for minimising the overall distance between the line and the data points
15. The linear trend observed within the data range may not continue to hold true indefinitely beyond it
16. It indicates how strongly the data actually follows a linear pattern, showing how much to trust the model
17. Little to no linear relationship between the variables
18. The least-squares method is sensitive to points far from the overall pattern, which can pull the line toward them
19. The relationship between the variables may not hold the same way across a group the model wasn't built from
20. A weak correlation means a wide range of possible outcomes for any given input, limiting predictive precision
21. It shows the actual spread and variability in the data, not just the summarised trend