Daniela Stan Raicu School of CTI, DePaul University CSC 323 Quarter: Spring 02/03 Daniela Stan Raicu School of CTI, DePaul University 11/28/2018 Daniela Stan - CSC323

Outline Normal Distributions The Empirical (68-95-99.7) rule Standard Normal Distribution Normal Distributions Calculations Introduction to the Statistical Software SAS 11/28/2018 Daniela Stan - CSC323

The normal distributions Normal curves are density curves that are: Symmetric Unimodal Bell-Shaped 11/28/2018 Daniela Stan - CSC323

The normal distributions (cont.) A normal distribution is specified by: Mean  Standard Deviation  Notation: N(, ) The equation of the normal distribution ( gives the height of the normal distribution) : f(x) Example of two normal curves specified by their mean and standard deviation 11/28/2018 Daniela Stan - CSC323

The Empirical Rule for Any Normal Curve 68% 95%  +1*  -1*  -2*   +2*  99.7% -3*   +3*  11/28/2018 Daniela Stan - CSC323

The 68-95-99.7 (empirical) rule In the normal distribution N(, ): Approximately 68% of the observations are between -  and +  Approximately 95% of the observations are between - 2 and + 2 Approximately 99.7% of the observations are between - 3 and + 3 11/28/2018 Daniela Stan - CSC323

Example The heights of adult women in the United States follow, at least approximately, a bell-shaped curve. What do you think that means? The most adult women are clumped around the average, with numbers decreasing the farther values are from the average in either direction. Health and Nutrition Examination Study of 1976-1980 (HANES) The average of the heights of adult women is  = 65 and the standard deviation is  = 2.5. What does the 68-95-97.7 rule imply? From Seeing Through Statistics, 2nd Edition by Jessica M. Utts. 11/28/2018 Daniela Stan - CSC323

68% of adult women have heights between 62.5 and 67.5 inches; The empirical rule: 65-2.5 65 65+2.5 68% of adult women have heights between 62.5 and 67.5 inches; 95% of adult women have heights between 60 and 70 inches; 99.7% of adult women have heights between 67.5 and 72.5 inches 11/28/2018 Daniela Stan - CSC323

? Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of individuals fall into any range of values? Example: What proportion of men are less than 68 inches tall? At what percentile a given individual falls, if you know their values What value corresponds to a given percentile ? 68 70 (height values) 11/28/2018 Daniela Stan - CSC323

Standardized score A “standardized score” is simply the number of standard deviations an individual falls above or below the mean for the whole group. Values above the mean have positive standardized scores; values below the mean have negative ones. Example: Females (ages 18-24) have a mean height of 65 inches and a standard deviation of 2.5 inches. What is the standardized score of a a women who is 67.5 inches tall? Standardized score: = (67.5 – 65)/2.5=1 From Seeing Through Statistics, 2nd Edition by Jessica M. Utts. 11/28/2018 Daniela Stan - CSC323

Standardized Scores standardized score = (observed value - mean) / (std dev) z is the standardized score x is the observed value m is the population mean s is the population standard deviation 11/28/2018 Daniela Stan - CSC323

The standard normal distribution The standard normal distribution N(0,1) is the normal distribution with mean 0 and standard deviation 1 If a variable X has any normal distribution N(, ), then the standardized variable Z has the standard normal distribution N(0,1). Why are normal distributions so important? Many statistical inference procedures based on normal distributions work well for other roughly symmetric distributions. They are good descriptions for real data 11/28/2018 Daniela Stan - CSC323

Normal distribution calculations Example: The heights of young women are approximately normal with mean =64.5 inches and =2.5 inches. What is the proportion of women how are less than 68 inches tall? 1. State the problem: X = height, X < 68 2. Standardize: 68 standardized to 1.4 X<68 Z < 1.4 11/28/2018 Daniela Stan - CSC323

Normal distribution calculations 3. What proportion of observations/women on the standard normal variable Z take values less than 1.4? Table entry is area to the left of z Area (Z<1.4)=.9192 Table A at the end of the book gives areas (proportions of observations) under standard normal curve. 11/28/2018 Daniela Stan - CSC323