# Square roots and √n: SD and the standard error > The standard deviation as √s², the standard error s/√n and why 4× the data halves it, the standard error of a proportion, the minimum detectable effect, the paired t and the finite-population correction. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Squares, roots and distances - Lesson 14/36, ~25 min - Source: https://productovi.com/courses/math-basics/square-roots-and-root-n ## Back to the original units **Alex Tutor:** Back at Klasna with last lesson's learner data: the variance is computed, but it isn't ready for a report yet. **Sofia:** Last time the variance came out as 14.5 "lessons squared". Nobody thinks in squared lessons. What do I report? **Alex Tutor:** Take the square root: s = √14.5 ≈ **3.81 lessons**. That's the **standard deviation**, back in the original units. Think of the variance as the area of a square; the SD is its side. ![A square of area 14.5 lessons squared has a side of √14.5 ≈ 3.81 lessons: the standard deviation is the side of the variance square.](https://productovi.com/courses/math-basics/diagrams/m4-l2-area-side.png) *A square root turns an area back into a length.* **Alex Tutor:** Read the SD as a typical distance of a learner from the mean of 8: most learners are within a few lessons of it, roughly 4 either way. **Sofia:** So I could describe the list as 8 ± 3.81 lessons? **Alex Tutor:** As a description, yes: mean ± SD. Just don't confuse it with a confidence interval, which uses a different number, the standard error. That's next. > **Exercise:** A sample of weekly lesson counts has variance s² = 20.25. What is the standard deviation s? **Alex Tutor:** Before we move on, one statement to test. > **Exercise:** A report describes five learners as "8 ± 3.81 lessons" (mean ± SD). This range is a confidence interval for the true average number of lessons. ## The standard error and √n **Alex Tutor:** Now a different question: not how much learners vary, but how much the **mean** of a sample would wobble if you drew another sample. That's the **standard error**: SE = s ÷ √n. **Sofia:** Why √n and not just n? **Alex Tutor:** Because random errors partly cancel: some learners are above the mean, some below. The cancelling grows with √n, not with n. With s = 3.81 and n = 5, SE = 3.81 ÷ 2.24 ≈ 1.70. With n = 20, SE ≈ 0.85. **Alex Tutor:** So four times the data only **halves** the error, because √4 = 2. To cut the error by ten, you need a hundred times the data. **Sofia:** So the first hundred users teach me a lot, and each next batch less. **Alex Tutor:** Exactly. From 100 to 400 users the error halves; from 400 to 1,600 it halves again, at four times the cost. ![The standard error falls as 1/√n: with s = 10 it is 1.0 at n = 100, 0.5 at n = 400 and 0.25 at n = 1,600; each fourfold increase in n halves it.](https://productovi.com/courses/math-basics/diagrams/m4-l2-root-n.png) *More data helps, but with diminishing returns.* > **Exercise:** Lesson ratings have a standard deviation s = 12 points, and you average n = 36 ratings. What is the standard error of the mean, s ÷ √n? > **Exercise:** A test's standard error is too large. How much more data do you need to cut it in half? ## Shares and test sizes **Alex Tutor:** For a share there's no separate s: the spread comes from p itself. SE_p = √(p(1 − p) ÷ n). At p = 5% with n = 1,900: √(0.0475 ÷ 1,900) = √0.000025 = **0.005**, or 0.5 pp. **Sofia:** That's the sample-size formula from module 2, the other way round. **Alex Tutor:** Yes. And the smallest effect a test can reliably detect, the **MDE**, is about 2.8 standard errors (module 7 shows why). So the same rule holds: with 2,000 users per group you can see 2 pp; to see 1 pp you need 8,000. > **Exercise:** A conversion rate is p = 0.20, measured on n = 400 users. What is its standard error SE_p = √(p(1 − p) ÷ n), in percentage points? **Alex Tutor:** And the MDE in numbers. > **Exercise:** A test's standard error for the difference between its two groups is 0.7 pp. Using MDE ≈ 2.8 × SE, roughly what is the smallest lift the test can reliably detect, in percentage points? ## Paired tests and small populations **Alex Tutor:** A paired test uses the same idea on differences. 16 tutors were rated before and after a training; the mean change is d̄ = 0.3 points, with s_d = 0.4. SE = 0.4 ÷ √16 = 0.1, so t = d̄ ÷ SE = **3**: a change three standard errors from zero. **Alex Tutor:** One correction: if your sample is a big part of a small population, the SE shrinks. Surveying 400 of Klasna's 1,120 learners, multiply by √((N − n) ÷ (N − 1)) = √(720 ÷ 1,119) ≈ 0.80. **Sofia:** And if I survey 400 of a million, the factor is about 1. > **Exercise:** 36 learners took a test before and after a course. The mean improvement is d̄ = 0.5 points with s_d = 1.2. What is the paired t = d̄ ÷ (s_d ÷ √n)? **Alex Tutor:** Now the correction factor in numbers. > **Exercise:** Klasna surveys 500 of the 1,000 parents who renewed this term. What is the finite-population correction √((N − n) ÷ (N − 1))? Give two decimals. ## Your turn: the price of a smaller effect **Alex Tutor:** Leo asks: "The test with 2,000 users per group can spot a 2 pp lift. Let's double it to 4,000 and spot 1 pp." Reply to him. > **Exercise:** Leo, Klasna's CEO, says: "With 2,000 users per group the test can detect a 2 pp lift. Doubling to 4,000 per group will let it detect 1 pp." Is he right? Give the correct number of users per group for 1 pp and what 4,000 per group would actually detect, and explain why in one sentence. ## Where you will meet this > **In short** > > - **SD** s = √s²: the side of the variance square, back in the original units. > - **SE** = s ÷ √n: how much a sample mean would wobble. 4× the data halves it; 100× cuts it tenfold. > - For a share, **SE_p** = √(p(1 − p) ÷ n). The **MDE** scales with the SE, so halving it needs 4× the users. > - **Paired t** = d̄ ÷ (s_d ÷ √n). A big sample of a small population: multiply SE by **√((N − n) ÷ (N − 1))**. > **These lessons use it** > > - [Measures of spread](/courses/statistics-for-pm-and-marketers/spread): s as the square root of the variance. > - [Sample vs population and the normal distribution](/courses/statistics-for-pm-and-marketers/samples-and-normal-distribution): SE = s/√n. > - [95% confidence intervals](/courses/statistics-for-pm-and-marketers/confidence-intervals): SE and SE_p inside the interval. > - [Paired t-test](/courses/statistics-for-pm-and-marketers/paired-tests): t = d̄ / (s_d/√n). > - [How big and how long](/courses/product-analytics/test-size): a smaller effect needs many more users. **Alex Tutor:** Next time: Pythagoras in statistics, where errors and distances add up as squares. - Previous lesson: [Why we square: variance, sums of squares and n − 1](https://productovi.com/courses/math-basics/why-we-square.md) - Next lesson: [Pythagoras in statistics: distances and errors in quadrature](https://productovi.com/courses/math-basics/pythagoras-in-statistics.md) --- Productovi Learning. 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