1 2 22S 30 105 Statistical Methods and Computing Using density curves to describe the distribution of values of a quantitative variable Normal Distributions Imagine the heights of 100 000 men who completed physical exams as part of a national health survey Lecture 4 January 27 2006 We might make relative frequency histograms of these height data using successively smaller width intervals Kate Cowles 374 SH 335 0727 kcowles stat uiowa edu 3 4 Density curve a curve that describes the overall pattern of a distribution total area under a probability density curve is 1 0 the curve never drops below the horizontal axis 5 Measures of center and spread can be used to describe density curves 6 Normal distributions characterized by a symmetric smooth bell shape To distinguish between these measures in the idealized curve vs in actual sample data we use different symbols also called Gaussian distributions after Karl Gauss for the mean of a density curve for the standard deviation of a density curve The normal distribution is a mathematical model that provides a good representation of the values of many kinds of real quantitative variables Analogy No room is perfectly rectangular in shape but the geometric model of a rectangle is good enough to enable you to measure the room and buy the right amount of carpet 7 Some characteristics of the normal distribution 1 Normal distribution is symmetric a The proportion of the values of a normal random variable that are less than z is equal to the proportion of the values that that are greater than z b The proportion of the values of a normal random variable that are less than z is equal to the proportion of the values that that are greater than z c The mean is equal to the median 8 2 There are lots of different normal distributions defined by different values of and The values of and completely determine the normal distribution When and are known the proportion of population values in any interval can be evaluated 3 If remains fixed but changes the density of the random variable remains the same shape but its location changes 4 If remains fixed but changes the density of the normal random variable has the same location but its shape changes 9 10 The 68 95 99 7 Rule Example of the 68 95 99 7 rule In the normal distribution with mean and standard deviation The distribution of systolic blood pressure in 18 to 74 year old males in the US is approximately normal with mean 129 mm of mercury and standard deviation 20 mm of mercury 68 of the observations fall within of the mean 95 of the observations fall within 2 of the mean 99 7 of the observations fall within 3 of the mean 11 The standard normal distribution The standard normal distribution is the normal distribution with 0 1 The name Z is often used for a variable that has the standard normal distribution For this particular normal distribution 12 Using tables of the standard normal distribution What if we wanted to know what proportion of values of a standard normal variable Z were less than some particular value Suppose we live in a particular Scandinavian city where temperature is measured in Centigrade Weather records kept for many years indicate that the temperature at 11 00 a m on Jan 28 follows a standard normal distribution 13 We want to know in what proportion of years we can expect the temperature at this time to be less than or equal to 1 5 C 14 What if we instead wanted to know the proportion of years with temperature 1 75 Remember that the total area under the normal curve is 1 0 We could use Table A in your textbook The proportion is 0 0668 Similarly the proportion of years we can expect the temperature at this time to be 1 75 C is 0401 15 Standardizing values from other normal distributions All normal distributions would be the same if we measured in units of size around the mean as center If x is an observation from a distribution that has mean and standard deviation the standardized value of x is z x Standardized values are often called z scores What if we instead wanted to know the proportion of years with temperature 1 75 Remember symmetry of the normal distribution 16 z scores tell how many standard deviations the original observation is away from the mean of the distribution and in which direction If the z score is positive the original observation was larger than the mean If the z score is negative the original observation was smaller than 17 Example of z scores Recall that the distribution of systolic blood pressure of men aged 18 74 is approximately normal with 129 mm Hg and 20 mm Hg The standardized height is z sbp 129 20 If a man has sbp 157 mm Hg his standardized sbp is z Using the standard normal distribution to compute proportions for other normal distributions Let s use the symbol X for a variable representing the systolic blood pressure of men What proportion of men have sbp 100 If a man has sbp 100 his standardized sbp is z 157 129 1 4 20 If a man has sbp 93 mm Hg his standardized sbp is z 18 93 129 1 8 20 100 129 1 45 20 According to Table A the proportion of values of a standard normal variable that are less than or equal to 1 45 is 0 0735 This proportion is the same as the proportion of X values that will be less than 100 19 General procedure for finding normal proportions 1 State the problem in terms of the observed variable X 2 Standardize the value of interest x to restate the problem in terms of a standard normal variable Z You may then wish to draw a picture to show the area under the standard normal curve 3 Find the required area under the standard normal curve using Table A and remembering The total area under the curve is 1 0 The normal distribution is symmetric