Mode (statistics)

This article describes the measure of samples. For the mode of a probability distribution see mode (stochastics) , further meanings can be found under mode .

The mode , also called the modal value , is a position parameter in descriptive statistics . It is defined as the most common value that occurs in the sample. If, for example, exam grades are recorded for a school class, the mode corresponds to the grade or grades that were awarded most frequently.

In contrast to other position measures, the mode has the advantage that it always exists. However, it is generally ambiguous.

definition

Each characteristic expression that occurs most frequently in a sample is called a mode of the sample. A mode is therefore exactly one peak of the corresponding frequency distribution . The mode is a number that most of the data will match.

The most common notations for the mode are or . ${\displaystyle D}$ ${\displaystyle {\overline {x}}_{M}}$

Examples

Nominal scale

The sample is given

${\displaystyle ({\text{Zebra}},{\text{Elefant}},{\text{Giraffe}},{\text{Zebra}},{\text{Giraffe}},{\text{Antilope}})}$

The characteristic values and appear. Occurs once, as well as . Both and occur twice. Furthermore, there is no trait that occurs three times or more. So result as modes ${\displaystyle {\text{Zebra}},{\text{Elefant}},{\text{Giraffe}}}$ ${\displaystyle {\text{Antilope}}}$ ${\displaystyle {\text{Antilope}}}$ ${\displaystyle {\text{Elefant}}}$ ${\displaystyle {\text{Zebra}}}$ ${\displaystyle {\text{Giraffe}}}$

${\displaystyle D_{1}={\text{Zebra}}}$

and

${\displaystyle D_{2}={\text{Giraffe}}}$

Ordinal scale

During a class test, the grades were

${\displaystyle ({\text{befriedigend}},\;{\text{sehr gut}},\;{\text{befriedigend}},\;{\text{gut}},\;{\text{gut}},\;{\text{ausreichend}},\;{\text{mangelhaft}},\;{\text{ungenuegend}},\;{\text{gut}})}$

forgive. The grades and were each awarded once, the grade twice and the grade three times. No further grade was given four times, so the mode is ${\displaystyle {\text{sehr gut}},\;{\text{ausreichend}},\;{\text{mangelhaft}}}$ ${\displaystyle {\text{ungenuegend}}}$ ${\displaystyle {\text{befriedigend}}}$ ${\displaystyle {\text{gut}}}$

${\displaystyle D={\text{gut}}}$.

Cardinal scale

Looking at the sample

${\displaystyle (1,1,1,2,10,11,12,67,72)}$

So all values ​​except for each occur only once, but three times. So is the mode ${\displaystyle 1}$ ${\displaystyle 1}$

${\displaystyle D=1}$

Properties and comparison

Comparison between mode, median and "mean" (actually: expected value ) of two log-normal distributions

The mode is always defined, but in general not unambiguous either. The example under the nominal scale shows both : none of the common measures of location can be used in such a general framework, but two modes occur in this sample. The extreme case occurs when all characteristic values ​​in the sample are different from one another: Then each occurs only once and thus each is also a mode.

For samples with an order structure, the median can be defined in addition to the mode . The two do not have to match, in the example below the ordinal scale would be the median

${\displaystyle M={\text{befriedigend}}}$,

whereas the mode as

${\displaystyle D={\text{gut}}}$

was determined. If a cardinal scale is available, the arithmetic mean can also be determined. However, the mode, median and arithmetic mean can differ widely. The mode in the example under #Cardinal scale is as

${\displaystyle D=1}$

has been determined. For the median results

${\displaystyle m=10}$

and for the arithmetic mean

${\displaystyle {\overline {x}}=19{,}67}$.

Constructive terms

Frequency distributions with two or more modes are called multimodal distributions . Distributions with two modes are referred to as bimodal . Distributions with only one mode are called unimodal .

Characterization of the slope

In observation series with ordinal and metrically scaled features, the mode value can be referred to as the density mean. In comparison with the median and arithmetic mean, the mode can characterize the inclination of the distribution - similar to statistical skewness . For example, Karl Pearson's mode skewness is defined as

${\displaystyle {\frac {{\text{Arithmetisches Mittel}}-{\text{Modus}}}{\text{Standardabweichung}}}}$.

The following rule of thumb relates the mode, median and arithmetic mean:

• right-skewed (left-hand) frequency distribution: mode <median <arithmetic mean
• left-skewed (right-hand) frequency distribution: mode> median> arithmetic mean
• unimodal symmetrical frequency distribution: mode ≈ median ≈ arithmetic mean

Individual evidence

1. ^ Thomas Cleff: Descriptive Statistics and Exploratory Data Analysis . A computer-aided introduction with Excel, SPSS and STATA. 3rd, revised and expanded edition. Springer Gabler, Wiesbaden 2015, ISBN 978-3-8349-4747-5 , p. 37 , doi : 10.1007 / 978-3-8349-4748-2 .
2. ^ Karl Bosch: Elementary introduction to applied statistics . 8th edition. Vieweg, Wiesbaden 2005, p. 20 .
3. ^ Reinhold Kosfeld, Hans Friedrich Eckey, Matthias Türck: Descriptive statistics . Basics - methods - examples - tasks. 6th edition. Springer Gabler, Wiesbaden 2016, ISBN 978-3-658-13639-0 , p. 68 , doi : 10.1007 / 978-3-658-13640-6 .
4. ↑ Prof. Dr. Roland Schuhr: Statistics and probability calculation . Leipzig 2017, p. 37 , urn : nbn: de: bsz: 15-qucosa2-159363 .
5. ↑ Markus Wirtz, Christof Nachtigall: Descriptive Statistics - Statistical Methods for Psychologists . 5th edition. Juventa, 2008.
6. ^ Paul T. von Hippel: Mean, Median, and Skew: Correcting a Textbook Rule. Journal of Statistics Education Volume 13, Number 2, 2005.