Empirical probability

The relative frequency is a measure of the descriptive statistics. It gives the proportion of elements of a set back, at which there is a certain characteristic value. It is calculated by the absolute frequency of a feature in an underlying quantity by the number of objects is divided into this set. Thus, the relative frequency is a fractional number and has a value between 0 and 1

  • 4.1 frequentist concept of probability
  • 4.2 Law of Large Numbers

General mathematical definition

Relative frequencies are calculated with respect to an underlying quantity. This amount can be both a population and a sample. To define the relative frequency, we assume that the underlying quantity has elements. Among these elements occurs n times the event A. The relative frequency is calculated as the number of observations of the characteristic A is divided by the total number of elements in the underlying volume.

The relative frequency gives this as

Is also referred to as an absolute frequency. In contrast to the relative frequency are meaningful comparisons between samples ( or populations ) of different size with the absolute frequency is usually not possible.

Examples

Proportion of girls in a school class

In a class A are 24 students, including 12 girls. In class B are 18 students, 9 girls. That is in class A are more girls (12 ) than B (9 ), if one considers the absolute frequency. Considering the frequency of girls but relative to the class size, you can see that in both classes is the same proportion of girls. In class A is the relative frequency in girls 0.5 (12/24 ) and class B also 0.5 (9 /18). The relative frequency can also be easily converted into a percentage by multiplying by 100%. Thus, both classes are 50 % (0.5 * 100) of girls.

Election poll

In an election poll 600 voters in Bavaria are interviewed, and 200 voters in Berlin. In Bavaria passed to 120 respondents to choose party A. In Berlin, 100 respondents say that they would vote for party A. The absolute frequency for voters of the party A is thus in Bavaria higher than in Berlin, namely 120 respondents in Bavaria over 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 allows a comparison with respect to the popularity of the party A between Bavaria and Berlin. In Bavaria, the relative frequency is 0.2 ( = 120/600 ). For Berlin is calculated as the relative frequency of 0.5 ( = 100 /200). Party A is in Berlin so much more popular than in Bavaria.

Properties

In contrast to the absolute frequency, the relative frequency is always moving between 0 and 1 This way you can compare different relative frequencies, although they refer to different reference. In the descriptive statistics relative frequencies are therefore used frequency distributions to (ie, independent of the sample size ) compare regardless of the number of elements in the population.

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

For the relative frequency following calculation rules apply:

  • Because of the normalization to the number of repetitions.
  • For the certain event.
  • For the sum of events.
  • For the complementary event.

Relative frequency and probability

Frequentist concept of probability

The frequentist concept of probability interprets the probability of an event as the relative frequency with which it occurs in a large number of identical, repeated, mutually independent random experiments. This is the so-called limit - definition ' according to von Mises. Prerequisite for this concept of probability is the random repeatability of the experiment; the individual passages must be independent of each other.

Example: You roll the dice 100 times, with the following distribution: 1 falls 10 times (this corresponds to a relative frequency of 10% ) falling 2 15 times ( 15%), the 3 is also 15 times ( 15%), 4 in the 20%, 5 to 30% and 6 to 10 % of cases. After 10,000 rounds, the relative frequencies have - if a fair die is present - in stabilized near the probabilities, so that eg the relative frequency for dicing a 3 is approximately 16.6 %.

The axiomatic definition of probability used today as the basis of probability theory does not require recourse to the concept of relative frequency. However, there is also using this definition of probability (using the law of large numbers) a close relationship between probability and relative frequency.

Law of large numbers

As laws of large numbers certain convergence theorems for random variables are called. In their simplest form, these rates imply that the relative frequency of a random result are generally the probability of this coincidence result approximates when the underlying random experiment is performed repeatedly. The laws of large numbers can be proved starting from Kolmogorov's axiomatic definition of probability. Thus, a close correlation between the relative frequency and probability exists even if you are not a representative of the objectivist conception probability.

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