Two Averages, Two Questions
A single-cell experiment may recover very different numbers of cells from different donors. If a measurement is summarized within each donor, there are then at least two natural ways to report an overall average: average the donor summaries equally, or pool the cells so that donors with more recovered cells contribute more. Neither calculation is automatically correct for every purpose. They answer different descriptive questions.
The key is to identify the unit that should receive equal weight in the target quantity. An equal-donor average describes the average of the donor-specific summaries. A pooled-cell average describes the average across the cells represented in the pooled data. When donors contribute unequal numbers of cells, these quantities generally differ. This is a question about the target of the summary, not a reason to treat cells as independent biological replicates.
These definitions assume that the donor-specific summaries refer to the same measurement, on the same scale, for the same kind of cell or other defined analysis unit. For example, comparing average expression in a particular cell type requires calculating each donor's mean over that cell type, not mixing cell types unintentionally. The formulas describe observed data; interpreting either average as representative of a broader population also depends on how donors and cells were sampled.
What Each Weighting Targets
In the equal-donor average, each donor contributes \(1/D\), regardless of cell yield. It summarizes the donor-specific averages for the donors included. If donors are sampled to represent a donor population, this may align with a question about the average donor in that population. Such an interpretation requires a suitable donor-sampling design and comparable donor summaries; equal weighting alone does not ensure representativeness.
In the pooled-cell average, donor \(i\) receives weight \(n_i/N\), where \(N=\sum_i n_i\) is the total number of analyzed cells. Thus, a donor contributing 100 cells receives ten times the weight of a donor contributing 10 cells. This average directly describes the cells in the pooled data, with each included cell receiving equal weight. It may also describe a broader cell population under an appropriate sampling design. It does not, just by being computed over many cells, become an average donor quantity.
Worked Example: Average Expression Across Donors or Cells
Suppose three hypothetical donors contribute cells of one defined cell type. The measured expression values are on the same scale for every donor. Donor A contributes \(n_1=10\) cells with mean expression \(\bar{Y}_1=2\) units; donor B contributes \(n_2=30\) cells with mean \(\bar{Y}_2=4\) units; donor C contributes \(n_3=10\) cells with mean expression \(\bar{Y}_3=10\) units. Calculate both averages.
The equal-donor average is
For the pooled-cell average, weight each donor mean by its cell count. The corresponding donor totals are \(10(2)=20\), \(30(4)=120\), and \(10(10)=100\) expression units across cells. Therefore,
The equal-donor estimate is the average of the three donor means; the cell-weighted estimate is the mean across the 50 observed cells. They differ because donor B, whose mean is below the equal-donor average, contributes the most cells. Neither number is a calculation error: the interpretation depends on whether the intended unit of equal weighting is donors or cells.
Why Unequal Yields Can Change the Result
The relationship between cell yield and donor-specific outcome determines the direction of the difference. If donors with higher cell yields tend to have higher values, the pooled-cell average tends to exceed the equal-donor average. If higher-yield donors tend to have lower values, the pooled-cell average tends to be lower. Equal yields, or particular patterns of yields and donor means, can make the two averages coincide even when donors differ.
This relationship can be stated exactly using the empirical covariance across the \(D\) donors. Define \(\bar{n}=\frac{1}{D}\sum_i n_i\), \(\bar{Y}_{D}=\frac{1}{D}\sum_i\bar{Y}_i\), and \(\operatorname{Cov}_{D}(n,\bar{Y})=\frac{1}{D}\sum_i(n_i-\bar{n})(\bar{Y}_i-\bar{Y}_{D})\). The subscript indicates that this covariance is calculated over the observed donors using divisor \(D\).
To verify the result, expand the covariance definition:
Since \(\sum_i n_i=D\bar{n}\), the pooled-cell average is
This is an algebraic identity for the observed summaries, not a causal claim. It does not establish that cell yield causes a donor's outcome, or explain why yields differ. It makes precise how the two weighting rules can produce different descriptive averages.
Worked Example: A Pooled Proportion Uses the Relevant Cell Counts
Consider a hypothetical gene-detection summary for a specified cell type. Three donors have the following numbers of detected cells out of analyzed cells: donor A has 2 out of 10, donor B has 18 out of 30, and donor C has 8 out of 10. Calculate the average donor proportion and the pooled proportion.
The donor-specific proportions are \(0.200\), \(0.600\), and \(0.800\). Giving the three donors equal weight gives
For the pooled-cell proportion, add the detected cells and divide by the total number of analyzed cells:
The equal-donor proportion is 53.33% per donor on average; the pooled proportion is 56.00% of the 50 analyzed cells. Here the pooled proportion is higher because the donor with proportion 0.600 contributes the largest number of cells. The denominator must match the stated question: for a gene-detection proportion within this cell type, use the analyzed cells of that type, not each donor's total number of cells across unrelated cell types.
A Second Calculation From Cell-Level Values
Worked Example: Checking the Pooled Mean Directly
Suppose three hypothetical donors have expression values, in arbitrary units, of \([1,3,5]\), \([7,9]\), and \([2,4,6,8,10]\), respectively, for the same gene in the same defined cell type. The donor cell counts are \(3\), \(2\), and \(5\). Calculate each donor mean before comparing the two overall averages.
The donor means are \((1+3+5)/3=3\), \((7+9)/2=8\), and \((2+4+6+8+10)/5=6\) units. The equal-donor average is
Pooling the ten cell values gives a total of \(9+16+30=55\) units, so
The two results are close but not identical. This example also checks the calculation from the cell-level observations: calculating the pooled mean directly gives the same answer as weighting the donor means by \(3\), \(2\), and \(5\).
Choosing and Reporting an Average
Choose the weighting rule by stating the target quantity before calculating. If the question concerns the average of donor-specific expression summaries, give each donor equal weight. If it concerns the average expression among the cells represented in the pooled sample, weight by the number of analyzed cells from each donor. State the target cell population, the measurement scale, and the weighting in the report.
| Question | Summary | What receives equal weight? |
|---|---|---|
| What is the average of the donor-specific means? | Donor-weighted average | Each donor |
| What is the average measurement across analyzed cells? | Cell-weighted average | Each analyzed cell |
| What is the pooled fraction of cells positive for a feature? | Total positive cells divided by total eligible cells | Each eligible cell |
For cell proportions, the denominator is particularly important. A pooled proportion is a weighted average of donor proportions, with weights given by the number of cells eligible for that proportion. If the outcome is positivity among a particular cell type, those eligible cells—not all cells recovered from each donor—supply the weights. The equal-donor alternative averages the donor-specific proportions and answers a different question.
Diagnostics, Limitations, and a Common Mistake
Before interpreting either summary, check that every donor mean refers to the same outcome definition and scale, and that the cell counts used as weights correspond to the cells included in those means. A mismatch—for example, weighting a cell-type-specific expression mean by a count of all cell types—does not implement the stated pooled-cell average. Confirm that the calculated pooled mean also equals total measured value divided by total analyzed cells, when the cell-level values are available.
A useful descriptive check is to compare the donor means and their cell counts. If the averages are far apart, inspect which donor means receive the greatest weight in the pooled calculation. A zero-cell donor has no observed cell mean for that analysis and cannot enter the pooled-cell average; its absence should not be silently interpreted as a measured value of zero. Also consider whether the measured scale is meaningful for the intended summary: averaging values after a transformation generally targets a different quantity from averaging them before the transformation.
Common mistake: “The pooled average uses thousands of cells, so it is automatically the best estimate of the average donor and has thousands of independent replicates.” This reasoning confuses the target of an average with the unit of biological replication. The pooled average weights donors according to their analyzed cell yields, so it need not estimate an equal-donor quantity. Moreover, cells from one donor are not independent biological replicates. The number of cells does not replace the number of donors when the intended inference is about donors.
The cell-weighted average is not inherently invalid, and the equal-donor average is not universally preferable. Each is limited by the data and design: the former describes a cell-yield-weighted collection, while the latter describes an equally weighted set of donor summaries. Neither alone explains why cell yields differ or resolves how much donors should contribute to a broader target population. Report the estimand and weighting explicitly rather than presenting an unlabeled “overall mean.”
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two donors have means of 2 and 8 units, based on 5 and 15 cells, respectively. Calculate the equal-donor average and the pooled-cell average.
- For the data in question 1, which average describes the mean across the 20 observed cells, and why?
- Three donors have cell-type-specific proportions of 0.10, 0.50, and 0.90, based on 10, 20, and 10 eligible cells. Calculate both the equal-donor and pooled proportions.
- Does a pooled-cell average based on many cells establish that there are many independent biological replicates? Explain briefly.
Solutions. 1. The equal-donor average is \((2+8)/2=5.000\) units. The pooled-cell average is \([5(2)+15(8)]/(5+15)=130/20=6.500\) units. 2. The pooled-cell average, 6.500 units, gives each of the 20 observed cells equal weight; the equal-donor average gives each donor mean equal weight instead. 3. The equal-donor proportion is \((0.10+0.50+0.90)/3=0.500\). The pooled proportion is \([10(0.10)+20(0.50)+10(0.90)]/40=20/40=0.500\). They coincide because the middle donor's proportion and cell count balance the two other donors in this example. 4. No. The cells are clustered within donors; a large cell count does not create additional independent biological replicates.