More on Normal Distributions

First, this section talks about the history of the normal distribution and the central limit theorem and the relation of normal distributions to errors. Then, it discusses how to compute the area under the normal curve. It then moves on to the normal distribution, the area under the standard normal curve, and how to translate from non-standard normal to standard normal. Finally, it addresses how to compute (cumulative) binomial probabilities using normal approximations.


  1. 68% of the distribution is within one standard deviation of the mean. 40 + 5 = 45, 40 - 5 = 35

  2. 95% of the distribution is within 1.96 standard deviations of the mean. You can round 1.96 to 2 to approximate. 20 - 2(3) = 14, 20 + 2(3) = 26

  3. Use the "Calculate Area for a given X" calculator and enter Mean = 5, SD = 2, Above 3. You will get 0.8413.

  4. Var = 100, so SD = 10. Use the "Calculate X for a given Area" calculator and enter Mean = 120, SD = 10, Shaded area = .35. Click below, and you will get 116.15.

  5. Use the "Calculate X for a given Area" calculator and enter Mean = 38, SD = 6, Shaded area = .80. Click below, and you will get 43.05, meaning a score of 43.

  6. Use the "Calculate Area for a given X" calculator and enter Mean = 38, SD = 6, Between 30 and 45. You will get 0.787, meaning 78.7%.