Chapter 2: The Normal Distributions 2.1c

1. Chapter 2: The Normal Distributions2.1c Target Goal: I can determine the relationship between the mean and median of a density curve. h.w: pg 108: 25 - 38

2. 2.1 Density Curves and the Normal Distributions Mathematical model: Density curve • Describes the overall pattern of the distribution. • Always on or above the horizontal axis, and • Has an area exactly 1 underneath it. (Area under curve represents all of the observations.)

3. Density Curve • If we count the number if students with scores 6 or less of the histogram, we find the proportion 287/947 = 0.303 of all students. • The density curve reveals that area with score 6 or less is 0.293.

4. The Median and Mean of a Density Curve • Mean: the balancing point of the curve if it were made of solid material. • Median: the equal areas point, the point with half the area under and half above it.

5. The Median and Mean of a Density Curve The mean and median of a right skewed density curve. • The mean of a skewed curve is pulled away from the median in the direction of the long tail. • The median and mean are the same for a symmetric curve.

6. We need to distinguish between the mean ( ) and standard deviation scomputed from the actual observations. • A density curve is an idealized description of data. We will use (mu:μ) and σ (sigma) for mean and standard deviation.

7. Example 1: A Uniform Distribution • The figure displays the density curve of a uniform distribution. The curve takes the constant value 1 over the interval from 0 to 1 and is zero outside the range of values. This means that data described by this distribution takes values that are uniformly spread between 0 and 1

8. Why is the total area under the curve equal to 1? Curve is a rectangle: 1 x1 = 1 b. What percent of the observations lie about 0.8? 20%: This rectangle has base 0.2 and height 1 c. What percent of the observations lie below 0.6? 60%

9. d. What percent of the observations lie between 0.25 and 0.75? 50% e. What is the mean μ of this distribution? The mean or balancing point is 0.5

10. Example 2: Finding Means and MediansWhich of these points on each curve do the mean and median fall? • mean = , median = C B (b) mean = , median = A A (c) mean = , median = A B

11. Ex 3: Baseball Salaries Did Ryan Madson, who was paid $1,400,000, have a high salary or a low salary compared with the rest of the team? Small group work: justify your answer by calculating and interpreting Madson’s percentile and z-score. Report out.

12. Class answers:

13. Complete answer: There were 14 salaries less than Madson’s putting his salary in the 48th %tile. We could also argue that since Madson’s salary was the median, his percentile would be 50 %. Madson had a typical salary compared with the rest of the team because his percentile was 50%.