Lesson 3 of 9
The standard error
A measured average is itself uncertain. For 67 US utilities the average P/E is 22.94 give or take about 3, which is enough to erase a comparison.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- The standard error is the standard deviation divided by the square root of the sample size.
- It measures how far a measured average is likely to sit from the one you would get with the whole population.
- It shrinks with the square root of n, so quadrupling the sample halves the error. Doubling it barely helps.
A standard deviation says how spread out the members of a cohort are. A standard error says something different and more useful: how much the cohort's own average would move if you had measured a different cohort of the same size.
Measured average P/E, with the range it could plausibly sit in
- Utilities, 67 listings · 19.80 to 26.0822.94x
- Consumer Defensive, 98 listings · 21.53 to 28.3524.94x
- Technology, 243 listings · 32.17 to 38.0535.11x
- Financial Services, 471 listings · 16.04 to 18.0017.02x
The formula, and one division
Standard error equals Standard deviation divided by The square root of the sample size.
Example13.11 / the square root of 67 = 1.60
The utilities cohort has 67 listings, a mean P/E of 22.94 and a standard deviation of 13.11. The square root of 67 is 8.19, so the standard error is 1.60. An interval of roughly two standard errors each way runs from 19.80 to 26.08.
That is the honest form of the measurement: not 22.94, but about 23, give or take three.
Why the square root
Errors in a sample partly cancel. Some members sit above the population average and some below, and the more members you have the more of that cancelling happens. The cancelling is proportional to the square root of the count, not to the count itself.
The consequence is brutal for anyone hoping to fix a weak result with more data. Going from 67 observations to 134 cuts the error by 29 percent. Cutting it in half needs 268. Cutting it to a tenth needs 6,700, which is more listings than the US market has.
What it does to a comparison
The standard error of a difference is not the two errors added. It is the square root of the sum of their squares, because two independent errors partly offset. For the first row: the square root of 1.60 squared plus 1.74 squared, which is 2.36.
Where this is skipped
Almost everywhere. A sector average, a factor return, a fund's outperformance and a backtest's edge are all measured quantities with a standard error, and the error is almost never printed beside them.
The question to ask of any market statistic is the one this lesson supplies: how many observations, and how spread out were they. Without those two numbers the error cannot be computed, and without the error the statistic cannot be told from noise.
Check yourself
A cohort of 67 has a standard deviation of 13.11. What is the standard error of its mean?
- 13.11
- About 1.60
- About 0.20
Show the answer
Answer: B.
Divide the standard deviation by the square root of the sample size. The square root of 67 is 8.19, and 13.11 divided by 8.19 is 1.60.You want to halve the standard error of a measured average. What do you need?
- Twice as many observations
- Four times as many observations
- A smaller standard deviation
Show the answer
Answer: B.
The error falls with the square root of the sample size, so halving it takes four times the data. That is why marginal extra observations help so little.Two cohort means differ by 2.00 and the standard error of that difference is 2.36. What has been shown?
- That the second cohort is more expensive
- Nothing: the gap is smaller than the uncertainty in measuring it
- That the cohorts are identical
Show the answer
Answer: B.
A difference inside its own error is consistent with no difference at all. It is also consistent with a real one, which is exactly why the honest statement is that nothing has been established.Sources
- Sector means, standard deviations and cohort sizes computed across active US equity listings with a positive P/E below 100, read from this product's own store on 16 September 2026.
What this leaves out
- The two standard error intervals here assume the sampling distribution of the mean is roughly symmetric. On a heavily skewed cohort that assumption is imperfect, which makes these intervals approximate rather than exact.
- Every cohort is restricted to positive ratios below 100, and a sector's companies are not a random sample of anything. They are the whole sector on one date, which makes the standard error a statement about measurement rather than about drawing from an urn.
- These count listings, not deduplicated companies.
- Nothing here is advice about any security.