Klassenmitten

Klassenmitten are the midpoints of class intervals in a frequency distribution of grouped data. They provide representative values for all observations within a class when the raw data are not available or impractical to use.

To compute a Klassenmitte, take the lower class limit L and the upper class limit U of a class interval and calculate m = (L + U) / 2. These midpoints are then combined with the corresponding frequencies f to analyze the data. A common use is to estimate the overall mean of a dataset by the formula x̄ ≈ (Σ f_i m_i) / (Σ f_i), where m_i are the class midpoints and f_i the class frequencies. Variance and other moments can be approximated similarly, for example using Σ f_i m_i^2 to derive second moments.

An example: class intervals 0–9, 10–19, 20–29 with midpoints 4.5, 14.5, 24.5 and frequencies 6, 12, 8 yield a mean ≈ (6×4.5 + 12×14.5 + 8×24.5) / (6+12+8) = 397 / 26 ≈ 15.3.

Klassenmitten are widely used in descriptive statistics for grouped data, in constructing histograms, and in quick data summarization when only grouped data are available. They rely on the assumption that values within each class are roughly evenly distributed around the midpoint, and the accuracy depends on the class width and placement. Open-ended classes may require assumptions about class width to assign a plausible midpoint.