1/27/11 Lecture 5 1 STOR 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL1/27/11 Lecture 5 2 A problem about Standard Deviation A variable has 5 possible values. The mean is 125 and the standard deviation is 25. If a sixth number of 125 is added in, what is the new standard deviation?1/27/11 Lecture 5 3 To describe distribution of variable X • Plot your data: – Stemplot, histogram, boxplot • Look for overall pattern (shapes) + striking deviations (outliers) • Calculate appropriate numerical summaries about center and spread – Mean, median, mode – Range, percentiles, quartiles, standard deviation – Five-number summary, boxplot – IQR and outliers • Make things easier with a smooth curve1/27/11 Lecture 5 41/27/11 Lecture 5 5 Density Curve --- A Math Approximation • To calculate probabilities, we define a probability density function f(x). • The curve that plots f(x) is called the corresponding density curve. • f(x) satisfies: -- f(x) >= 0; – The total area under the curve representing f(x) equals 1.1/27/11 Lecture 5 6 Density Curves • Describe the overall shape of distributions • Idealized mathematical models for distributions • Show patterns that are accurate enough for practical purposes • Always on or above the horizontal axis • The total area under the curve is exactly 1 • Areas under the curve represent relative frequencies of observations1/27/11 Lecture 5 71/27/11 Lecture 5 8 Histograms vs. Density Curves • Histograms show either frequencies (counts) or relative frequencies (proportions) in each class interval. • Density curves show the proportion of observations in any region by areas under the curve. • You can think of density curve as an approximation to refined histograms when there are a lot of observations.1/27/11 Lecture 5 9 Histogram vs Density Curve1/27/11 Lecture 5 10 Center of a Density Curve • The mode of a distribution is the point where the curve is the highest. Highest Point. • The median is the point where half of the area under the curve lies on the left and the other half on the right. Equal Areas Cut-off Point. • The mean is the point at which the curve would balance if made out of solid material. Balance Point. 1/27/11 Lecture 5 11 Mean of a Density Curve1/27/11 Lecture 5 12 Spread of a Density Curve • Quartiles can be found by dividing the area under the curve into four equal parts – ¼ of the area is to the left of the 1st quartile, Q1 – ¾ of the area is to the left of the 3rd quartile, Q3 • The standard deviation of a density curve is denoted by . – Not easy to calculate 1/27/11 Lecture 5 131/27/11 Lecture 5 14 Normal Distribution • Pictorially speaking, a Normal Distribution is a distribution that has a symmetric, unimodal and bell-shaped density curve. • The mean and standard deviation completely specify the curve. • The mean, median, and mode are the same.1/27/11 Lecture 5 15 • The height of a normal density curve at any point x is given by 2)(2121)(xexf is the mean is the standard deviation 1/27/11 Lecture 5 16 Change of Curvature1/27/11 Lecture 5 17 Example: The normal distribution is the most important distribution in Statistics. Typical normal curves with different sigma (standard deviation) values are shownbelow.1/27/11 Lecture 5 18 Examples with approximate Normal distributions • Height • Weight • IQ scores • Standardized test scores • Body temperature • Repeated measurement of same quantity • …1/27/11 Lecture 5 19 The 68-95-99.7 Rule 231/27/11 Lecture 5 20 The 68-95-99.7 Rule1/27/11 Lecture 5 21 Example: Young Women’s Height • The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches.1/27/11 Lecture 5 22 Example: Young Women’s Height • % of young women between 62 and 67? • % of young women lower than 62 or taller than 67? • % between 59.5 and 62? • % taller than 68.25?1/27/11 Lecture 5 23 Standardizing and z-Scores • an observation x comes from a distribution with mean and standard deviation • The standardized value of x is defined as which is also called a z-score. • A z-score indicates how many standard deviations the original observation is away from the mean, and in which direction. • Mean and S.D. of the distribution of z? ,xz1/27/11 Lecture 5 24 Example: Young Women’s Height • The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches. • In our class, there is a female student who is 68.25 inches tall, what is her z-score?1/27/11 Lecture 5 25 Effects of Standardizing • Standardizing is a linear transformation. What are a and b? • Effects on shape, center and spread. • The standardized values for any distribution always have mean 0 and standard deviation 1. • Linear transformation: normal into normal.1/27/11 Lecture 5 26 The standard normal distribution • The standard normal distribution is the normal dist. with mean 0 and standard deviation 1, denoted as N(0,1). • N(0,1) can be treated as a benchmark. • Any normal distribution can be related to N(0,1) by a linear transformation. • Z: N(0,1) • What is the distribution for X=a+bZ?1/27/11 Lecture 5 27 Take Home Message • Density curve – Center, spread • Normal distributions and normal curves • The 68-95-99.7 rule for normal distributions • Standardizing observations • The standard normal
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