---
title: Count by Number or by Volume
url: https://oosioo.com/en/p/%EA%B0%9C%EC%88%98%EB%A1%9C-%EC%84%B8%EB%8A%90%EB%83%90-%EB%B6%80%ED%94%BC%EB%A1%9C-%EC%84%B8%EB%8A%90%EB%83%90
date: 2026-10-13T12:25:34+00:00
author: SYSOP
summary: What is the vertical axis of a particle-size graph a percentage of? The same ground coffee peaks at 13 µm if you count by particle number, but past 800 µm if you count by weight. Using the actual 1,578 particles from a public example photo and a model that adds back in the fines, I worked out how much the graph changes depending purely on how you count — and why a camera-measured count graph ends up looking more like the camera than the coffee.
---
# Count by Number or by Volume

In the previous pieces, I'd been assigning a size to each individual particle: flattening the photo, separating particles from the background, deciding how to define diameter. Now it's time to gather those thousands of numbers onto a single graph.

A particle-size distribution graph usually looks like this. The horizontal axis is particle size, the vertical axis is percentage. But percentage of what? Depending on whether it's a percentage of particle count or a percentage of powder weight, the same powder produces a completely different graph. That one word — tucked in small print in the graph's title, or missing entirely — is what this piece is going to examine.

## Same Powder, Different Peak

A 2016 paper in *Scientific Reports* shows this difference clearly. Uman, Hendon, and colleagues ground four types of beans on a Mahlkönig EK43 and measured particle size with a laser diffraction instrument. They then plotted the same measurement results two different ways: a distribution counted by particle number, and one counted by particle surface area.

![A three-panel graph. The top panel shows particle count distribution (light blue) and cumulative curve (gray) over a 0–100 µm horizontal axis, with a purple dashed line at 70 µm and an arrow labeled 1% to its right. The bottom two panels use a 1–1000 µm log-scale horizontal axis; the solid line (count) has a single peak at 13–14 µm, while the dashed line (surface area) has two peaks, at 27 µm and at 257–282 µm](/uploads/245ce8c99d7556e1.webp =720)

*Top: counted by particle number, 99% of particles are 70 µm or smaller. Bottom two panels: count distribution (solid) and surface-area distribution (dashed) for four bean types. Image: Uman et al. 2016, Scientific Reports 6:24483, Figure 3 (CC BY 4.0)*

The solid line, counted by number, has a single peak, with a mode of 13–14 µm. The dashed line, counted by surface area, has two peaks — one at 27 µm, the other at 257–282 µm. Looking at the top panel, 99% of particles by count are 70 µm or smaller. The coarse particles barely register by count — under 1% — but by surface area they're large enough to form a second peak. Had it been counted by weight, that coarse-side peak would have been even higher.

All three graphs are the same powder, the same measurement. The measurement isn't wrong — only the counting unit differs.

## One Coarse Particle Equals 24,000 Fines

The reason the difference is this large is simple: weight scales with the cube of diameter.

![On the left, a large brown circle (one coarse particle, 870 µm); on the right, a very small dot (one fine particle, 30 µm). To the right, in red text: "By volume (weight), equivalent to 24,389 fines," and below it in gray: "(870 ÷ 30)³ = 29³"](/uploads/a46a669539b8d653.webp =720)

A particle 870 µm in diameter is 29 times larger than a fine particle 30 µm in diameter. Surface area is 29 squared — 841 times larger — and volume is 29 cubed — 24,389 times larger. If we assume the density of a coffee fragment doesn't depend on its size, then volume is weight. Filling the weight of a single coarse particle with fines requires more than 24,000 of them.

So when you count by particle number, countless fines dominate the graph. When you weigh, a handful of coarse particles dominate the graph. Surface area sits in between.

There's even a standard notation for this distinction. The international standard for representing particle size results (ISO 9276-1) writes the number-based distribution as q0, the length-based one as q1, the area-based one as q2, and the volume-based one as q3. The subscript indicates which power of diameter was used as the weighting. If that number is missing when you look at a distribution graph, you're only reading half of it.

## Counting from an Actual Photo

I checked this directly using Gagné's example photo from the previous piece. Just as before, I flattened the background, applied a threshold cut, removed merged clumps, and used the remaining 1,578 particles. Diameter is the equivalent circular diameter. I left these 1,578 particles untouched and changed only what was being counted.

![A cumulative distribution graph with four curves, horizontal axis equivalent circular diameter (300–2000 µm, log scale), vertical axis the fraction (%) smaller than that size. The blue solid line (count) sits furthest left, with a median of 572 µm. The green solid line (projected area) is at 797 µm, the orange dashed line (available mass, the coffeegrindsize default) at 817 µm, and the red solid line (volume = mass) at 867 µm, clustered toward the right in that order](/uploads/e1d7f592a2c68612.webp =760)

| What's being counted | Median | D10–D90 |
|---|---|---|
| Count | 572 µm | 361–946 µm |
| Projected area | 797 µm | 446–1,126 µm |
| Available mass (coffeegrindsize default) | 817 µm | 475–1,148 µm |
| Volume (= mass) | 867 µm | 533–1,224 µm |

The same 1,578 particles, yet the median went from 572 µm to 867 µm — a factor of 1.52. That's a similar magnitude to the gap (1.53×) that came purely from changing the diameter definition in the previous piece. Change both the diameter definition and the weighting, and the gap grows larger still.

A weight-based median of 867 µm means this: particles larger than that make up half the powder's weight. I located and colored those particles in the photo.

![A zoomed-in crop of Gagné's example photo. Coffee particles scattered on cream-colored paper; the large particles are colored red, and the remaining particles 330 µm and above are colored blue. Many tiny uncolored black dots are also visible. A 5 mm scale bar is in the lower left](/uploads/65145bd07ed6040d.webp =760)

*Red: 869 µm and above, 16% of all particles (252 of them), making up half the weight. Blue: 330–869 µm. Uncolored dots: particles excluded from measurement for being too small or merged together. Original photo: Jonathan Gagné, coffeegrindsize (MIT license)*

The 252 largest particles — 16% of the total — hold half the weight. Conversely, even combining all of the smaller half (789 particles), they amount to only 13% of the weight. The middle of the count graph is, by weight, nearly empty.

### What Does coffeegrindsize Plot by Default?

Gagné's coffeegrindsize can plot twelve different graphs — count, surface area, and mass basis, each plotted against diameter, surface area, or volume axes. Looking at the code, the graph selected by default when the program opens is the eighth one: "Available mass vs Surface." It treats each particle as a sphere of equal volume and assumes water only reaches a depth of 0.1 mm from the surface, counting only the volume of that shell. Particles with a radius smaller than 0.1 mm are counted in full; larger particles only have their shell counted.

This bakes in the idea that the core, which water never reaches, doesn't contribute to flavor. That's why, in the table above, the median for available mass (817 µm) falls between volume (867 µm) and area (797 µm). The coarser the particle, the more core gets excluded, pulling it slightly left of the volume figure. The 0.1 mm depth is a value fixed in the code, and the code comments themselves note that it "could be done better analytically."

### Can the Weight Graph Be Trusted Without Knowing Thickness?

The previous piece noted that thickness doesn't show up in a photograph. For a particle whose thickness is half its width, the volume computed from the equivalent circular diameter comes out 2.4 times the true value. Does that make the weight graph useless?

Not necessarily. The weight graph is plotted as each particle's volume divided by total volume. If every particle's thickness is the same proportion of its width, then every volume is off by the same factor, and that factor cancels out in the division. Indeed, recalculating with thickness ÷ width set to 0.3, 0.5, and 1.0, the weight-based median came out to the same 867 µm in all three cases. The powder's total weight would be wrong, but the shape of the graph would be correct.

The problem arises when the thickness ratio varies by size. Assuming larger particles are flatter (thickness = width at 330 µm, thickness = 0.4× width at 1,500 µm) and recalculating, the median dropped from 867 µm to 822 µm, a 5% decrease. Giving less weight credit to the large particles pulled the graph to the left. Whether coffee particle thickness actually varies with size this way can't be determined from this photo alone.

## Adding In the Fines the Photo Doesn't Show

Up to this point, only particles of 330 µm or larger — the ones captured in the photo — were counted. But ground coffee contains fines much smaller than that. Since coffee bean cell size is 25–50 µm, fragments broken along cell-wall boundaries form a separate peak around 30–40 µm. Mo, Ellero, and colleagues measured four types of ground coffee with laser diffraction in a 2023 paper and reported that particles under 100 µm made up 9.7% to 29.2% of volume.

I appended these fines to the photo data to build a synthetic powder. The coarse peak is the photo's actual particle distribution extended down below 330 µm. The fine peak was set to sit at 35 µm by volume. Its spread was tuned so that, counted by number, that fine peak's mode comes out to 13 µm — the count-based mode Uman and colleagues measured. The fines were set at 20% by weight, roughly the midpoint of the range reported in the literature.

![A three-panel graph. Horizontal axis is particle diameter (1–4000 µm, log scale), with light shading to the right of 330 µm labeled "range measured by photo (Gagné example)." The top panel, "counted by number," has a single peak at 13 µm and a median of 13 µm. The middle panel, "counted by surface area," has a large peak around 25 µm and a small peak around 1000 µm, with a median of 28 µm. The bottom panel, "counted by volume (mass)," has a small peak at 35 µm and a large peak at 970 µm, with a median of 832 µm](/uploads/a2f89573c97dba15.webp =760)

The same powder has a median of 13 µm by count, 28 µm by surface area, and 832 µm by weight. The count-based and weight-based medians differ by a factor of 64. The count graph shows no trace of coarse particles at all, while in the weight graph the fines remain only as a small bump.

![Two horizontal bar charts. On the left, "share of particles under 100 µm (fines)": 99.97% by count, 87.2% by surface area, 19.3% by volume. On the right, "share of particles 330 µm and above (the range the photo captured)": 0.009% by count, 11.6% by surface area, 79.1% by volume](/uploads/9dd66d02a72b8b66.webp =720)

By particle count, fines make up 99.97%. By surface area, 87%. By weight, 19%. If this model is correct, the particles of 330 µm or larger that Gagné's photo measured amount to roughly 1 in 10,000 by count, but 79% by weight. The photo missed most of the particles, but captured most of the weight.

## The Count Graph Resembles the Camera

This result reveals something important for measurement: a count-based distribution shifts entirely depending on the smallest size the instrument can detect.

Suppose we photograph this synthetic powder with a camera. As we vary the smallest particle size the camera can resolve, the count-based median moves as follows.

| Smallest size camera resolves | Count-based median | Weight-based median |
|---|---|---|
| 1 µm (everything visible) | 13 µm | 832 µm |
| 10 µm | 17 µm | 834 µm |
| 30 µm | 36 µm | 884 µm |
| 100 µm | 126 µm | 963 µm |
| 330 µm | 555 µm | 975 µm |

The count-based median jumped from 13 µm to 555 µm — a 43-fold increase. Over the same range, the weight-based median moved by 17%. That's because a count-based distribution always clusters around the smallest thing that's visible. Change the camera, and the graph changes — and that change has nothing to do with the coffee.

This can also be confirmed in the actual photo. Keeping only particles 500 µm or larger out of the 1,578, the count-based median rises 28%, from 572 µm to 734 µm. The weight-based median rises only 4%, from 867 µm to 902 µm. Simply shooting the photo from a bit farther away, losing a few small particles, moves the count graph substantially while leaving the weight graph nearly unchanged.

So if photo-based measurement results are shown only as a count graph, the reader ends up seeing the camera's resolution more than the coffee. It's consistent with this reasoning that coffeegrindsize chose available mass, not count, as its default graph.

## Different Instruments Count in Different Units

It's also worth knowing that different measurement methods start out counting in different units entirely.

- **Sieves** weigh the powder retained on each screen. They are weight-based from the start.
- **Laser diffraction** works backward from the pattern of scattered light to find the combined volume of spheres that would produce the same pattern. The international standard ISO 13320's default result is also a volume-based distribution. The count distribution a laser instrument displays is usually this volume result converted to a count basis.
- **Photo analysis** counts particles one by one. Its default is count, and a weight graph only emerges after assuming a thickness and constructing a volume.

It's not strange at all for a volume graph from a laser instrument and a count graph from photo analysis to have peaks dozens of times apart in position — even when both measurements are correct.

Average diameter suffers the same issue. There are several ways to compute an average, and each hides a different weighting.

| Mean | Definition | Actual photo, 1,578 particles |
|---|---|---|
| D[1,0] number mean | Σd ÷ count | 622 µm |
| D[3,2] Sauter mean | Σd³ ÷ Σd² | 800 µm |
| D[4,3] volume mean | Σd⁴ ÷ Σd³ | 887 µm |

D[3,2] is the diameter of a sphere whose volume-to-surface-area ratio matches that of the whole powder. It's frequently used in extraction or permeation calculations, where the area in contact with water matters. In Mo and colleagues' paper, the D[3,2] for ground coffee containing fines was 81–214 µm. Compared to this photo's 800 µm, that's a difference of roughly four to ten times. Computing D[3,2] from a photo that can't see fines produces this large an error — because the fines, which hold nearly all the surface area, are missing.

## Which Counting Method Is Closer to Flavor?

So which graph should you actually look at? It depends on the question.

Extraction is the process by which water dissolves compounds out of the powder's surface. The rate of dissolution depends on surface area; the total amount of dissolvable material depends on weight. In the model above, fines are 19% by weight but 87% by surface area — meaning they dissolve out far faster relative to their weight. Conversely, coarse particles hold most of the weight, but water has a hard time reaching their interior. coffeegrindsize's choice of available mass as its default graph is an attempt to capture both of these in a single graph.

In the end, the count graph is useful for assessing instrument performance, the weight graph for seeing how the powder's mass is distributed, and the surface-area graph for seeing what will dissolve out quickly. Which weighting best predicts actual flavor differences is something no settled research has established yet.

## Try It Yourself

In the simulation below, you can count the same powder four different ways. In the "model with fines" mode, you can adjust the weight fraction of fines, the position of both peaks, and the smallest size the camera can resolve. The graph on the left shows the distribution for the chosen counting method: the solid line is the overall median, the dashed line is the median computed only from particles the camera can see. The graph on the right shows what fraction the fines and the camera-visible portion make up under each counting method. Clicking "actual photo, 1,578 particles" switches to the real particles from Gagné's photo, showing a cumulative weight curve built up from the largest particles.

Drag the camera's minimum size from 10 µm up to 330 µm while toggling between "count" and "volume," and you'll immediately see which dashed line moves the most.

```map /embed/coffee-weights-2026-en.html 16:12
A simulation for counting the same ground coffee by number, surface area, volume, and available mass. The left side shows the particle size distribution (horizontal axis 1–4000 µm, log scale) for the chosen counting method, with the portion the camera can't see shaded, and both the overall median (solid line) and the median of only camera-visible particles (dashed line) plotted. The right side is a bar chart showing what percentage the fines under 100 µm and the camera-visible portion make up under count, surface area, volume, and available mass bases. You can adjust the fines' volume fraction, the fine peak's position, the coarse peak's position, and the smallest size the camera can see. In actual-photo mode, it shows the histogram of the 1,578 particles from Gagné's example photo and a cumulative volume curve built up from the largest particles.
```

## What This Model Leaves Out

- **The fines were not actually measured.** The fine peak is a model fit to figures from the literature (volume-mode 30–40 µm, count-mode 13 µm, under-100-µm volume share 9.7–29.2%), not something measured from a photo. Changing the fines fraction to 10% or 30% changes the share figures. You can adjust this directly in the simulation
- **Thickness is still an assumption.** The weight graph's shape is correct if the thickness ratio doesn't depend on size, but shifts by a few percent if it varies by size
- **Density was assumed equal.** In converting volume to weight, I assumed coarse particles and fines have the same density. Coarse particles contain empty cellular space while fines are cell-wall fragments, so the actual densities may differ
- **The 0.1 mm for available mass is a fixed value.** I used the value as-is from the coffeegrindsize code. The actual depth water penetrates would depend on extraction time and water temperature
- **This is one photo, one grind.** The shape of the coarse peak would differ for a different bean or a different grinder

## Summary

- The vertical axis of a particle-size graph is a percentage of one of three things: particle count, surface area, or weight. For the same powder, the peak position can shift by dozens of times depending on which one is used
- Weight scales with the cube of diameter. A single 870 µm particle weighs as much as 24,000 fines of 30 µm
- Counting the 1,578 particles from Gagné's example photo gave a median of 572 µm by count and 867 µm by weight. The largest 16% held half the weight
- coffeegrindsize's default graph is not count but "available mass" — counting only the outer 0.1 mm shell
- The shape of the weight graph is correct even without knowing thickness, as long as every particle's thickness ratio is the same
- In the model that includes fines, fines were 99.97% by count, 87% by surface area, and 19% by weight. The particles of 330 µm or larger that the photo captured were 0.009% by count but 79% by weight
- A count graph is highly sensitive to the smallest size the instrument can detect. Raising the camera's limit from 1 µm to 330 µm moved the count-based median by 43-fold, and the weight-based median by 17%
- Sieves count by weight, lasers count by volume, photos count by number. Before comparing graphs from different methods, you first need to match up what they're each counting

Once you're holding a particle-size graph, there's now one more question to ask beyond how diameter was defined: a percentage of what?

## References

- Uman E, Colonna-Dashwood M, Colonna-Dashwood L, Perger M, Klatt C, Leighton S, Miller B, Butler KT, Melot BC, Speirs RW, Hendon CH (2016). The effect of bean origin and temperature on grinding roasted coffee. *Scientific Reports* 6, 24483. [doi:10.1038/srep24483](https://doi.org/10.1038/srep24483) — EK43, laser diffraction (Beckman Coulter LS13 320 MW), count mode 13.0/14.3 µm, surface-area peaks at 27.4 µm and 256.9/282.1 µm, 99% of particles 70 µm or smaller. Some authors affiliated with the grinder manufacturer and the measurement instrument maker
- Mo C, Johnston R, Navarini L, Liverani F, Ellero M (2023). Exploring the link between coffee matrix microstructure and flow properties using combined X-ray microtomography and smoothed particle hydrodynamics simulations. *Scientific Reports* 13, 16374. [doi:10.1038/s41598-023-42380-y](https://doi.org/10.1038/s41598-023-42380-y) — fine peak at 30–40 µm (cell size 25–50 µm), under-100-µm volume fraction 9.7–29.2%, D[3,2] 81–214 µm
- Gagné J. *coffeegrindsize*. GitHub. [link](https://github.com/jgagneastro/coffeegrindsize) — 12 graph types, default is "Available mass vs Surface" (8th in the list), 0.1 mm penetration depth confirmed in code
- ISO 13320:2020. Particle size analysis — Laser diffraction methods — volume-based distribution
- ISO 9276-1:1998. Representation of results of particle size analysis — Part 1: Graphical representation — q0–q3 notation
- Microtrac. *Coffee powder* (application note, instrument manufacturer material). [link](https://www.microtrac.com/applications/coffee-powder/) — differences in basis between sieve, laser, and image analysis

## Image Credits

- Coffee particle distribution by count and surface area: Uman et al. 2016, *Scientific Reports* 6:24483, Figure 3. [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/) (reduced, unmodified)
- Coffee particle photo: Jonathan Gagné, [coffeegrindsize](https://github.com/jgagneastro/coffeegrindsize) Help/Decent_Example_Picture.png, MIT license. Particle coloring and cumulative distribution graph were created by analyzing this photo directly
- All other figures and the simulation: created directly