Location parameters

Location parameters are calculated to determine the central tendency of data. In everyday language, we refer to these as average values.

Objective
  • Where are the majority of values?
  • Where is the "center" of a distribution?
  • Where is the "middle" of the observations?
  • What is a "typical" observation?
berg_glossar_zielsetzung

Procedure:

Determine the data type

The three Location parameter have different requirements for the data.
▪ The mode can be determined for any data type.
▪ The median requires at least one ordinal characteristic.
▪ The arithmetic mean requires the data to be metric (quantitative).

Assess the distribution shape

The median is a stable measure of central tendency in the presence of outliers or skewed distributions. In symmetric distributions, the mode, median, and mean are almost identical.

Calculate Location parameter

To calculate the corresponding location parameter, the data must be counted, summed, divided, or simply identified.
Mode (xD): The most frequent value of the characteristic is the mode.
Median (~x): The value in the middle of a sorted list is the median.
Mean (x¯): The sum of all values divided by the number of values gives the mean.

Example:
In a company with ten employees, the current salary distribution is shown in the following table. The Human Resources department wants to hire a new employee and will use statistics to determine the salary.

Mode:
The mode is the most frequent value. Four out of ten employees earn xD= 2.500 €. If there are two values that occur most frequently, it is a bimodal distribution.

NameSalary/Month
Mr. Anton2.300 €
Mr. Berta1.700 €
Mr. Cäsar2.500 €
Mr. Dora2.500 €
Mrs. Emil1.700 €
Mrs. Friedrich2.500 €
Mrs. Gustav2.300 €
Mrs. Heinrich2.300 €
Mrs. Ida2.500 €
(Mrs. Julius)(7.000 €)

Table 1: Salary table, sorted alphabetically by name

Median:
The median is the value in the middle of a sorted data set. Without Mrs. Julius’s salary, Mrs. Heinrich is the median with ~x = 2.300 €. With Mrs. Julius included, the median is the middle value, so:
~x = 2.400 € ( 2.500 2+ 2.300 ).

NameSalary/Month
Mr. Berta1.700 €
Mrs. Emil1.700 €
Mr. Anton2.300 €
Mrs. Gustav2.300 €
Mrs. Heinrich2.300 €
Mr. Cäsar2.500 €
Mr. Dora2.500 €
Mrs. Friedrich2.500 €
Mrs. Ida2.500 €
(Mrs. Julius)(7.000 €)

Table 2: Salary table, sorted in ascending order.

Arithmetic Mean:
The mean is the sum of all salaries divided by the number of salaries. With Mrs. Julius’s salary included, the mean increases significantly.
x¯1 = 2.300 + 1.700 + 2.500 + 2.500 + 1.7900 + 2.500 + 2.300 + 2.300 + 2.500 = 2.255,55 €
x¯2 = 2.300 + 1.700 + 2.500 + 1.700 + 2.500 + 2.500 + 1.700 + 2.500 + 2.300 + 2.300 + 2.500 + 7.000 = 2.730,00 €

The Human Resources Department decides to offer the new employee a median salary, as this is fairer.

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