Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 113

5.1 Normal Distribution Distribution:  Describe how the data might be skewed.  The incomes of people sitting in luxury boxes at the World Series.  The salaries of NFL football players on any particular team. Right skewed, very wealthy (outlier?) Right skewed, some unique players make much more than the majority of their teammates

5.1 Normal Distribution Normal Distribution-  symmetric…  Bell shaped distribution…  Single peak…  Its peak corresponds to the mean, median, and mode of the distribution.  Its variation can be characterized by the standard deviation.  Also called the Gaussian distribution, after the famous mathematician Carl Friedrich GaussCarl Friedrich Gauss

5.1 Normal Distribution Examples Normal Distribution-  Any unimodal symmetric data…  Chest Circumference  Height  Weight  SAT scores  Statistics from large populations…?

5.1 Normal Distribution Thinking test….is the data normal?  11 th Grade Language Arts Test scores  Skewed left  Weights of Great Danes compared to all canines  Skewed right  Ages of individuals in Florida  Relatively normal possibly skewed left  Heart rates of high school students  Normal

5.1 Normal Distribution What is the total relative frequency for any data set? 100% or 1 How does relative frequencies relate to normal distribution? Because the total relative frequency must be 100% or 1, the total area under the normal distribution curve must be 1. Therefore the area that lies under the normal distribution curve corresponding to a range of values on the horizontal axis is the relative frequency of those values. Area Under Curve = 100% or 1

5.1 Normal Distribution ESTIMATING AREAS: In the normal distribution below estimate the percentage under the curve highlighted in red. ≈ 85% to 90% Hence, approximate the area highlighted in blue. ≈ 10% to 15%

5.1 Normal Distribution  More with what makes a variable normally distributed ? 1. The variable is determined by many different factors (i.e. physiological variables or performance variables) 2. The frequencies of the variable cluster around a single peak that is near the mean of the distribution. 3. The frequencies of the variable are symmetric about the peak. 4. Large deviations from the mean are increasingly rare.

5.1 Normal Distribution Let’s try some more… WHAT ABOUT…. Scores on a very difficult exam.. Right Skew, outlier 100% Heights of random women… Normal In remembrance of Apple Crunch day…the number of apples in each of 100 full bushel baskets… Normal

5.1 Normal Distribution SOME FUN…(TECHNOLOGY IS AWESOME!) BellCurves.nbp AreaOfANormalDistribution.nbp SampleVersusTheoreticalDistribution.nbp on=1.6.0_07&browser=MSIE&vendor=Sun_Microsystems_Inc. on=1.6.0_07&browser=MSIE&vendor=Sun_Microsystems_Inc

5.1 Normal Distribution HOMEWORK:  Pg 201 Exercises: # 1-20 all WORK