The normal curve and its properties

Babies’ birth weights are normally distributed. When very many babies are weighed and the histogram’s classes are made narrower, the tops of the bars look like a smooth bell-shaped curve – the normal curve.

The mean of a normal distribution is written $\displaystyle \mu$ (mu), and the standard deviation $\displaystyle \sigma$ (sigma). For the birth weights $\displaystyle {\mu}={3300}$ grams and $\displaystyle {\sigma}={500}$ grams:

• The curve is symmetric round the mean: the mean, the median and the mode are equal – 3,300 grams.
• Most of the data is gathered in the middle, and the curve falls quickly further from the mean: very few babies weigh less than 2,000 grams or more than 4,600 grams.
• The curve never touches the axis: a value even further out is always possible, but very rare.
• The axis is marked at the mean, and at one, two and three standard deviations above and below it.

The area under the whole curve is 1: it stands for all the data, 100%. So the area under the curve between two values is the part of the data between those values.

The shaded area is the part of the babies born weighing less than 2,800 grams:

The same area can be read in three ways:

• the proportion of babies born weighing less than 2,800 grams – a number between 0 and 1;
• the percentage of those babies;
• the probability that a baby chosen at random weighs less than 2,800 grams.

Two machines in a factory fill 500 ml bottles, and a third fills too much:

• The mean sets where the curve is: machine C’s curve is machine A’s, moved right to 508 ml.
• The standard deviation sets the width: machine B’s curve has a larger standard deviation – it is wide and low, and its bottles are more spread out.
• The area under each curve is 1, so a wider curve is also lower.

In every normal distribution the part of the data in each band of one standard deviation is fixed, roughly:

• between $\displaystyle \mu$ and $\displaystyle \mu+\sigma$: about 34%, and the same between $\displaystyle \mu-\sigma$ and $\displaystyle \mu$;
• between $\displaystyle \mu+\sigma$ and $\displaystyle \mu+2\sigma$: about 13.5%;
• above $\displaystyle \mu+2\sigma$: about 2.5%.

$\displaystyle X$ stands for a value chosen at random. The rule works only at a whole number of standard deviations; in the next lesson you will learn to work out the part between any two values.

The life of batteries is normally distributed.
15.9% of the batteries work for less than 185 hours.
15.9% of the batteries work for more than 215 hours.
What is the mean life, and what is the standard deviation?

• Both sides have the same percentage, so by the symmetry the mean is in the middle: $\displaystyle {\mu}={\frac{185+215}{2}}={200}$ hours.
• Above $\displaystyle \mu+\sigma$ there is about $\displaystyle {13.5\%+2.5\%}={16\%}$, and 15.9% is very close to that: 215 is $\displaystyle \mu+\sigma$.
• $\displaystyle {\sigma}={215-200}={15}$ hours.

Here are three normal curves, of three different populations.
Compare the three populations: what is the mean of each, and which is more spread out?

The heights of 17-year-old boys and girls are normally distributed.
The mean height of the boys is greater than the mean height of the girls.
Which curve describes the boys’ heights, and which the girls’? Explain.

The weight of loaves of bread is normally distributed.
2.5% of the loaves weigh less than 480 grams.
2.5% of the loaves weigh more than 520 grams.

What is the mean weight?

What is the standard deviation?

IQ test scores are normally distributed, with mean 100 and standard deviation 15.
Answer by the 34/13.5/2.5 rule.

What percentage of those tested scored between 85 and 115?

What percentage scored more than 130?

2,000 people took the test. How many of them scored between 70 and 130?