Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 1 of 11 Chapter 7 Section 4 Assessing Normality

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 2 of 11 Chapter 7 – Section 4 ●Learning objectives  Draw normal probability plots to assess normality 1

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 3 of 11 Chapter 7 – Section 4 ●Many real world variables have bell shaped histograms, so we would say that they should or could have normal probability distributions ●We need methods to assess whether this is a good assumption or not

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 4 of 11 Chapter 7 – Section 4 ●The main method used to assess whether sample data is approximately normal is the normal probability plot ●This plot graphs the observed data, ranked in ascending order, against the “expected” Z-score of that rank

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 5 of 11 Chapter 7 – Section 4 ●The chart compares  The lowest observed value with where it is expected to be (according to the normal)  The second lowest observed value with where it is expected to be (according to the normal)  Etc.  The highest observed value with where it is expected to be (according to the normal)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 6 of 11 Chapter 7 – Section 4 ●The expected lowest value, the expected second lowest value, etc. are not easy to derive ●Technology should be used to construct these graphs ●If the sample data was taken from a normal random variable, then this plot should be approximately linear

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 7 of 11 Chapter 7 – Section 4 ●Excel  The PHStat add-in to Excel includes this plot  It is also an option under the Regression package, but the axes are linear percents (unlike MINITAB and StatCrunch) … that can be changed manually ●Excel  The PHStat add-in to Excel includes this plot  It is also an option under the Regression package, but the axes are linear percents (unlike MINITAB and StatCrunch) … that can be changed manually ●StatCrunch  The option Graph – QQ Plot in StatCrunch creates normal probability plots (also called QQ plots)  The StatCrunch axes are switched compared to the MINITAB axes

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 8 of 11 Chapter 7 – Section 4 ●The two plots below are for the data in Table 6  One using MINITAB (from the text) ●The two plots below are for the data in Table 6  One using MINITAB (from the text)  One using StatCrunch (the axes are switched)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 9 of 11 Chapter 7 – Section 4 ●Both of these show that this particular data set is far from having a normal distribution  It is actually considerably skewed right

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 10 of 11 Chapter 7 – Section 4 ●The plot below is from Excel’s Data Analysis – Regression package, with the horizontal axes modified to be normal quantiles instead of linear percents

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 11 of 11 Summary: Chapter 7 – Section 4 ●We can assess whether sample data is approximately normal by using the normal probability plot ●If the data is approximately normal, then the normal probability plot (a.k.a. the QQ plot) should be approximately normal also

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 12 of 11 Example: Chapter 7 – Section 4 ●Would this be approximately normal?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 13 of 11 Example: Chapter 7 – Section 4

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 4 – Slide 14 of 11 Example: Chapter 7 – Section 4