Mean Particle Size Calculator for Range Bins

How to Calculate Mean Particle Size for Range Bins

When particle size data is reported in bins (for example, 0–10 µm, 10–20 µm, 20–40 µm), you do not have every individual particle measurement. Instead, you have grouped ranges with associated weights such as mass fraction, volume fraction, or count fraction. The standard way to estimate mean particle size from this type of grouped data is to assign a representative size to each bin and then compute a weighted mean across bins.

This page provides a practical method used in laboratories and process environments: the midpoint-weighted mean. It is fast, transparent, and easy to audit. For most routine reporting, it gives a robust estimate of average size when full raw data is unavailable.

Core Concept

Each particle size bin has two boundaries and one weight:

• Lower boundary: L_i
• Upper boundary: U_i
• Weight: w_i (mass %, count %, volume %, or fraction)

Choose a midpoint m_i for each bin, then calculate:

If bins are linear and narrow, arithmetic midpoint is common:

If bins are logarithmic (common in PSD work), geometric midpoint is often better:

Step-by-Step Procedure

• List all size bins and weights in a table.
• Pick midpoint method: arithmetic for linear bins, geometric for log-spaced bins.
• Compute midpoint for each bin.
• Multiply each midpoint by its bin weight.
• Sum all midpoint × weight products.
• Divide by total weight.

The result is your estimated mean particle size in the same unit as bin boundaries (µm, mm, nm, etc.).

Worked Example

Suppose your sieve or laser diffraction report provides:

• 0–10 µm: 15%
• 10–20 µm: 35%
• 20–40 µm: 30%
• 40–80 µm: 20%

Using arithmetic midpoints (5, 15, 30, 60):

So the estimated mean particle size is 27 µm.

Choosing the Right Midpoint Method

Midpoint choice influences the mean, especially for broad bins. Use arithmetic midpoint when bins are equally spaced in linear units. Use geometric midpoint when bins are spaced by ratio (for example, each upper limit is roughly double the lower limit). In many particle sizing contexts, especially with wide logarithmic bins, geometric midpoint better represents central tendency within each class.

Mass-Weighted vs Count-Weighted Mean

The computed mean depends on what your weights represent:

• Mass-weighted mean: influenced by larger particles.
• Volume-weighted mean: often similar to mass-weighting for constant density.
• Number-weighted mean: influenced by finer particles due to high counts.

Always label your result clearly, for example: “mass-weighted mean particle size from binned data.”

Common Mistakes to Avoid

• Mixing units (µm and mm in the same table).
• Using cumulative percentages as if they were bin percentages.
• Ignoring open-ended bins without defining a representative limit.
• Using arithmetic midpoint on strongly logarithmic bins without checking bias.
• Forgetting to verify whether percentages are retained, passing, or differential values.

How to Handle Open-Ended Bins

Some datasets include bins like “<5 µm” or “>150 µm.” To compute a midpoint-based mean, you need finite lower and upper values. Practical options include:

• Set a technical bound based on instrument range.
• Use process-relevant cutoffs from product specification.
• Run sensitivity checks with multiple plausible bounds to estimate uncertainty.

Document assumptions whenever open-ended bins are present.

Quality Control and Reporting Tips

• Keep at least 3 significant figures during calculation, round at the end.
• Report the midpoint method used (arithmetic or geometric).
• Include the weighting basis (mass, volume, count).
• Archive the original binned distribution for traceability.
• If possible, compare with D10, D50, D90 metrics for fuller interpretation.

Why This Method Is Useful

In production and research environments, raw particle-by-particle data is often unavailable or unnecessary for routine decisions. A binned weighted mean is fast, interpretable, and sufficient for trend tracking, lot release checks, and process adjustments. It is especially effective when used consistently with a fixed binning protocol.

FAQ: Mean Particle Size from Range Bins

Use this calculator as a practical estimator for grouped PSD data. For high-stakes modeling or regulatory analysis, complement midpoint methods with raw distribution analysis and instrument-specific validation.