Density Curve (p52) Density curve is a curve that - is always on or above the horizontal axis.

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1 1.3 Density curves p50 Some times the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. It is easier to work with a smooth curve, because the histogram depends on the choice of classes. Density Curve (p52) Density curve is a curve that - is always on or above the horizontal axis. - has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. 1

2 Example: The curve below shows the density curve for scores in an exam and the area of the shaded region is the proportion of students who scored between 60 and 80. 2

3 The median of a distribution is the point that divides the area under the curve in half. (p54) A mode of a distribution described by a density curve is a peak point of the curve. (p53) Quartiles of a distribution can be roughly located by dividing the area under the curve into quarters as accurately as possible by eye.(p53) 3

4 (Hint: Recall the formula to calculate the area of a triangle) Normal distributions p54 4

5 An important class of density curves are the symmetric unimodal bell-shaped curves known as normal curves. They describe normal distributions. All normal distributions have the same overall shape. The density curve for a particular normal distribution is specified by giving the mean and the standard deviation. The mean is located at the center of the symmetric curve and is the same as the median (and mode). Changing without changing moves the normal curve along the horizontal axis without changing spread. 5

6 The standard deviation controls the spread of a normal curve. P55 There are other symmetric bell- shaped density curves that are not normal. P55 The normal density curves are specified by a particular equation. The height of a normal density curve at any point x is given by 1 1 e 2 2 x 2 6

7 The rule p56 In the normal distribution with mean m and std. deviation s, Approx. 68% of the observations fall within of the mean µ. Approx. 95% of the observations fall within 2 of the mean µ. Approx. 99.7% of the observations fall within 3 of the mean µ. Example (1.34, p57) The distribution of heights of women aged is approx. normal with mean µ = 64.5 inches and std. dev. = 2.5 inches. 2 = 5 inches. The rule says that the middle 95% (approx.) of women are between to inches tall. The other 5% have heights outside the range from to inches. 7

8 2.5% of the women are taller than Ex. 1) The middle 68% (approx.) of women are between to inches tall. 2) % of the women are taller than 67 3) % of the women are taller than 72 8

9 Notation: A normal distribution with mean and std. dev. is denoted by N(,). The distribution of women s heights is N(64.5, 2.5). Standardizing and z-scores p58 If x is an observation from a distribution that has mean and std. dev., the standardized value of x is z x A standardized value is often called a z-score. A z-score tells us how many standard deviations the original observation falls away from the mean. 9

10 Example (1.35 p58) heights of women are approx.. normal with mean = 64.5 inches and std. dev. = 2.5 inches. The standardized height is z height64.5 The standardized value 2.5 (z-score) of height 68 inches is z or 1.4 std. dev. above 2.5 the mean. A woman 60 inches tall has standardized height z or 1.8 std. dev. 2.5 below the mean 10

11 The Standard Normal distribution p59 The standard normal distribution is the normal distribution N(0, 1). If a random variable X has normal distribution N(, ), then the standardized variable z x has the standard normal dist. 11

12 The standard normal tables p61 Examples What proportion of observations of a std. normal variable Z takes values a) less than 1.4? b) greater than 1.4? c) greater than 1.96 d) between 0.43 and 2.15 e) between 0.92 and

13 Ex. X~N(65, 15). What proportion of observations of X takes values? less then 50 greater than 80 between 50 and 80 13

14 Example Scores on SAT verbal test follow approx. the N(505, 110) dist. How high must a student score in order to place in the top 10% of all students taking the SAT? 14

15 Normal quantile plots p65 A histogram or stem plot can reveal distinctly nonnormal features of a distribution. If the stemplot or histogram appears roughly symmetric and unimodal, we use another graph, the normal quantile plot as a better way of judging the adequacy of a normal model Use of normal quantile plots. P65 If the points on a normal lie close to a straight line, the plot indicates that the data are normal. Outliers appear as points that are far away from the overall pattern of the plot. 15

16 Histogram and the nscores plot for data generated from a normal distribution ( N(500, 20)) (for 1000 observations) 16

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18 Histogram and the nscores plot for data generated from a right skewed distribution 18

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20 Histogram and the nscores plot for data generated from a left skewed distribution 20

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