The normal distribution: the bell curve

Science and society, grade 12 (3 units) Lesson 9 of 14Learning

The normal distribution: the bell curve

The heights of a thousand girls

In a Ministry of Health survey the heights of 1,000 girls aged 17 were measured.
The histogram shows how many girls are in each class of 3 cm:

Most of the girls are near the middle, between 158 and 170 cm.
The further from the middle – up or down – the fewer the girls, and very few are below 149 or above 179.
The two sides are nearly mirror images of each other.

The bell curve

If more and more girls are measured and the classes made smaller, the tops of the bars look like a smooth bell-shaped curve. It is called a normal curve, and data that this curve describes are normally distributed.

For the girls’ heights the mean is $\displaystyle {\mu}={164}$ cm and the standard deviation $\displaystyle {\sigma}={6}$ cm:

The curve is symmetric about the mean: the mean, the median and the mode are equal – 164.
It is highest in the middle: that is where most of the data are.
The axis is marked at the mean, and one, two and three standard deviations above and below it: $\displaystyle 158$ and $\displaystyle 170$ , $\displaystyle 152$ and $\displaystyle 176$ , $\displaystyle 146$ and $\displaystyle 182$ .
The Greek letters $\displaystyle \mu$ (mu) and $\displaystyle \sigma$ (sigma) are the usual names for the mean and the standard deviation of a normal distribution.

The standard deviation: the bell’s width

In grade 11 you learnt that the standard deviation measures spread: roughly how far the data are from the mean. In a normal distribution it sets the bell’s width.

Two machines in a factory fill 500 ml bottles of water. In both the mean is 500 ml, but in machine A the standard deviation is 2 ml, and in machine B 5 ml:

Machine A’s bell is narrow and tall: nearly all its bottles are close to 500 ml. It is more precise.
Machine B’s bell is wide and low: some bottles hold 490 ml, others 510.
The area under each curve is the same – it stands for all the bottles, 100%. So a wider bell is also lower.

Which data are normally distributed

A normal distribution fits a feature affected by many small causes, each adding a little or taking a little away:

the height and weight of people of the same age;
the weight of babies at birth;
scores in a national exam, and IQ tests;
the amount a machine puts in a bottle, the weight of a loaf of bread.

Some data are not normally distributed. Monthly pay, for example, is not symmetric: most people earn a middling wage, and a few a very high one, which “pulls” a long tail to the right:

In such data the mean is larger than the median, and the rules of the normal curve must not be used.

Exercise 1

The weight of babies at birth is normally distributed:

What are the mean and the standard deviation of the birth weights?

The mean

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The standard deviation

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Work out the weight that is two standard deviations above the mean.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Work out the weight that is one standard deviation below the mean.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The times of two groups in a 5 km race, in minutes, are normally distributed:
group A – members of a running club, and group B – pupils in the school race.

Which group ran faster on average? In which group are the times more spread out? How do the curves show it?

Exercise 3

Which of these data are probably normally distributed? Explain.

The length of leaves from the same tree.
The number of children in a family.
House prices in a city.
The weight of 1 kg bags of flour coming out of a machine.
The marks of all the pupils in the country in the English matriculation exam.

Formulas: science and society, grade 12 (3 units)

Open as a page

A quadratic model

A quadratic model – $\displaystyle a<0$ : it has a highest point, $\displaystyle a>0$ : a lowest point
$\displaystyle y=ax^{2}+bx+c$ $\displaystyle a\neq0$
The vertex’s $\displaystyle x$ – the axis of symmetry
$\displaystyle x=-\frac{b}{2a}$
The vertex’s $\displaystyle x$ – midway between two values of $\displaystyle x$ with the same $\displaystyle y$
$\displaystyle x=\frac{x_1+x_2}{2}$
The quadratic formula for $\displaystyle {ax^{2}+bx+c}={0}$ – the zeros
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Other models

A linear relation – the differences in the table are equal
$\displaystyle y=mx+b$
An exponential relation – the ratio of two neighbouring values in the table is fixed, $\displaystyle b$
$\displaystyle y=a\cdot b^{x}$

The normal distribution

The standard deviation of $\displaystyle n$ data – $\displaystyle \bar{x}$ their mean
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^{2}+(x_2-\bar{x})^{2}+\dots+(x_n-\bar{x})^{2}}{n}}$
In a normal distribution – $\displaystyle \mu$ the mean, $\displaystyle \sigma$ the standard deviation: about 68% of the data within one standard deviation of the mean (34% on each side)
$\displaystyle P(\mu-\sigma<x<\mu+\sigma)\approx68\%$
About 95% within two standard deviations of the mean (another 13.5% on each side)
$\displaystyle P(\mu-2\sigma<x<\mu+2\sigma)\approx95\%$
About 2.5% above $\displaystyle \mu+2\sigma$ , and about 2.5% below $\displaystyle \mu-2\sigma$
$\displaystyle P(x>\mu+2\sigma)\approx2.5\%$
How many of $\displaystyle n$ data are $\displaystyle p$ percent of them
$\displaystyle n\cdot p\%$

The standard score

The standard score of the value $\displaystyle x$ – how many standard deviations it is above the mean (or below it, when $\displaystyle z<0$ )
$\displaystyle z=\frac{x-\mu}{\sigma}$
The value whose standard score is $\displaystyle z$
$\displaystyle x=\mu+z\cdot\sigma$