CHAPTER 2 Modeling Distributions of Data

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1 CHAPTER 2 Modeling Distributions of Data 2.2 Density Curves and Normal Distributions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers

2 Density Curves and Normal Distributions Learning Objectives After this section, you should be able to: Density Curves: define and describe the mean and median location Normal Distribution: estimate areas/ proportion of values/ probability. Use Rule The Standard normal distribution: find the proportion of z-values in a specified interval, or a z-score from a percentile in the standard Normal distribution. The Practice of Statistics, 5 th Edition 2

3 Exploring Quantitative Data Exploring Quantitative Data 1. Always plot your data: make a graph, usually a dotplot, stemplot, or histogram. 2. Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers. 3. Calculate a numerical summary to briefly describe center and spread. 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. The Practice of Statistics, 5 th Edition 3

4 Density Curves A density curve is a curve that is always on or above the horizontal axis, and has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion, percentile or probability of all observations that fall in that interval. Example The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars. Approximated area George Box (famous statistician Essentially all models are wrong but some are useful) The Practice of Statistics, 5 th Edition 4

5 Batting averages The histogram below shows the distribution of batting average (proportion of hits) for the 432 Major League Baseball players with at least 100 plate appearances in a recent season. The smooth curve shows the overall shape of the distribution. The Practice of Statistics, 5 th Edition 5

6 Describing Density Curves: measure of center Distinguishing the Median and Mean of a Density Curve median -> equal-areas point, the point that divides the area under the curve in half. mean -> balance point (µ (the Greek letter mu)) The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail. The Practice of Statistics, 5 th Edition 6

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8 Normal Distributions Described by Normal curves (symmetric, single-peaked, and bellshaped) Described by mean µ (right at the center) and its standard deviation σ. (from center up to the inflection point) Notation: N(µ,σ). The Practice of Statistics, 5 th Edition 8

9 Why are Normal Distributions important? Normal distributions are good descriptions for some distributions of real data.(scores on tests, repeated measurements on volleyball diameter. Characteristics of biological population) Normal distributions are good approximations of the results of many kinds of chance outcomes (number of heads in many tosses with fair coin). Many statistical inference procedures are based on Normal distributions. The Practice of Statistics, 5 th Edition 9

10 Discovery Applet Activity Follow instructions on page Summarize: For any normal density curve, the area under the curve within one, two or three standard deviations of the mean is about % % %. Page: 110 read, + example page 11 The Practice of Statistics, 5 th Edition 10

11 The Rule Although there are many Normal curves, they all have properties in common. The Rule In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ. The Practice of Statistics, 5 th Edition 11

12 Steps to Draw a Normal Distribution 1. Draw a number line with 7 evenly spaced stick marks (mean, μ ± σ, μ ± 2σ, μ ± 3σ) 2. Draw a dot above the mean value where the peak will be 3. Above the mark where we have one standard deviation, 60% as high draw a dot 4. Above the two standard deviation mark, draw dots about 15% as high draw a dot 5. Above the three standard deviation mark draw two dots right above the horizontal axis The Practice of Statistics, 5 th Edition 12

13 Practice: Batting Average In the previous alternate example about batting averages for Major League Baseball players, the mean of the 432 batting averages was with a standard deviation of Suppose that the distribution is exactly Normal with = and = Problem: (a) Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard deviations from the mean. (b) What percent of the batting averages are above 0.329? Show your work. (c) What percent of the batting averages are between and 0.295? Show your work. The Practice of Statistics, 5 th Edition 13

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15 The Standard Normal Distribution All Normal distributions are the same if we measure in units of size σ from the mean µ as center. The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable z = x - m s has the standard Normal distribution, N(0,1). The Practice of Statistics, 5 th Edition 15

16 The Standard Normal Table The standard Normal Table (Table A) is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z. Suppose we want to find the proportion of observations from the standard Normal distribution that are less than We can use Table A: P(z < 0.81) =.7910 Z The Practice of Statistics, 5 th Edition 16

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20 Page 114 problem Practice (you ll be asked to find L, R tail area and in between z- scores) Finding areas under the standard Normal curve Problem: Find the proportion of observations from the standard Normal distribution that are between 0.58 and The Practice of Statistics, 5 th Edition 20

21 Homework Calculator Activity (in the AP exam choose whichever method is easiest: table A or calculator) Page 128 # 33 to 51 The Practice of Statistics, 5 th Edition 21