Central limit theorem

1. Gaussians everywhere Central limit theorem Gaussians in physics Slightly-disguised Gaussians in biology 0 1 -1 0

2. Central limit theorem H T -3 -2 -1 0 1 2 3

3. Gaussians everywhere Central limit theorem Gaussians in physics Slightly-disguised Gaussians in biology 0 1 -1 0

4. Physics lab: Engineered for tightly-controlled noise

5. Physics lab: Engineered for tightly-controlled noise

6. Physics lab: Engineered for tightly-controlled noise x1 Hook vibration “Small” noise: Neglect quadratic terms in Taylor expansion x2 0 Uneven air flow x3 Thermal expansion y V x4 Twisting t Laser pointer vibration x5

7. Gaussians everywhere Central limit theorem Gaussians in physics Slightly-disguised Gaussians in biology 0 1 -1 0

8. Biology: Law of mass action and logarithms -1 +1 x2 x1 x3 y y y y y y y y y y Fluctuations in x1, x2, x3, etc. are not necessarily engineered to be small. First-order Taylor-expansion might be inaccurate.

9. Biology: Law of mass action and logarithms A histogram of the logarithm of the concentration of y displays a normal distribution e5 = 150 yST = e4 = 55 e6 = 400 0 100 200 300 2 3 4 5 6 7 8 3000 0 150 300 7 150