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MathChapter 8: Statistics

Notes

Scatter Plots Notes

Describe association, interpret regression lines, make responsible predictions, distinguish observed and predicted values, and recognize nonlinear patterns.

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A scatter plot displays paired numerical measurements. Its pattern can suggest direction, form, and strength of association. A regression line summarizes a linear pattern and supports prediction, but association alone does not establish cause.

Direction and strength of association

Three association patterns

Compare the overall direction of the point cloud rather than requiring every point to move perfectly.

Positive associationPoints generally rise from left to right.
Positive association03691202.557.510InputResponse
View chart data
SeriesInputResponse
Observed points12
Observed points23
Observed points33.5
Observed points45
Observed points56.5
Observed points66
Observed points78.5
Observed points89
Observed points910.5
Negative associationPoints generally fall from left to right.
Negative association03691202.557.510InputResponse
View chart data
SeriesInputResponse
Observed points110.5
Observed points29
Observed points39.5
Observed points47.5
Observed points56.5
Observed points66
Observed points74
Observed points83.5
Observed points92
No clear associationThe points show no consistent upward or downward pattern.
No clear association03691202.557.510InputResponse
View chart data
SeriesInputResponse
Observed points14
Observed points29
Observed points33
Observed points48
Observed points55
Observed points610
Observed points74.5
Observed points87
Observed points93.5
Language for scatter-plot patterns
FeatureQuestion to askPossible descriptions
DirectionAs \(x\) increases, what generally happens to \(y\)?Positive, negative, or no clear association
FormDoes the point cloud follow a line or curve?Linear or nonlinear
StrengthHow tightly do points follow the pattern?Strong, moderate, or weak
Unusual featuresAre any points far from the pattern?Possible outliers or clusters

Line of best fit and regression

Study time and practice scoreObserved points with a line of best fit used for prediction.
Study time and practice score4055708510002468Study time (hours)Practice scorey = 6x + 48

Predicted values lie on the model line; observed values are the plotted measurements.

View chart data
SeriesStudy timePractice score
Observed scores153
Observed scores261
Observed scores362
Observed scores472
Observed scores578
Observed scores681
Observed scores791
Linear prediction model
\[\hat{y}=a+bx\]

The symbol \(\hat{y}\) denotes the predicted response. The slope \(b\) is predicted change in \(y\) for each one-unit increase in \(x\); \(a\) is the predicted response at \(x=0\).

Worked example

Interpret the line of best fit

The model is \(\hat{y}=6x+48\), where \(x\) is study time in hours and \(\hat{y}\) is predicted practice score.

  1. Slope

    For each additional hour, the predicted score increases by \(6\) points.

  2. Intercept

    At \(x=0\), the model predicts a score of \(48\). Interpret this only if \(0\) hours is meaningful in context.

  3. Prediction

    At \(x=5\), \(\hat{y}=6(5)+48=78\).

The slope is \(6\) score points per hour, and the predicted score at \(5\) hours is \(78\).
Residual
Observed response minus predicted response: \(y-\hat{y}\). A positive residual places a point above the model line; a negative residual places it below.

Observed versus predicted

Observed value \(y\)

The response actually measured and shown as a plotted point.

Predicted value \(\hat{y}\)

The response read from or calculated with the fitted model.

Prediction and model limits

  • Interpolation predicts within the observed \(x\)-range and is generally more defensible.
  • Extrapolation predicts beyond the observed range and may fail if the pattern changes.
  • A line of best fit summarizes a pattern; it does not pass through every point.
  • An outlier can influence a fitted line and should be examined in context.
  • The model intercept may be mathematically valid but contextually meaningless if \(x=0\) lies outside the relevant range.
Worked example

Observed, predicted, and residual

At \(x=7\), the model \(\hat{y}=6x+48\) predicts \(90\), while the observed score is \(91\).

  1. Predict

    \(\hat{y}=6(7)+48=90\).

  2. Compare

    The observed value \(91\) lies one point above the line.

  3. Residual

    \(y-\hat{y}=91-90=1\).

The residual is \(1\), so the model underpredicts by \(1\) point.

Recognize when a line is not enough

Curved associationPoints follow an upward-curving pattern that a straight line would not model well.
Curved association01020304001.83.55.37InputResponse

A visible curve suggests a nonlinear model, such as a quadratic or exponential pattern, may be more appropriate.

View chart data
SeriesInputResponse
Observed points02
Observed points13
Observed points26
Observed points310
Observed points417
Observed points526
Observed points638

Common scatter-plot mistakes

  • Calling an association positive because all values are positive.
  • Treating one local decrease as proof that no positive association exists.
  • Interpreting slope without units.
  • Confusing a plotted observation with the point on the regression line at the same \(x\).
  • Reversing residual subtraction.
  • Extrapolating far beyond the observed data without caution.
  • Claiming causation from a scatter plot alone.
  • Forcing a linear model onto an obviously curved pattern.
Mini check

Check your understanding

For \(\hat{y}=4x+30\), the observed value at \(x=8\) is \(65\). What is the residual?

  1. \(-3\)
  2. \(3\)
  3. \(62\)
  4. \(65\)
Show answer and explanation

Answer: \(3\)

The predicted value is \(4(8)+30=62\), so residual \(=65-62=3\).

Key takeaways

What to remember

  • Describe direction, form, strength, and unusual points.
  • A strong association is not automatically causal.
  • Regression slope is predicted response change per one explanatory-variable unit.
  • Residual equals observed minus predicted.
  • Interpolation is safer than unsupported extrapolation.
  • Use nonlinear models when the plotted pattern is systematically curved.
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Apply the ideas with SAT-style questions, then reinforce key details with flashcards.