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

Learning Objectives After this section, you should be able to: The Practice of Statistics, 5 th Edition2 2.2 What is a density curve? How do you use the normal curve to calculate percentages? Density Curves and Normal Distributions

The Practice of Statistics, 5 th Edition3 Exploring Quantitative Data In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy. 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. Exploring Quantitative Data 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 Edition4 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 of all observations that fall in that interval. 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 of all observations that fall in that interval. 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. Example

The Practice of Statistics, 5 th Edition5 Describing Density Curves Our measures of center and spread apply to density curves as well as to actual sets of observations. The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. 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. Distinguishing the Median and Mean of a Density Curve

The Practice of Statistics, 5 th Edition6 Describing Density Curves A density curve is an idealized description of a distribution of data. We distinguish between the mean and standard deviation of the density curve and the mean and standard deviation computed from the actual observations. The usual notation for the mean of a density curve is µ (the Greek letter mu). We write the standard deviation of a density curve as σ (the Greek letter sigma).

The Practice of Statistics, 5 th Edition7 Normal Distributions One particularly important class of density curves are the Normal curves, which describe Normal distributions. All Normal curves have the same shape: symmetric, single- peaked, and bell-shaped Any specific Normal curve is completely described by giving its mean µ and its standard deviation σ.

The Practice of Statistics, 5 th Edition8 Normal Distributions A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. The mean of a Normal distribution is the center of the symmetric Normal curve. The standard deviation is the distance from the center to the change-of-curvature points on either side. We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ). A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. The mean of a Normal distribution is the center of the symmetric Normal curve. The standard deviation is the distance from the center to the change-of-curvature points on either side. We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ). Why are the Normal distributions important in statistics? Normal distributions are good descriptions for some distributions of real data. Normal distributions are good approximations of the results of manykinds of chance outcomes. Many statistical inference procedures are based on Normal distributions.

The Practice of Statistics, 5 th Edition9 In the first graph below, the bars in red represent the proportion of players who had batting averages of at least There are 177 such players out of a total of 432, for a proportion of In the second graph below,the area under the curve to the right of is shaded. This area is 0.391, only away from the actual proportion of

The Practice of Statistics, 5 th Edition10 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 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 Edition11 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 has the standard Normal distribution, N(0,1). 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 has the standard Normal distribution, N(0,1).

The Practice of Statistics, 5 th Edition12 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. Z P(z < 0.81) =.7910 Suppose we want to find the proportion of observations from the standard Normal distribution that are less than We can use Table A:

The Practice of Statistics, 5 th Edition13 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 Edition14 Finding areas under the standard Normal curve Problem: Find the proportion of observations from the standard Normal distribution that are between −0.58 and 1.79.

The Practice of Statistics, 5 th Edition15 Serving speed In a recent tournament, tennis player Rafael Nadal averaged 115 miles per hour (mph) on his serves. Assume that the distribution of his serve speeds is Normal with a standard deviation of 6 mph. Problem: About what percent of Nadal’s serves would you expect to exceed 120 mph? Problem: What percent of Rafael Nadal’s serves are between 100 and 110 mph?

The Practice of Statistics, 5 th Edition16 Assignment 33, 35, 39, 41, 43, 45, 47, 49, 51