Relative frequency

Calculation of the relative frequency as a quantity diagram

The relative frequency is a division number and a measure of descriptive statistics . It shows the proportion of elements in a set that have a certain characteristic value. It is calculated by dividing the absolute frequency of a feature in an underlying set by the number of objects in this set. The relative frequency is therefore a fraction and has a value between 0 and 1.

General mathematical definition

Relative frequencies are calculated in relation to an underlying quantity. This set can be either a population or a sample . To define the relative frequency, let's assume that the underlying set has elements. The event occurs under these elements . The relative frequency is calculated as the number of observations with the characteristic divided by the total number of all elements in the underlying set. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle H_{n}(A)}$ ${\displaystyle A}$ ${\displaystyle A}$

The relative frequency is therefore found as

${\displaystyle h_{n}(A)={\frac {H_{n}(A)}{n}}}$.

${\displaystyle H_{n}(A)}$is also known as the absolute frequency . In contrast to the relative frequency , meaningful comparisons between samples (or populations) of different sizes with the absolute frequency are usually not possible. ${\displaystyle h_{n}(A)}$ ${\displaystyle H_{n}(A)}$

Examples

Proportion of girls in a school class

Class A has 24 students, 12 of them girls. In class B there are 18 students, 9 of them girls. This means that there are more girls in class A (12) than in class B (9) if you look at the absolute frequency. On the other hand, if you look at the frequency of girls in relation to the respective class size, you can see that the proportion of girls is the same in both classes: in class A the relative frequency of girls is 0.5 (= ¹² ⁄ ₂₄ ) and in class B it is also 0 , 5 (= ⁹ ⁄ ₁₈ ). The relative frequency can also be easily converted into a percentage by multiplying it by 100%. Thus, both classes consist of 50% (= 0.5 × 100%) girls.

Polls

In an election poll, 600 eligible voters in Bavaria are interviewed, as well as 200 eligible voters in Berlin. In Bavaria passed to 120 respondents, the party A to choose. In Berlin, 100 respondents say they would vote for party A. The absolute frequency for voters of Party A is thus higher in Bavaria than in Berlin, namely 120 respondents in Bavaria versus 100 respondents in Berlin. However, this is due to the fact that three times as many people were interviewed in Bavaria as in Berlin. A comparison of the absolute frequencies is therefore not useful.

In contrast, the relative frequency enables a comparison of the popularity of Party A between Bavaria and Berlin. In Bavaria the relative frequency is 0.2 (= ¹²⁰ ⁄ ₆₀₀ ). The relative frequency for Berlin is calculated as 0.5 (= ¹⁰⁰ ⁄ ₂₀₀ ). Party A is much more popular in Berlin than in Bavaria.

properties

In contrast to the absolute frequency, the relative frequency is always between 0 and 1. This allows different relative frequencies to be compared with one another, although they refer to a different reference variable. In descriptive statistics , relative frequencies are therefore used to be able to compare frequency distributions regardless of the number of elements in the population (i.e. regardless of the sample size ).

In the context of inferential statistics and stochastics , the relative frequency is used as a maximum likelihood estimator for the parameter probability of success of a binomial distribution .

The following calculation rules apply to the relative frequency:

• ${\displaystyle 0\leq h_{n}(A)\leq 1}$due to the normalization to the number of repetitions. ${\displaystyle n}$
• ${\displaystyle h_{n}(\Omega )=1\,}$for the safe event .
• ${\displaystyle h_{n}(A\cup B)=h_{n}(A)+h_{n}(B)-h_{n}(A\cap B)}$ for the sum of events.
• ${\displaystyle h_{n}({\bar {A}})=1-h_{n}(A)}$ for the complementary event.

Relative frequency and probability

Frequentistic concept of probability

→ Main article : Perception of probability

The frequentistic concept of probability interprets the probability of an event as the relative frequency with which it occurs in a large number of identical, repeated, independent random experiments . This is the so-called 'Limes definition' according to von Mises . The prerequisite for this concept of probability is that the experiment can be repeated as required; the individual rounds must be independent of each other.

Example: You roll the dice 100 times and get the following distribution: the 1 falls 10 times (this corresponds to a relative frequency of 10%), the 2 falls 15 times (15%), the 3 also 15 times (15%), the 4 in 20%, the 5 in 30% and the 6 in 10% of the cases. After 10,000 runs, the relative frequencies - if there is a fair dice - have stabilized in the vicinity of the probabilities. For example, the relative frequency of rolling a 3 is approximately 16.6%.

The axiomatic definition of probability used today as the basis of probability theory manages without recourse to the concept of relative frequency. Even using this definition of probability, however, there is a close relationship (by means of the law of large numbers ) between probability and relative frequency.

Law of Large Numbers

Laws of large numbers denote certain convergence theorems for the almost certain convergence and the convergence in probability of random variables. In their simplest form, these sentences say that the relative frequency of a random result usually approaches the probability of this random result if the underlying random experiment is carried out over and over again. The laws of large numbers can be proven from Kolmogorov's axiomatic definition of probability. Thus there is a close connection between relative frequency and probability even if one is not a representative of the objectivistic conception of probability.

literature

• Bernhard Rüger: Inductive Statistics. Introduction for economists and social scientists . R. Oldenbourg Verlag, Munich Vienna 1988, ISBN 3-486-20535-8 .

Individual evidence

1. ↑ Bernhard Rüger (1988), p. 8 ff.
2. ↑ Bernhard Rüger (1988), p. 11 ff.
3. ↑ ^{a b c} Bernhard Rüger (1988), p. 79 ff.