How Sure Are You?

Uncertainty, Slopes, and Predictions

Five Drops, Five Answers

You drop your ball from 1.00 m, five times, and read the bounce:

Trial 1 2 3 4 5
Bounce (m) 0.58 0.63 0.60 0.65 0.59

Same ball. Same height. Same person reading.

So which one is the answer?

None of Them. All of Them.

The honest answer is not a single number — it's a center and a spread.

Center: the mean → m

Spread: half the range → m

Scatter Is Not a Mistake

Your five readings disagreeing is not sloppy work.

It is the measurement telling you how precisely it can be made.

Report the spread. Never hide it by writing down one "good" trial.

So Why Graph It At All?

You could stop here: and call it done.

Don't. A single ratio has three problems:

  • It rests on two measurements — you took forty
  • It assumes the line goes through the origin. It can't check
  • One misread apex moves it a lot. One bad point barely tilts a line

The graph is how you spend all your data at once.

Drawing the Best-Fit Line

  • Plot mean bounce height vs. drop height
  • Do not connect the dots — you're not tracing, you're modeling
  • One straight line, roughly as many points above as below
  • Extend it across the full range

Finding the Slope

Pick two points that sit on your line — not two of your data points.

Units? Meters over meters. They cancel — the slope is a pure number.

Reading slope off two data points throws away the whole point of fitting a line.

But How Sure Is That?

You drew one line. Your data would have tolerated others.

Steepest and Shallowest

Draw the steepest line your data still supports, and the shallowest.

Line Slope
Steepest 0.65
Best fit 0.60
Shallowest 0.55

Your Model

That is the whole lab in one line: an equation, built from your data, carrying your precision.

Write it in physics symbols. Never .

Now Make It Earn Its Keep

An equation that only describes data you already have proves nothing.

Predict a bounce you have not measured.

Pick a drop height inside your range that you skipped — say 1.40 m.

A Prediction Is a Range

Run your best slope, then both edges:

Using slope Predicted bounce
0.55 (shallowest) m
0.60 (best fit) m
0.65 (steepest) m

Then Test It

Write the prediction before you drop the ball. Then run five trials.

  • Lands inside your range → your model works, and your uncertainty was honest
  • Lands just outside → you were probably too confident — your range was too narrow
  • Lands far outside → something is wrong with the model, not the arithmetic

A prediction written after the measurement tests nothing at all.

Two Kinds of Error

Random — scatter, different every time

  • Misjudging the apex, slight release differences
  • More trials and averaging shrink it

Systematic — same direction every time

  • Measuring drop to the ball's bottom but bounce to its top
  • Averaging does nothing. A thousand trials stay wrong

Only one of these is fixed by working harder. The other is fixed by thinking harder.

In Your Notebook

  • Mean and spread at each height
  • A graph with a best-fit line, plus your steepest and shallowest*
  • Slope uncertainty
  • Your model equation in physics symbols
  • Prediction and range, written before the test drop
  • Measured result, and whether it landed inside