Why a Mean Depth Does Not Describe Every Position
Sequencing produces a collection of short reads, and those reads do not necessarily cover every position equally. Some positions receive many reads; others receive few or none. This variation matters because evidence at a position is based on the reads that actually cover it, not on the average across an entire target. In particular, low depth means that the observed reads may provide a limited and variable sample of the alleles present.
Three related quantities help describe coverage. Depth at a position is a count of reads covering that position. Average depth summarizes the total number of aligned bases relative to the length of the target. Coverage breadth is the fraction of target positions that meet a specified minimum depth. These quantities answer different questions: a large average depth does not guarantee that every position has substantial depth.
If \(B\) aligned read bases fall within a target of length \(G\), then average depth is \(B/G\), conventionally reported with an “x” suffix. For example, 20x average depth means that the total aligned bases equal 20 times the target length. It does not mean that every position has exactly 20 reads. The ratio is a useful summary, but it contains no direct information about the spread of the individual \(D_i\) values.
Worked Example: Calculating Average Depth
Suppose a hypothetical sequencing run yields 1,200,000 reads of 150 bases each. Assume every base aligns within a target of 6,000,000 bases, with no bases lost or counted more than once. Calculate average depth.
The total number of aligned bases is \(1{,}200{,}000\times150=180{,}000{,}000\) bases. Therefore,
This is a target-wide average. It does not establish that all 6,000,000 positions have depth 30. Some positions could have greater depth and others less, while the average remains 30x. The calculation also depends on the stated assumptions: if some reads do not align to the target, the aligned-base total within the target is smaller and so is the target's average depth.
A Simple Model for Random Coverage
To isolate sampling variation, consider a simplified uniform-coverage model. Suppose there are \(N\) independent read-base placements, each equally likely to land at any one of \(G\) positions. This ignores some features of real reads, including the fact that a read covers neighboring positions together. It is a useful starting model for understanding why depth varies even when no position is favored.
At any fixed position, each placement either covers that position or does not. The probability of coverage from one placement is \(1/G\), so the depth has a binomial distribution with \(N\) trials and success probability \(1/G\). Its mean is \(N/G\). When \(G\) and \(N\) are large, with \(N/G\) held near a fixed value \(\lambda\), the binomial distribution is well approximated by a Poisson distribution with mean \(\lambda\).
The standard deviation explains why a fixed average does not imply fixed depth. Under this model, the coefficient of variation is \(\sqrt{\lambda}/\lambda=1/\sqrt{\lambda}\). Thus, the relative spread is larger at lower expected depth. Increasing the expected depth reduces relative sampling variation, but it does not make the coverage perfectly uniform.
From Depth to Coverage Breadth
A position is uncovered exactly when its depth is zero. For a Poisson variable with mean \(\lambda\), the probability of zero reads is \(P(D_i=0)=e^{-\lambda}\). Consequently, the probability of at least one read is \(1-e^{-\lambda}\). If positions have the same coverage probability, this is also the expected fraction of positions covered at least once. For a threshold above 1, breadth is found by adding the probabilities of all depths meeting that threshold.
Worked Example: Expected Breadth at Fourfold Depth
Suppose a hypothetical target has 10,000 positions, and use a uniform Poisson model with mean depth \(\lambda=4\). Estimate the expected number and fraction of positions covered by at least one read.
The probability that a particular position has zero reads is \(P(D_i=0)=e^{-4}\approx0.01832\). Thus, the probability that it has at least one read is
The expected breadth is approximately \(0.9817\), or 98.17%. For 10,000 positions, the expected number covered at least once is \(10{,}000\times0.9817\approx9{,}817\) positions. The remaining approximately 183 positions are expected to have zero reads. This is an expectation under the model, not a guarantee that exactly 183 positions will be uncovered in a particular run.
The mean depth is 4x, but the depth at a specific position can be 0, 1, 2, or substantially more. The breadth calculation makes that distinction explicit: it measures how many positions meet a coverage threshold, not the average number of reads.
Why Random Allocation Creates Unevenness
The model also gives a direct justification for uneven depth. If the number of placements at a position is approximately Poisson with mean \(\lambda\), its variance is \(\lambda\), not zero. Therefore, even equal chances of coverage at every position produce a distribution of depths rather than one common depth. At \(\lambda=4\), for example, the standard deviation is \(\sqrt{4}=2\) reads, a substantial spread relative to the mean.
The Poisson model is a baseline, not a complete description of every sequencing data set. Actual reads span adjacent positions, so depths at neighboring positions are not independent. Real data can also depart from the equal-probability assumption, producing more or less variation than this model predicts. A measured depth profile should therefore be examined rather than inferred from average depth alone.
Low Depth and Sampling of Alleles
Depth matters because the reads at a position form a finite sample of the molecules being sequenced. To illustrate, suppose a diploid individual is heterozygous at a position and each read independently samples either allele with probability \(1/2\). Assume for this calculation that reads are unbiased and error-free. If depth is \(d\), the number \(A\) of reads carrying one specified allele then follows \(A\sim\operatorname{Binomial}(d,1/2)\). This model describes sampling variability in the allele counts; it does not by itself determine whether a variant should be called.
Worked Example: Alternate-Allele Evidence at Low and Higher Depth
Consider a hypothetical heterozygous site under the unbiased, independent, error-free sampling assumptions above. Compare the probability of observing at least two reads carrying the alternate allele at depth 4 and depth 12.
At depth 4, \(A\sim\operatorname{Binomial}(4,1/2)\). The probability of at least two alternate reads is
At this depth, the probability of observing no alternate reads is also \(P(A=0)=(1/2)^4=1/16=0.0625\). Even with a true heterozygous genotype and unbiased sampling, a small read count can produce an allele count that does not closely reflect the underlying one-half fraction.
At depth 12, the probability of fewer than two alternate reads is the probability of zero or one: \(P(A=0)+P(A=1)=1/4096+12/4096=13/4096\). Hence
Under these assumptions, at least two alternate reads are much more likely at depth 12 than at depth 4. This comparison illustrates why a low-depth observation can be weak or unstable allele evidence. It is not a universal calling rule: read errors, mapping, and other data features are not represented by this simple binomial calculation.
How to Check Whether the Summary Is Adequate
A useful coverage summary reports more than the target-wide mean. Inspect the distribution of per-position depth, the fraction of positions at depth zero, and the fractions meeting relevant minimum-depth thresholds. A histogram or depth quantiles can show whether a small number of high-depth positions are lifting the mean while many positions remain shallow. For local questions, summaries by position or by genomic window can reveal regions that are poorly represented.
Compare observed breadth with the Poisson expectation only as a diagnostic baseline. A large discrepancy signals that equal-probability random coverage is not capturing the observed pattern; it does not, by itself, identify the cause. The simple model can also be a poor approximation when reads are dependent, the target is small, or local coverage probabilities differ. In those settings, empirical counts from the reads provide the direct description of observed coverage.
| Summary | Question answered | What it cannot establish alone |
|---|---|---|
| Average depth | How many aligned bases per target position on average? | Whether each position has adequate depth |
| Breadth at depth 1 | What fraction of positions has any read? | Whether covered positions have enough reads for a particular inference |
| Breadth at a chosen threshold | What fraction meets that minimum depth? | Why the other positions fall below it |
| Per-position depth distribution | How variable is coverage across positions? | Whether the reads are error-free or biologically representative |
Common Pitfall and Practical Interpretation
Wrong reasoning: “The sample has 30x average depth, so every position was read about 30 times and any variant should have strong support.” This confuses an average with a per-position guarantee and treats depth as if it established the accuracy of each read.
Correction: Average depth is the total aligned bases divided by target length. To assess coverage at a locus, use its observed depth; to describe target-wide completeness, report breadth at a stated threshold. Even at a position with several reads, allele counts vary because the reads are a sample. Depth improves the amount of sampling information but does not alone establish that the reads are correct or unbiased.
A deterministic calculation of average depth is useful for planning and concise reporting. The Poisson model adds a simple account of random allocation and expected breadth. Direct empirical coverage summaries are preferable when the goal is to describe a particular data set, because they do not require every position to share one coverage probability. Each approach has limits: a mean hides spread, a simple probability model relies on assumptions, and observed coverage alone does not resolve the reliability of every read.
Check Your Understanding
Answer each question, then compare with the solutions.
- A hypothetical target has 2,000,000 aligned read bases across 100,000 positions. What is its average depth, and what does that number not tell you?
- Under a Poisson model with mean depth \(\lambda=2\), what fraction of positions is expected to have at least one read?
- At a hypothetical heterozygous site, suppose four independent, unbiased, error-free reads are sampled. What is the probability that none carries a specified allele?
- Why can two targets with the same average depth have different coverage breadth?
Solutions. (1) Average depth is \(2{,}000{,}000/100{,}000=20\) reads per position, or 20x. It does not show the depth at any individual position or the fraction meeting a threshold. (2) Breadth at depth 1 is \(1-e^{-2}\approx1-0.1353=0.8647\), or about 86.47%. (3) Each read has probability \(1/2\) of not carrying the specified allele, so the probability for all four is \((1/2)^4=1/16=0.0625\). (4) Average depth fixes only the total read bases relative to target length. The positions can have different depth distributions; one target may distribute reads more unevenly, leaving more positions uncovered or below a chosen threshold.