# The normal curve > Area as probability, the 68–95–99.7 rule, where 1.96 comes from, one- and two-tailed p-values, and the z-values for α and power. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Probability and distributions - Lesson 29/36, ~25 min - Source: https://productovi.com/courses/math-basics/the-normal-curve ## Area is probability **Alex Tutor:** Back on Klasna's product team, where a mock math exam runs every spring. Last year’s scores looked like a bell: most learners near 150, fewer and fewer towards the edges. The mean was 150 and the standard deviation 15. **Sofia:** And the curve is a smooth version of that histogram? **Alex Tutor:** Yes, the **normal curve**. Its key trick: the area under it between two points is the share of learners there, or the probability that a random learner lands there. The whole area is 1. **Alex Tutor:** First, make sure the idea of area as a share sticks. > **Exercise:** Klasna’s mock exam scores follow a normal curve with mean 150 and SD 15. The area under the curve between 140 and 160 is 0.495. What does that number mean? **Alex Tutor:** Three areas are worth remembering. Within 1 SD of the mean lies about 68%: here 135 to 165. Within 2 SD, about 95%: 120 to 180. Within 3 SD, 99.7%: 105 to 195. ![Scores with mean 150 and SD 15: about 68% between 135 and 165, 95% between 120 and 180, 99.7% between 105 and 195.](https://productovi.com/courses/math-basics/diagrams/m7-l5-68-95-997.png) *The 68–95–99.7 rule: shares of the area, not of the width.* **Sofia:** So what’s outside 2 SD is split between the two ends? **Alex Tutor:** Equally, because the curve is symmetric. 100% − 95% = 5% outside, so 2.5% above 180 and 2.5% below 120. **Sofia:** Does every metric look like this bell? **Alex Tutor:** No. Revenue per learner or lesson length are skewed, with a long tail on one side. The curve matters most for averages and shares measured on many people: those come out bell-shaped even when single values don’t. That’s why tests rely on it. > **Exercise:** Mock exam scores are roughly normal with mean 150 and SD 15. By the 68–95–99.7 rule, about what share of learners score above 165, in percent? **Alex Tutor:** One more check on when the bell is the right picture. > **Exercise:** Revenue per learner at Klasna has a long right tail: most pay for one package, a few pay for many. With a mean of 9,000 UAH and an SD of 6,000 UAH, the 68–95–99.7 rule says about 95% of learners pay between −3,000 and 21,000 UAH. ## Where 1.96 comes from **Alex Tutor:** To use the curve for any data, measure in SDs from the mean: z = (x − mean) ÷ SD. A score of 180 is z = (180 − 150) ÷ 15 = 2. Then one table works for everything. **Alex Tutor:** Quick practice with z before we get to 1.96. > **Exercise:** Mock exam scores have mean 150 and SD 15. A learner scored 171. What is the score’s z, the distance from the mean in SDs? **Sofia:** And 1.96 is just the more precise version of 2? **Alex Tutor:** Exactly. "95% within 2 SD" is rounded. The exact z that leaves 95% in the middle and 2.5% in each tail is **1.96**. That’s why intervals and tests at the 5% level use 1.96. > **Exercise:** Why does a 95% confidence interval use 1.96? **Alex Tutor:** A confidence interval is then just x̄ ± 1.96 × SE. Take the trial lessons from earlier: 30% bought, SE ≈ 2.3 pp. 30% ± 1.96 × 2.3 = 30% ± 4.5, so from 25.5% to 34.5%. > **Exercise:** A survey finds that 42% of parents want weekend lessons, with a standard error of 3 pp. What is the upper end of the 95% interval, x̄ + 1.96 × SE, in percent, to one decimal? ## One tail or two **Alex Tutor:** Now a test. A new lesson reminder lifted attendance, and the z statistic came out 1.87. The area beyond 1.87 on one side is 0.031. **Alex Tutor:** The usual question is "is the reminder different, better or worse?". Then extreme results on both sides count: p = P(|Z| ≥ 1.87) = 0.031 + 0.031 = 0.062, above 0.05. Asked only "is it better?", p = 0.031. ![For z = 1.87, the two-tailed p counts both tails, 0.031 + 0.031 = 0.062; the one-tailed p counts one, 0.031. The two-tailed 5% cut-off is 1.96.](https://productovi.com/courses/math-basics/diagrams/m7-l5-tails.png) *The same z, two different p-values.* **Sofia:** So I could just pick one-tailed and get significance? **Alex Tutor:** Only if you decided that before seeing the data, and would truly ignore a drop. Switching after the fact doubles your real false-alarm rate. Most product tests are two-tailed. > **Exercise:** A test gives z = −2.10, and the area beyond 2.10 on one side is 0.018. What is the two-tailed p-value? ## z for α and for power **Alex Tutor:** Sample-size formulas use two z-values. z_{α/2} = 1.96 for a two-tailed 5% test. z_β = 0.84 for 80% power: 80% of the curve lies below 0.84. Together, (1.96 + 0.84)² = 2.8² = 7.84. $$ n \propto (z_{\alpha/2} + z_{\beta})^2 $$ *The sample size grows with the square of the two z-values added together.* > **Exercise:** For 90% power, z_β = 1.28. Raising power from 80% to 90%, with everything else the same, makes the required sample about a third larger. ## Your turn **Alex Tutor:** The reminder test from before: z = 1.87. A teammate writes: "One-tailed p is 0.031, so it’s significant. Let’s ship." > **Exercise:** A new lesson reminder lifted attendance with z = 1.87; the area beyond 1.87 on one side is 0.031. A teammate writes: "One-tailed p is 0.031, so it’s significant. Let’s ship." In two or three sentences, reply to the teammate: is the result significant at 5%, and when would the one-tailed p be acceptable? ## Where you will meet this > **In short** > > - Under the **normal curve**, area = probability; the whole area is 1. > - **68–95–99.7:** shares within 1, 2 and 3 SD of the mean. > - z = (x − mean) ÷ SD. ±**1.96** leaves 95% in the middle and 2.5% per tail; interval x̄ ± 1.96 × SE. > - **Two-tailed** p counts both tails (0.062 for z = 1.87); one-tailed counts one (0.031). Choose before the data. > - Sample size: (z_{α/2} + z_β)² = (1.96 + 0.84)² = 7.84 for 5% and 80% power. > **These lessons use it** > > - [Sample vs population and the normal distribution](/courses/statistics-for-pm-and-marketers/samples-and-normal-distribution): z and the 68–95–99.7 rule. > - [p-value and statistical significance](/courses/statistics-for-pm-and-marketers/p-value): one- and two-tailed p-values. > - [95% confidence intervals](/courses/statistics-for-pm-and-marketers/confidence-intervals): x̄ ± 1.96 × SE. > - [Power analysis and sample size](/courses/statistics-for-pm-and-marketers/power-and-sample-size): z_{α/2} and z_β. > - [How big and how long](/courses/product-analytics/test-size): (1.96 + 0.84)² = 7.84 in the users-per-variant formula. > - [Case: reading a finished test](/courses/product-analytics/reading-a-test): an interval for the effect. **Alex Tutor:** Next time: other curves, t, chi-square and F, and how to compute the power of a test. - Previous lesson: [Many tests at once](https://productovi.com/courses/math-basics/many-tests-at-once.md) - Next lesson: [Other curves and power](https://productovi.com/courses/math-basics/other-curves-and-power.md) --- Productovi Learning. 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