The Normal Curve

• If we look at histograms of many data sets, some patters begin to emerge
  • The uniform distribution, or rectangular distribution
  • A J shaped distribution, either constantly increasing or constantly decreasing
  • A skewed left, or skewed right distribution.
  • A normal, or bell shaped distribution.
• The normal distribution shows up all over the place.
  • Distribution of scores on many standardized tests.
  • Sizes of things (height, weight, ...)
  • Failure rates, life spans, ...
• This is the bell curve, also known as the Gaussian distribution.
• In a normal distribution, the mean median and mode are all the same value.
• The empirical rule:
  • If a population is normally distributed, 68% of the data is within one standard deviation of the mean.
  • 95% is within two standard deviations
  • 99.7% is within three standard deviations.
• If SAT scores are normally distributed, with a μ=500 and σ=100
  • What scores are one, two and three standard deviations away from the mean.
  • If 100 students take the test, how man students should have scores in each of these ranges?
• The z score gives us a way to compute with the normal curve when we have a value other than a standard deviation away.
• z = (x-μ)/σ
• If the μ=100, σ =10, find the z-scores for 110, 115, 97 and 84
• Using the table on pages 888 and 889, we can find the percentage to the left of a given z-score.
• Example Problems, p 895: 32, 34, 35, 43, 49, 50, 52, 53, 81, 82