Calculator
Formula used
For each class, use midpoint xᵢ = (Lᵢ + Uᵢ)/2.
Mean: x̄ = Σ(fᵢxᵢ) / Σfᵢ.
Population variance: σ² = Σ[fᵢ(xᵢ−x̄)²] / N.
Sample variance: s² = Σ[fᵢ(xᵢ−x̄)²] / (N−1).
Choose assumed mean A and width h.
Compute uᵢ = (xᵢ − A)/h.
Mean: x̄ = A + h·(Σfᵢuᵢ / N).
Variance: σ² = h²[(Σfᵢuᵢ²/N) − (Σfᵢuᵢ/N)²].
About mean: MAD = Σ[fᵢ|xᵢ−x̄|] / N.
About median (estimate): MDₘ = Σ[fᵢ|xᵢ−Median|] / N.
How to use this calculator
Paste each class and frequency on its own row.
Set rounding, deviation focus, and optional cross-checks.
Review results, then download a CSV or PDF.
Data structure and bin quality
Grouped datasets summarize many records into class intervals. Each row provides a lower bound, upper bound, and frequency. The calculator converts every class into a midpoint xi, then weights it by frequency fi. Uniform widths improve interpretability, but varying widths still work. If classes are open ended, choose credible midpoints from domain knowledge. When using midpoint mode, supply trusted midpoints and frequencies only. For interval mode, ensure boundaries increase and do not overlap, so cumulative frequency grows predictably for clean estimation.
Weighted mean as a reference center
The weighted mean xbar equals sum(fi*xi)/N, where N is total frequency. It represents the best single value center under squared error loss. A high frequency class near xbar will pull the mean toward that region. If xbar lies close to an edge, the distribution is likely skewed. Use the computation table to confirm that sum(f) equals N and sum(f*x) matches expectations.
Mean deviation for robust variability
Mean deviation uses absolute distances, sum(fi*abs(xi-center))/N, reported in the same unit as xi. Because it avoids squaring, it is less dominated by extreme classes and is often easier to explain. Choosing the mean center yields MAD, while selecting the estimated median yields median deviation. The median estimate uses the median class and cumulative frequency, then interpolates within the class width for a smoother center.
Variance and standard deviation for risk
Population variance equals sum(fi*(xi-xbar)^2)/N and standard deviation is the square root. Squared deviations emphasize large departures, useful for risk, reliability, and forecasting. The sample versions divide by N minus 1 to reduce bias when frequencies represent a sample. Compare CV percent equals 100*SD/abs(xbar) to benchmark dispersion across different scales. In operational dashboards, stable CV helps detect process drift even when levels change.
Computation table for validation and export
The computation table lists midpoint, frequency, cumulative frequency, f*x, f*x^2, and f*(xi-xbar)^2. These fields support rapid validation, sensitivity checks, and peer review. Step deviation cross check can be used with assumed mean A and class width h, improving hand calculation transparency. CSV export fits spreadsheets and pipelines, while the PDF summary captures inputs, rounding, and key metrics. This supports reproducible reporting and reduces rework.
FAQs
1) What data formats are accepted?
Use interval rows as lower-upper,frequency or lower,upper,frequency. Use midpoint rows as midpoint,frequency. Blank lines are ignored, and lines starting with # act as comments.
2) Why does the calculator use midpoints?
Grouped data hides raw values, so the midpoint approximates each class. Weighting midpoints by frequency produces practical estimates for mean and dispersion when individual observations are unavailable.
3) When should I use sample standard deviation?
Use the sample version when your grouped table represents a sample from a larger population. It divides by N minus 1 to reduce downward bias in the variance estimate.
4) How is the median based deviation handled?
The tool estimates the median from class intervals using cumulative frequency and linear interpolation inside the median class. It then computes average absolute deviations from that estimated median using midpoints.
5) What if class widths are not equal?
Direct midpoint calculations still work. Non uniform widths mainly affect simplified hand methods, so the step deviation cross check may be less tidy. Keep boundaries accurate to preserve meaningful midpoints.
6) What do the CSV and PDF downloads include?
Both exports include the summary metrics and the computation table fields used in the calculation. CSV is best for spreadsheets and models, while PDF is useful for sharing a fixed report.