Comparing two normal distributions

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

Comparing two normal distributions

Different means: the heights of girls and boys

Two distributions drawn on the same axis show at a glance how they differ: the bell’s place is the mean, and its width the spread.

The height of 18-year-old girls and boys is normally distributed:
for the girls the mean is 163 cm and the standard deviation 6 cm.
For the boys the mean is 176 cm and the standard deviation 7 cm.

The boys’ bell is further right: on average they are taller.
The curves overlap a great deal: many girls are taller than many boys.

Michal is 175 cm tall, and Yoav 183 cm. Which of them stands out more in height, compared with their own group?

Michal: $\displaystyle {z}={\frac{175-163}{6}}={2}$ – taller than about 97.5% of the girls.
Yoav: $\displaystyle {z}={\frac{183-176}{7}}={1}$ – taller than about 84% of the boys.

Yoav is taller than Michal, but Michal stands out more in her group.

The same mean, different spreads

In two classes the mean mark in a test is 75.
In class A the standard deviation is 5, and in class B 15.
What percent of each class scored more than 90?

In class A: $\displaystyle {z}={\frac{90-75}{5}}={3}$ . Above three standard deviations there is almost nobody.
In class B: $\displaystyle {z}={\frac{90-75}{15}}={1}$ . Above one standard deviation – 16% of the class.

The mean is the same, but class B has many very high marks – and many very low ones too. A mean alone does not tell the whole story: the standard deviation is needed as well.

Exercise 1

An exam had two versions, and the marks in each are normally distributed:
in version A the mean is 68 and the standard deviation 8.
In version B the mean is 76 and the standard deviation 4.
Yoav sat version A and got 80. Maya sat version B and got 84.

a

Work out Yoav’s standard score.

/ – 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

Work out Maya’s standard score.

/ – 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

Which of them did better compared with those who sat the same version? Which version was probably harder?

Exercise 2

Back to the two versions of Exercise 1.

a

What percent of those who sat version A got more than 76?

/ – 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 those who sat version B got more than 76?

/ – 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 working life of two kinds of battery, in hours, is normally distributed:
kind A – mean 20 hours, standard deviation 2 hours.
Kind B – mean 22 hours, standard deviation 4 hours.

a

What percent of each kind works for more than 26 hours?

b

What percent of each kind works for less than 18 hours?

Kind A

/ – 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

Kind B

/ – 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

A doctor needs a battery for a medical device, which must work for at least 16 hours. Which kind should she choose? Explain.

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