# Standardizing: z, CV, effect size and r > Put different scales side by side: z-scores, the coefficient of variation, t as signal over noise, Cohen's d as an effect in SD units, and correlation as a covariance scaled to between −1 and 1. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Squares, roots and distances - Lesson 16/36, ~35 min - Source: https://productovi.com/courses/math-basics/standardizing ## One ruler for different scales **Alex Tutor:** Back on Klasna's product team for the last lesson of the module, and the school has an award to give. **Sofia:** Klasna wants a "tutor of the term". An English tutor averages 4.7 and a math tutor 4.8. The math tutor wins, right? **Alex Tutor:** Not so fast: subjects rate differently. English tutors average 4.5 with an SD of 0.1; math tutors average 4.6 with an SD of 0.25. Ask how far each tutor stands from their own peers, in units of their spread. $$ z = \frac{x - \mu}{\sigma} $$ *x: the value; μ: the mean of its group; σ: the group's SD. z counts SDs above (+) or below (−) the mean. With sample data use x̄ and s.* **Alex Tutor:** English: (4.7 − 4.5) ÷ 0.1 = **2.0**. Math: (4.8 − 4.6) ÷ 0.25 = **0.8**. The English tutor is two SDs above her peers; the math tutor less than one. On a common ruler, she stands out more. **Sofia:** And z = 0? **Alex Tutor:** Exactly average for the group. A negative z is below average: a math tutor at 4.35 would have (4.35 − 4.6) ÷ 0.25 = **−1.0**. ![English ratings: mean 4.5, SD 0.1, tutor at 4.7 is z = 2.0. Math: mean 4.6, SD 0.25, tutor at 4.8 is z = 0.8. On a common z ruler the English tutor stands further out.](https://productovi.com/courses/math-basics/diagrams/m4-l4-z-scores.png) *z moves every group onto the same ruler, measured in SDs.* **Sofia:** And an outlier check is the same thing: a z of 3 or more is suspicious. > **Exercise:** A learner scored x = 78 on a practice test where the mean is μ = 70 and σ = 5. What is the z-score? **Alex Tutor:** Same ruler, two learners. > **Exercise:** Learner A scored 82 on a test where the mean is 75 and the SD is 10. Learner B scored 68 on a harder test where the mean is 60 and the SD is 4. Who did better relative to their own group? ## Spread relative to size **Alex Tutor:** The **coefficient of variation**, CV = s ÷ x̄, compares spread across things of different size. English tutors teach 40 lessons a week on average with an SD of 8: CV = 20%. Programming tutors: 10 lessons, SD 4: CV = 40%. **Sofia:** So programming has a smaller SD but is twice as unpredictable for its size. **Alex Tutor:** Right. Use CV only for things that can't go below zero, like lessons or money. For a change that can be negative, the mean can be near zero and the CV explodes. > **Exercise:** Monthly spend per learner averages x̄ = 1,250 UAH with an SD of s = 375 UAH. What is the coefficient of variation, in percent? **Alex Tutor:** Where the CV stops working. > **Exercise:** The coefficient of variation is a good way to compare how unstable the weekly change in bookings is, a number that can be positive or negative. ## Signal over noise **Alex Tutor:** The test statistics you'll meet are standardized too. **t** = difference ÷ its standard error: a signal measured in units of noise. And **Cohen's d** = difference ÷ s_p measures an effect in units of the spread between people. **Alex Tutor:** A new homework format raises test scores by 6 points; s_p = 15. So d = 6 ÷ 15 = **0.4**: four tenths of an SD. Unlike t, d doesn't grow with the sample, so it tells you how big the effect is, not how sure you are. **Alex Tutor:** Compute one d yourself. > **Exercise:** A new homework format raises average test scores by 9 points; the pooled SD of scores is s_p = 20. What is Cohen's d? **Sofia:** So a huge test could make a tiny effect look convincing? **Alex Tutor:** Yes. With enough users t grows large even for d = 0.05. Report both: is it real (t) and is it big (d)? **Alex Tutor:** So which sentence belongs in the report? > **Exercise:** A test with 100,000 users per group shows a lift with t ≈ 6.7 and Cohen's d = 0.03. What should the report say? ## Correlation as a scaled covariance **Alex Tutor:** Covariance says whether two metrics move together, but its size depends on the units. Divide by both SDs and you get **r**, always between −1 and 1. Homework hours and test scores: cov = 24, s_x = 3, s_y = 10, so r = 24 ÷ 30 = **0.8**. ![Three scatter plots: points rising tightly (r ≈ 0.8), a shapeless cloud (r ≈ 0) and points falling (r ≈ −0.6). Strength is the size of |r|; the sign is the direction.](https://productovi.com/courses/math-basics/diagrams/m4-l4-correlation.png) *The sign gives the direction, |r| the strength.* **Sofia:** So r = −0.6 is a stronger link than r = 0.4? **Alex Tutor:** Yes: compare |r|, the size without the sign. The minus only says that one goes down as the other goes up. > **Exercise:** Days since the last lesson and the chance of booking the next one have cov = −18; s_x = 4 and s_y = 9. What is r = cov ÷ (s_x · s_y)? > **Exercise:** Which correlation shows the strongest linear link? > **Exercise:** Test A raised the average lesson rating by 0.2 points (pooled SD s_p = 0.5). Test B raised weekly practice time by 12 minutes (s_p = 60 minutes). Compute Cohen's d for each and say in one or two sentences which effect is bigger relative to how much people differ, and why raw units can't answer that. ## Case: the tutor of the term **Alex Tutor:** Now the module case. Three finalists, three subjects, and a decision that needs every tool from this module: z, the standard error and a sense of how reliable each number is. **Sofia:** Raw averages first, then the common ruler. > **Exercise:** Recommend a tutor of the term, with the numbers behind it. ## Where you will meet this > **In short** > > - **z** = (x − μ) ÷ σ: how many SDs a value sits from its group's mean. It puts different groups on one ruler. > - **CV** = s ÷ x̄: spread as a share of the mean, unit-free. > - **t** = difference ÷ SE: signal over noise. **Cohen's d** = difference ÷ s_p: effect size in SD units, independent of sample size. > - **r** = cov ÷ (s_x · s_y), between −1 and 1. Compare **|r|** for strength; the sign is the direction. > **These lessons use it** > > - [Measures of spread](/courses/statistics-for-pm-and-marketers/spread): the coefficient of variation. > - [Outliers and skewed data](/courses/statistics-for-pm-and-marketers/outliers-and-skew): (x − x̄)/s to flag unusual values. > - [Sample vs population and the normal distribution](/courses/statistics-for-pm-and-marketers/samples-and-normal-distribution): z = (x − μ)/σ. > - [Two-sample t-test](/courses/statistics-for-pm-and-marketers/t-test): t as a difference over its standard error. > - [Correlation coefficient](/courses/statistics-for-pm-and-marketers/correlation): r between −1 and 1. > - [Clustering for customer segmentation](/courses/statistics-for-pm-and-marketers/clustering): standardize features before measuring distances. **Alex Tutor:** That's module 4. Next: lines and models, starting with the linear function behind trend lines and break-even. - Previous lesson: [Pythagoras in statistics: distances and errors in quadrature](https://productovi.com/courses/math-basics/pythagoras-in-statistics.md) - Next lesson: [The linear function](https://productovi.com/courses/math-basics/the-linear-function.md) --- Productovi Learning. Free, no sign-up needed to read.