What the bell curve is telling you

You only need three numbers: the average you expect (E), the typical distance from that average (σ), and how many σ's away your target sits (z). Set a business question below and watch how the three combine into a probability you can act on.

The idea in four steps

From a business question to a probability

1 · E, the centreexpected valueWhere results cluster. The peak of the curve.

2 · σ, the spreadstandard deviationThe typical distance of a result from E.

3 · z, the distancez = (target − E) ÷ σHow many σ's the target is from E.

4 · The tablez → areaTurns that distance into a probability.

Your question

Set up the business question

Start from an example

What is measured

Unit before

Unit after

Expected value E

Standard deviation σ

Target

Upper value

The picture

Two rulers, one curve

The bottom of the chart has two rulers that describe the same points. The upper one is in real units (rands, millilitres, days). The lower one counts standard deviations from E. A z-score is just a reading on the lower ruler. Drag across the chart or use the slider to move the target.

Move the target

Area asked about = probability E, the centre σ bands (±1, ±2, ±3)

The 68 – 95 – 99.7 rule

The "1 to 3 times σ" you half remember

For any bell-shaped result, about 68% of outcomes fall within 1σ of E, 95% within 2σ, and 99.7% within 3σ. Here is what that means for the figures you entered.

Exam technique

The working, step by step

Show the full standard normal table (z from 0,00 to 3,49)

Each cell is the area to the left of z: the probability of a result below that point. Find the row for the first decimal of z and the column for the second decimal. Printed tables list positive z only. For a negative z, look up the positive value and subtract the answer from 1.

Know the limits

When the bell curve can mislead

The curve is an assumption

The method assumes results spread smoothly and symmetrically around E. That fits well for things that come from many small, independent effects, such as fill volumes on a production line, monthly sales across many customers, or delivery times. It fits poorly when there are only a few possible outcomes, or when one extreme outcome is possible (a "boom" scenario, a large bad debt).

Check it against the scenarios

Experiments to try

See what moves, and why

Double σ while keeping E and the target fixed. z halves, and the tail beyond the target grows. More spread means more chance of a result far from plan.
Put the target exactly on E. z is 0 and the probability is 50%, whatever the units and whatever σ is.
Load "Lotion filling line" and raise the average fill by 1 ml. Underfilled tubes fall sharply. This is why factories aim slightly above the legal minimum.
Load "Magasman vs R150 000". The blue spikes are the four actual scenarios, and four spikes cannot form a bell. Compare the curve's answer with the counted answer.