Mode (statistics)

The mode or modal value is

• With an empirical frequency distribution of the most frequent value,
• At a discrete random variable the expression with the highest probability and
• At a continuous random variable, the maximum point of the density function.

Number

Does the statistical variable or random variable at least ordinal scale level, the frequency distribution, probability density function, or function can also have several separate maxima (modes or modal values ​​). Depending on the number of modes can be found a unimodal distribution ( only one maximum ) or bimodal bimodal ( two maxima exactly ) or multi-modal or multimodal distribution (more than two peaks ).

Scale - dependence

For the determination of the mode nominal scale level is sufficient, whereas, for example, for the median ordinal and for the calculation of the arithmetic mean an interval scale is a prerequisite.

Characterization of the inclination

In series of observations with ordinal and metric scaled features of the modal value can be referred to as density means. Compared with the median and the arithmetic mean, the mode, the slope of the distribution - similar to the statistical skewness - characterize. The Karl Pearson mode skewness is defined as, for example,

The following rule of thumb is mode, median and arithmetic mean in relationship:

• Skewed to the right (left parts ) frequency distribution: mode < median < mean
• Left -skewed ( rather steep ) frequency distribution: mode > median > arithmetic mean
• Symmetric unimodal frequency distribution: mode ≈ ≈ median arithmetic mean

Insights

In distributions which can be described by means of a monotonous function such as exponential function, it fails to specify the mode, since this provides no gain in knowledge with it.

Examples

• Example with a common expression:

• Example with several common forms: