9/10/09 Lecture 5 1STOR 155 Introductory StatisticsLecture 5: Density Curves and Normal Distributions (I)The UNIVERSITY of NORTH CAROLINAat CHAPEL HILL9/10/09 Lecture 5 2A problem about Standard DeviationA 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?9/10/09 Lecture 5 3To 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 curve9/10/09 Lecture 5 49/10/09 Lecture 5 5Density 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 1.9/10/09 Lecture 5 6Density 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 observations9/10/09 Lecture 5 79/10/09 Lecture 5 8Histograms 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.9/10/09 Lecture 5 9Histogram vs Density Curve9/10/09 Lecture 5 10Center 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.9/10/09 Lecture 5 11Mean of a Density Curve9/10/09 Lecture 5 12Spread of a Density Curve• Quartiles can be found by dividing thearea 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 3rdquartile, Q3• The standard deviation of a density curve is denoted by . – Not easy to calculate9/10/09 Lecture 5 139/10/09 Lecture 5 14Normal Distribution• Pictorially speaking, a Normal Distribution is a distribution that has a symmetric, unimodaland bell-shaped density curve.• The mean and standard deviation completely specify the curve.• The mean, median, and mode are the same.9/10/09 Lecture 5 15• The height of a normal density curve at any point x is given by2)(2121)(xexfis the meanis the standard deviation 9/10/09 Lecture 5 16Change of Curvature9/10/09 Lecture 5 17Example: The normal distribution is the most important distribution in Statistics. Typical normal curves with different sigma (standard deviation) values are shownbelow.9/10/09 Lecture 5 18Examples with approximate Normal distributions• Height• Weight• IQ scores• Standardized test scores• Body temperature• Repeated measurement of same quantity• …9/10/09 Lecture 5 19The 68-95-99.7 Rule239/10/09 Lecture 5 20The 68-95-99.7 Rule9/10/09 Lecture 5 21Example: Young Women’s Height• The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches.9/10/09 Lecture 5 22Example: 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?9/10/09 Lecture 5 23Standardizing and z-Scores• an observation x comes from a distribution with meanand 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?,xz9/10/09 Lecture 5 24Example: 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?9/10/09 Lecture 5 25Effects 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.9/10/09 Lecture 5 26The standard normal distribution• The standard normal distribution is the normal dist. with mean 0 and standard deviation 1, denoted asN(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?9/10/09 Lecture 5 27Take 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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