Can two readings agree and still be wrong?
Compare an average and a difference when two measurements share error.
The question
Two fictional instruments measure the same fixed unknown quantity. Each reading is the true value plus an error.
| Mean | Variance | |
|---|---|---|
| Instrument A | 0 | 4 |
| Instrument B | 0 | 4 |
The two errors are independent.
“How would you use the mean and the difference of the two readings?”
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A strong answer, step by step
Open each step in turn. This is one strong answer, not the only one.
Step 1: What does each summary measure?
What would you do here? Open the move you’d make.
Use the difference to estimate the value
Tempting, but it misses something. Subtracting cancels the true value.
Use the mean for the value and the difference for agreement
The strong move. Each summary answers a different question.
Treat the readings as exact because the error means are zero
Tempting, but it misses something. An average error of zero does not make an individual error zero.
See how a strong candidate says it
“The mean estimates the quantity. Subtracting the readings cancels the true value, so the difference measures disagreement.”
Why it works: Both instruments measure the same fixed quantity. Their errors determine how far the summaries vary.
Step 2: What are the opening variances?
What would you do here? Open the move you’d make.
Mean variance 2, difference variance 8
The strong move. Both errors contribute and the scale of the mean is squared.
Mean variance 4, difference variance 8
Tempting, but it misses something. Dividing the sum by two divides its variance by four.
Mean variance 2, difference variance 0
Tempting, but it misses something. The true values cancel. The independent errors do not.
See how a strong candidate says it
“The variance of the mean error is (4 + 4)/4 = 2. The variance of the difference is 4 + 4 = 8.”
Why it works: The errors are independent. Scaling the sum by one half divides its variance by four.
Step 3: How do the variances change?
The errors keep their original means and variances. They share noise, and their covariance is +3.
What would you do here? Open the move you’d make.
Mean variance 2, difference variance 8
Tempting, but it misses something. The earlier calculation used zero covariance.
Mean variance 3.5, difference variance 14
Tempting, but it misses something. Fourteen is the variance of the sum. The difference uses the opposite sign.
Mean variance 3.5, difference variance 2
The strong move. Add twice the covariance for the sum and subtract it for the difference.
See how a strong candidate says it
“The mean variance is (4 + 4 + 2 × 3)/4 = 3.5. The difference variance is 4 + 4 − 2 × 3 = 2.”
Why it works: Positive covariance increases the variance of the sum and reduces the variance of the difference.
Step 4: What does agreement tell you?
What would you do here? Open the move you’d make.
Trust the average more because the readings agree
Tempting, but it misses something. The average now has greater error variance even though the difference has less.
Check for common error before treating agreement as accuracy
The strong move. The difference suppresses a shared error.
Use only one instrument because the average is now biased
Tempting, but it misses something. The original error means are still zero, so the mean remains unbiased.
See how a strong candidate says it
“A small difference shows the instruments gave close readings. A shared error can cancel in that difference while remaining in the average.”
Why it works: Agreement on a measurement can coexist with error in both readings.
The answer, in short
- Decision
- Use the covariance to revise both variances before interpreting agreement.
- Because
- Shared error lowers the variance of the difference and raises the variance of the mean.
- What would change it
- With different error variances or covariance, recalculate the mean and difference.
Common traps
- Assuming independence from the two marginal variances
- Confusing agreement with accuracy
Check your answer
- What quantity cancels in the difference?
- Why does the scale of the mean get squared?
- Where does covariance enter?
- What would help check accuracy?
Last reviewed Nov 11, 2026