Scatter diagrams
Section: Statistics | Syllabus: Cambridge IGCSE Mathematics (0580)
Correlation and Types of Scatter Diagram
- A scatter diagram plots pairs of values for two variables, one on each axis, to show whether there is a relationship (correlation) between them
- Positive correlation: as one variable increases, the other also tends to increase (points slope upward, left to right)
- Negative correlation: as one variable increases, the other tends to decrease (points slope downward, left to right)
- Zero (no) correlation: there is no clear pattern between the two variables
- Correlation can also be described by strength: points lying close to a clear trend show strong correlation; points more loosely scattered around a trend show weak correlation
Worked Example: Describing Correlation
- Question: The table shows the number of hours studied and the test score for 8 students.
Describe the correlation shown by this data, including its type and strength.Hours studied 1 2 3 3 4 5 6 7 Test score 40 45 50 55 60 65 75 85 - Step 1: As the number of hours studied increases, the test score also increases
- Step 2: This means the correlation is positive
- Step 3: The scores increase fairly consistently alongside the hours, staying close to a clear increasing pattern, so the correlation is strong
- Answer: There is strong positive correlation between hours studied and test score
Common Mistakes
MistakeConfusing which direction of slope matches positive and negative correlation
Fixpicture the trend from left to right: sloping upward is positive, sloping downward is negative - "both go the same way" is positive, "one goes up while the other goes down" is negative
MistakeAssuming that correlation between two variables always proves that one directly causes the other
Fixcorrelation only shows that two variables tend to change together - it does not by itself prove that a change in one variable causes the change in the other
Line of Best Fit and Prediction
- A line of best fit is a single straight line, drawn by eye, that follows the general trend of the plotted points, with roughly equal numbers of points above and below it
- It can be used to estimate a value of one variable for a given value of the other; estimating within the range of the plotted data is called interpolation, and is reasonably reliable
- Estimating outside the range of the original data is called extrapolation - this is much less reliable, since the trend may not continue in the same way beyond the data actually collected
Worked Example: Using a Line of Best Fit to Predict a Value
- Question: Using the line of best fit for the hours-studied and test-score data above, estimate the test score for a student who studied 4.5 hours. Explain why it would not be appropriate to use the line to estimate the score for a student who studied 20 hours.
- Step 1: 4.5 hours lies within the range of the plotted data (1 to 7 hours), so reading from the line of best fit gives a reasonable estimate, approximately 62
- Step 2: 20 hours lies far outside the range of the plotted data (1 to 7 hours)
- Step 3: Using the line to predict this far beyond the collected data is extrapolation, and is unreliable, since the relationship between hours studied and score might not continue in the same way at such extreme values
- Answer: estimated score 62; the estimate for 20 hours would be extrapolation and therefore unreliable
Common Mistakes
MistakeDrawing a line of best fit that joins the points like a dot-to-dot picture, instead of a single straight line following the overall trend
Fixa line of best fit is always a single straight line - it does not need to pass through every point, only follow their general direction
MistakeTreating an extrapolated estimate (far beyond the data range) as being just as reliable as an interpolated one
Fixalways check whether the value being estimated falls inside or outside the range of the original data before trusting the estimate
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