Scatter Plots and Correlation

Scatter Plots #

Each item one dot, two measurements.

A scatter plot shows each item as one dot, placed by two measurements

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Each dot is one person: hours practiced on the x-axis, score on the y-axis.

The dots rise from left to right: more practice goes with a higher score.

Correlation #

Rising together, falling apart, or neither.

Correlation says whether two measurements rise together, move in opposite directions, or neither

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The dots rise together, up and to the right. That is positive correlation.

As hours of TV go up, the score goes down. That is negative correlation.

A shapeless cloud is no correlation: one measurement tells you nothing about the other.

Line of Best Fit #

Through the middle of the drift.

A line of best fit follows the trend of the dots and passes through their mean point

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The dots trend upward but no straight line passes through them all.

Average the x values and the y values: 5 and 5.4. That is the mean point.

Draw the line through the mean point, with roughly as many dots above it as below.

To estimate at x = 6, go up to the line and across: y is about 6.

That line is y = 0.6x + 2.4, and substituting x = 6 gives the same estimate.

Choosing Between Two Models #

The one whose residuals show no pattern wins.

The better model is the one whose residuals are small and show no pattern

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Here are five weekly counts of plants, with the line of best fit drawn through them.

The gap at each point is the residual: the count minus what the line predicted.

In order, the residuals are positive, negative, negative, negative, positive: a curve the line misses.

A doubling model, $y=2^x$y = 2ˣ, fits the same five counts instead.

Its residuals are 0, 0, 1, −1 and 1: small, with no pattern left to explain.

Interpolation and Extrapolation #

Outside what you measured is a guess.

Reading outside the data you collected is a guess rather than a result

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The data runs from 2 to 6, so an estimate at 4 is inside it: that is interpolation.

Extending the line to 20 is extrapolation: no data there supports it.

Time Series #

Readings in order — trend under the wiggle.

A time series joins readings taken in order so trend and season become visible

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One reading per month, joined in time order — this is a time series.

It goes up and down, but each low point is higher than the one before: the overall trend is rising.

Sales peak every summer, so compare the same season: this July against last July.

Correlation Is Not Causation #

Rising together is a question, not an answer.

Two things can rise together because a third thing drives both

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Ice-cream sales and drownings rise together: the correlation is real.

Behind both sits a shared cause: hot weather sells ice cream and sends people swimming.

A correlation raises a question — it never answers one on its own.