Scatter diagrams and correlation
Introduced| English | Meaning |
|---|---|
| scatter diagram/ˈskætə ˈdaɪəɡræm/ | A plot of paired data as points, showing correlation. |
| correlation/ˌkɒrɪˈleɪʃn/ | A statistical association between variables, which does not by itself establish causation. |
| positive correlation/ˈpɒzɪtɪv ˌkɒrɪˈleɪʃn/ | positive correlation |
| negative correlation/ˈneɡətɪv ˌkɒrɪˈleɪʃn/ | negative correlation |
| line of best fit/laɪn ɒv best fɪt/ | A line chosen to summarise the overall relationship between two variables in a scatter plot. |
| extrapolation/ekˈstræpəleɪʃn/ | Estimating a value outside the range of observed data by extending a pattern or model. |
| outlier/ˈaʊtlaɪə/ | An observation unusually far from the rest of the data or from the pattern expected under a model. |
Do tall people have bigger feet?
- Plot height against shoe size for 30 people and you'll see a pattern: taller people tend to have larger feet.
- A scatter diagram 散点图 reveals whether two variables are linked — and how strongly.
Correlation 相关性
- Positive correlation 正相关: as one goes up, the other goes up (points slope upward).
- Negative correlation 负相关: as one goes up, the other goes down (points slope downward).
- Zero correlation: no clear pattern (points scattered randomly).
Correlation $\neq$ causation. Ice cream sales and drowning deaths both rise in summer — but ice cream doesn't cause drowning. Both are caused by a third factor: hot weather.
A line of best fit 最佳拟合线 is one straight line through the middle of the points, with about as many on each side; use it to predict
Scatter & correlation
r near ±1 = strong
Tighten the points to see strong correlation; scatter them for weak — read r and the line.
As one quantity increases, the other also increases. This is:
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Both rising together is positive correlation.
Negative correlation means as one quantity goes up, the other goes down.
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A downward trend on the scatter diagram is negative correlation.
If two variables are correlated, one must cause the other.
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Correlation does not prove causation — both could be caused by a third factor.
Line of best fit
- If there is correlation, draw one straight line through the middle of the points, with roughly equal numbers on each side.
- Use it to predict values.
A line of best fit through height-shoe size data: height $170$ cm → predicted shoe size $9$. But don't trust predictions far outside the data range (extrapolation 外推 is risky).
Positive correlation: both rise together. Negative: one rises as the other falls. Zero: no clear link
A line of best fit is mainly used to:
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It models the trend so you can predict one value from the other.
Using the line of best fit to predict values far outside the data range is called ______.
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Extrapolation is unreliable because the trend may not continue beyond the observed data.
Outliers 异常值 on scatter diagrams
- A point far from the general pattern is an outlier.
- Investigate before removing it — it might be a genuine unusual case, or a recording error.
Scatter diagrams reveal patterns: positive (like a rising line), negative (like a falling line), or zero (no pattern).
On a scatter diagram, most points follow a line, but one point is far away. This point is called an ______.
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An outlier is a data point that does not fit the general pattern.
Plot pairs and read an estimate
- Mark each pair with a small cross, not joined points. For $(1,12),(2,15),(3,19),(4,20),(5,24)$, a sensible by-eye line is close to $y=3x+9$; it spans all the data and roughly balances the points above and below.
- At $x=3.5$, this line estimates $y=19.5$. A nearby balanced line may give a slightly different answer. Predicting at $x=20$ is extrapolation and much less secure; correlation alone does not prove cause.
Use the line y = 3x + 9 to estimate y at x = 3.5.
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y = 3(3.5) + 9 = 19.5.
You've got it
- positive correlation rises together; negative falls as the other rises; zero $=$ no link
- a line of best fit is one straight line through the middle of the points
- use the line to predict — but don't trust it far outside the data