Weigh a thousand apples. What shape do the masses make?

Most sit near the average, fewer and fewer lie further out, and the two sides mirror each other. That bell is the Normal distribution.

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The masses of apples from an orchard are modelled by

in grams: symmetric about the mean, 150 g, with standard deviation 12 g setting the spread. The whole area under the curve is 1.

The probability that an apple weighs less than 160 g is the area to the left of 160:

No formula gives it exactly; a calculator's Normal function does.

Under the grams, a second scale counts standard deviations from the mean. 160 g is $\tfrac{10}{12} = 0.833$ of a standard deviation above 150, so the same area is $P(Z < 0.833)$ for $Z \sim N(0, 1)$: every Normal curve is the standard one, stretched and shifted.

The curve changes from bending down to bending up at 138 and 162, one standard deviation either side. About 68% of apples lie between those, 95% within two standard deviations, 126 to 174, and 99.7% within three.

One shape, measured in standard deviations.

Inverse Normal problems, finding an unknown mean or standard deviation, the Normal approximation to the binomial with a continuity correction, and twelve graded exam questions with full worked solutions are waiting.

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