The 34/13.5/2.5 rule of the normal distribution

Science and society, grade 12 (3 units) Lesson 10 of 14Learning

The 34/13.5/2.5 rule of the normal distribution

The 34/13.5/2.5 rule

In every normal distribution, whatever its mean and standard deviation, the same percent of the data lies in each band of one standard deviation.

Scores in an IQ test are normally distributed, with mean $\displaystyle {\mu}={100}$ and standard deviation $\displaystyle {\sigma}={15}$ :

Between the mean and one standard deviation above it – 34%: between 100 and 115.
Between one and two standard deviations – 13.5%: between 115 and 130.
More than two standard deviations above – 2.5%: above 130.
On the other side of the mean – the same percents, since the curve is symmetric.

Two useful rules follow:

\displaystyle P(\mu-\sigma<x<\mu+\sigma)\approx68\%

\displaystyle P(\mu-2\sigma<x<\mu+2\sigma)\approx95\%

And nearly all the data (99.7%) lie between $\displaystyle \mu-3\sigma$ and $\displaystyle \mu+3\sigma$ .

The percent between two values

What percent of people score between 85 and 130 in the test?

85 is one standard deviation below the mean, and 130 two above it.
Add up the bands between them: $\displaystyle {34\%+34\%+13.5\%}={81.5\%}$ .

The percent above a value

What percent of people score more than 115?

Above 115 there are two bands: $\displaystyle {13.5\%+2.5\%}={16\%}$ .
Another way: above the mean is half the people, 50%, less the 34% between 100 and 115: $\displaystyle {50\%-34\%}={16\%}$ .

How many people

In a year group of 600 pupils, how many are expected to score more than 115?

600\cdot16\%=600\cdot0.16=96

About 96 pupils. The number is an estimate: a real year group might have 90 or 101.

Exercise 1

Babies’ birth weight is normally distributed, with mean 3,300 grams and standard deviation 500 grams.

a

What percent of babies weigh between 2,800 and 3,800 grams?

/ – 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

b

What percent of babies weigh more than 4,300 grams?

/ – 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

c

What percent of babies weigh between 3,300 and 4,300 grams?

/ – 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

d

A hospital has 2,000 babies born a year. How many of them are expected to weigh less than 2,300 grams?

/ – 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

Exercise 2

The marks in a national exam are normally distributed, with mean 70 and standard deviation 10.
What percent of the pupils got a mark in the shaded range?

/ – 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

Exercise 3

The height of 18-year-old boys is normally distributed, with mean 176 cm and standard deviation 7 cm.

a

What percent of the boys are taller than 183 cm?

/ – 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

b

What percent of the boys are shorter than 162 cm?

/ – 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

c

What percent of the boys are between 169 and 190 cm tall?

/ – 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

d

Of 800 boys called up to the army, how many are expected to be taller than 190 cm?

/ – 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

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

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 $\displaystyle z$
$\displaystyle x=\mu+z\cdot\sigma$