A histogram of repeats
Suppose you don't weigh the coin five times but five hundred times, then sort the results into little bins and draw a bar for how many landed in each bin. A picture appears: a tall hump centred on the mean, falling away smoothly and symmetrically on both sides. Most readings sit near the middle; a few stray far out. The further from the centre, the rarer.
This bell shape is not a coincidence. Whenever a measurement is jostled by many small, independent random errors that add up, the results pile into this same curve. It is so universal that we give it a name: the Gaussian distribution, also called the normal distribution.
Two numbers draw the whole curve
Here is the beautiful part: a Gaussian is fully described by just the two numbers from the last guide. The mean sets where the peak sits — slide it left or right. The standard deviation sets how wide the bell is — a small s gives a tall, narrow, precise hump; a large s gives a low, fat, sloppy one. Change those two dials and you can draw any normal curve there is.
The width of the bell is exactly the variance story from before, just drawn out. A precise method has small variance and a sharp peak. So when someone reports a mean and a standard deviation, they have implicitly handed you the whole bell — its location and its width — in two tidy numbers.
The 68–95–99.7 rule
Because every Gaussian has the same shape, you can read off fixed fractions of the data by counting standard deviations away from the mean. About 68% of readings fall within one standard deviation of the mean; about 95% within two; about 99.7% within three. This is sometimes called the empirical rule.
For the coin (mean 4.010 g, s ≈ 0.016 g) the rule predicts about 95% of future weighings will land between 4.010 − 0.032 and 4.010 + 0.032 — that is, between 3.978 and 4.042 g. So the standard deviation isn't just a label for past scatter; it is a forecast of where the next reading will probably fall.
What the bell can't see: bias
The Gaussian describes random error beautifully, but it is blind to one thing. If your balance reads 0.05 g heavy on every single weighing, the whole bell simply shifts sideways — it stays the same sharp, narrow shape, just centred on the wrong value. That constant offset is systematic error, and it quietly poisons your accuracy without widening the curve at all.
This is why precision (a narrow bell) and accuracy (a bell centred on the truth) are different things. A small standard deviation tells you readings agree with each other; it says nothing about whether they agree with reality. Catching a shifted bell needs a known reference — something the next guides build toward.