Overview:

1. Notes - 2.1 Density Curves and Normal Distributions (Pages 64-79)"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." -- H.G. Wells, 1866-1946

2. Overview: • Sometimes the overall pattern of a distribution is such that we can describe it with a smooth curve. • It is remarkable how many natural phenomena appear to be related to a bell-shaped curve known as a normal distribution. • When appropriate, using a normal distribution model to represent distributions that occur in real-life situations can be extremely useful in statistical analysis.

3. Density Curve: • Displays the overall pattern (shape) of a distribution. • Has an area of exactly 1 sq. unit underneath it. • Is on or above the horizontal axis. • A histogram becomes a density curve if the scale is adjusted so that the total area of the bars is 1 sq. unit.

4. The median of a density curve is the point that divides the area under the curve into halves. The mean of a density curve is the "balance point" of the curve. (Think of a teeter-totter.) They are the same for a symmetric distribution.

5. As an illustration, consider the set {1,2,3,5,11,14}. If each box with an X has an area of 1/6, then the total area of the six boxes would be 1. The median of this set is 4, and the mean (the balance point of the teeter-totter) is 6. x x x x x x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 M M e e d a i n a n

6. Work problems 2.1 2.2 2.3 2.4 2.5 Read Pages 73 to 77 For next time:

7. "Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself." -- Claude Bragdon

8. Normal Distributions • A special type of density curves form normal distributions. • These distributions are bell-shaped, and a normal curve is determined by the mean (m) and standard deviation (s) of the data set. • While it will not be used directly in this course, the formula for the normal distribution function, which involves the two amazing numbers pi and e, is

9. The Normal Distribution

10. For the normal distribution: • 68% of the observations fall within 1 standard deviation of the mean. • 95% of the observations fall within 2 standard deviations of the mean. • 99.7% of the observations fall within 3 standard deviations of the mean.

11. Inflection points • A normal distribution curve has two points where curvature changes. • These are called points of inflection, and they are located 1 standard deviation on either side of the mean.

12. TI-83 tidbits • normalcdf(lowerbound,upperbound,mean,stan-dard deviation) can be very useful in statistical analysis. Press [2nd] [VARS] • Note that if a normal distribution has mean = 0 and st.dev. = 1, then • normalcdf(-1,1,0,1) = .6826894809 • normalcdf(-2,2,0,1) = .954499876 • normalcdf(-3,3,0,1) = .9973000656

13. More TI-83 tidbits • If you let y1 = normalpdf(x,0,1), and set • Xmin = -4 • Xmax = 4 • use ZoomFit • You will see a graphic representation of the normal distribution curve with mean = 0 and standard deviation = 1.

14. Percentiles • An observation's percentile is the percent of the distribution that is at or to the left of the observation. • If, for instance, you have a test score representing the 90th percentile, then only 10% of the test-takers scored higher than you did.

15. Work 2.6 2.7 2.8 2.9 Read Summary pages 78 & 79 Quiz 2.1 tomorrow For next time: