# How is absolute certainty achieved in mathematics

### Toooor !!

Football isn't just for boys. :-) Here are 2 girls who play soccer:

What do you think: which of the girls is more accurate?

Can you read that so quickly? Correct!

Ida scored more goals, namely 8. But it makes a difference whether she scores in 8 out of 20 appearances or in 8 out of 100 appearances.

So you are not interested in the absolute number 8, but how much these 8 hits make up of the total stakes.

Mathematicians call it **absolute** and **relative frequency**.

The **absolute frequency** here are the **Number of goals scored** at. Ida is ahead. She scored 8 goals, Carla only 6.

The **relative frequency** gives the **proportion of** of goals scored based on the **Total number** of the missions.

You calculate the relative frequency: $$ (T o r e) / (total number \ of \ units) $$.

Carla | Ida | |
---|---|---|

absolute frequency (goals) | ||

Total number (missions) | ||

relative frequency / proportion | $$ \ frac {6} {12} = \ frac {1} {2} = 0.5 $$ | $$ \ frac {8} {20} = \ frac {4} {10} = 0.4 $$ |

**Absolutely** Ida scored more goals than Carla.**Relative** Carla scored the larger percentage of goals based on the number of appearances.

So Carla is more accurate than Ida.

### Absolute and relative frequencies

If you want to compare data, you need not only the absolute but also the relative frequency.

As a formula it looks like this:

$$ relative \ frequency = frac {ab solute \ frequency} {total number} $$

**Example with balls:**

You have a vessel with these balls:

Find the absolute and relative frequency of the red and blue balls.

Total number of balls: $$ 10 $$

**Red balls:**

absolute frequency: $$ 6 $$

Relative frequency: $$ 6/10 = 0.6 $$

**Blue balls:**

absolute frequency: $$ 4 $$

Relative frequency: $$ 4/10 = 0.4 $$

The **absolute frequency** is a **number**. So that will **counted how often** something occurs.

The **relative frequency** is the **proportion of** on a total number. You write them as a fraction or a decimal fraction.

The relative frequency of a result is the quotient of the absolute frequency and the total number: $$ relative \ frequency = frac {ab solute \ frequency} {total number} $$

You can find relative frequencies both in **Fractions, decimal fractions** as well as in **Percent (%)** specify.

Example: $$ frac {1} {4} = frac {25} {100} = 0.25 = 25% $$

### Small repetition: convert fractions to decimal fractions

First write down relative frequencies as a fraction.

But sometimes decimal fractions are more practical.

You expand or shorten the fraction until you have a 10, a 100, or a 1000 at the bottom of the denominator. You can then write such a fraction as a decimal fraction.

**Examples:**

$$ 3/5 stackrel (2) = (3 * 2) / (5 * 2) = 6/10 = 0.6 $$

$$ 1/25 stackrel (4) = (1 * 4) / (25 * 4) = 4/100 = 0.04 $$

You already have some breaks in your head:

- $$1/2 = 0,5$$
- $$1/4=0,25$$
- $$3/4=0,75$$

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### Relative frequencies and charts

You can also find relative frequencies in **Charts** represent.

**Example:**

Have you ever played the wheel of fortune? And what did you win? :-)

This is what a wheel of fortune can look like:

If 200 people spin the wheel of fortune, this result can come out:

absolute frequency | relative frequency | |
---|---|---|

red | 50 | $$ frac {50} {200} = frac {1} {4} = 0.25 $$ |

orange | 50 | $$ frac {50} {200} = frac {1} {4} = 0.25 $$ |

blue | 100 | $$ frac {100} {200} = frac {1} {2} = 0.5 $$ |

This is how it looks **bar chart** in addition from:

### Sum sample

Anna rolls one dice 100 times. She wears that

Enter the results in a frequency list:

Result | absolute frequency | relative frequency |
---|---|---|

$$ frac {19} {100} = 0.19 $$ | ||

$$ frac {16} {100} = 0.16 $$ | ||

$$ frac {18} {100} = 0.18 $$ | ||

$$ frac {17} {100} = 0.17 $$ | ||

$$ frac {15} {100} = 0.15 $$ | ||

$$ frac {19} {100} = 0.15 $$ |

Calculate the sums of the absolute and relative frequencies:

Absolute frequencies: $$ 19 + 16 + 18 + 17 + 15 + 15 = 100 $$

You get the total number, here Anna's 100 litters.

Relative frequencies:

$$0,19+0,16+0,18+0,17+0,15+0,15=1$$

You receive 1.

It's always like that! This rule is called **Sum sample** and you can use it as a control.

If you add up all the absolute frequencies, you always get the total number.

If you add up all the relative frequencies, the result is always 1.

The **Sum sample** as a calculation control can deviate from 1 if the relative frequencies are rounded values.

If you have the **Sum of the relative frequencies** you will get the following result:

$$0,19+0,16+0,18+0,17+0,15+0,15=1$$

This result is general. You can therefore call this rule **Sum sample** also use it as a control.

The **Sum sample** as a calculation control can deviate from 1 if the relative frequencies are rounded values.

### Roundâ€¦

**Example:** 6 apps for kids

In a survey were **150 people** asked about popular apps for children. Here is the result. The decimal fractions are rounded.

App | absolute frequency | relative frequency |
---|---|---|

Moo box | ||

Little Winzki | ||

Mini piano | ||

Kids Paint | ||

memory kids | ||

English is easy |

Perform the sum test for the relative frequencies:

$$0,17+0,23+0,12+0,15+0,19+0,13=0,99$$

Oh, it doesn't get 1 at all !!

But if you have the relative frequencies in the **Fractional spelling** add up, you get the sum 1 again:

$$ frac {26} {150} + frac {35} {150} + frac {18} {150} + frac {23} {150} + frac {28} {150} + frac {20} {150} = frac {150} {150} = 1 $$

The problem is, you sometimes have to round off the decimal fractions. If you continue to calculate with the rounded numbers, you will get inaccurate results. It is safest to calculate with fractions.

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