Scatter Plots and Correlation

Scatter Plots #

Each item one dot, two measurements.

A scatter plot puts each item as one dot using two measurements

Each dot is one person: hours practiced across, score up the side.

The dots drift upwards together. That drift is the pattern.

Correlation #

Rising together, falling apart, or neither.

Correlation says whether the dots rise together fall together or neither

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

One climbs while the other falls. 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 runs through the middle of the drift of the dots

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 putting x = 6 in 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

Five weekly counts, with the best straight line laid over them.

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

In order the residuals run high, low, low, low, high — a curve the line missed.

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

Its residuals are about 1 and change sign, so no pattern is 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

The data runs 2 to 6, so a reading at 4 sits inside it: that is interpolation.

Stretch it out to 20 and that is extrapolation — nothing measured 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

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

It wiggles, yet every dip lands higher than the dip before — under the wiggle, it rises.

Summers spike every year, so compare like with like: 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

Ice-cream sales and drownings rise together — the dots genuinely climb.

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.