Normal Distribution• Recall how we describe a distribution of data:– plot the data (stemplot or histogram)– look for the overall pattern (shape, peaks, gaps) and departures from it (possible outliers)

– calculate appropriate numerical measures of center and spread (5-number summary and/or mean & s.d.)

– then we may ask "can the distribution be described by a specific model?" (one of the more common models for symmetric, single-peaked distributions is the normal distribution having a certain mean and standard deviation)

– can we imagine a density curve fitting fairly closely over the histogram of the data?• a density curve is a curve that is always on or above the horizontal axis (>= 0) and whose total area under the curve is 1

• An important property of a density curve is that areas under the curve correspond to relative frequencies - see figures below.

rel. freq=287/947=.303 area = .293

• Note the relative frequency of vocabulary scores <= 6 is roughly equal to the area under the density curve <= 6.

• We can describe the shape, center and spread of a density curve in the same way we describe data… e.g., the median of a density curve is the “equal-areas” point - the point on the horizontal axis that divides the area under the density curve into two equal (.5 each) parts. The mean of the density curve is the balance point - the point on the horizontal axis where the curve would balance if it were made of a solid material.

• For a normal density curve we see the characteristic “bell-shaped”, symmetric curve with single peak (at the mean value ) and spread out according to the standard deviation ()

x=seq(-6,6,.01)

plot(x,dnorm(x),col="red")

lines(x,dnorm(x,sd=2.5),col="black")

dnorm(x) gives the value of the standard normal density curve at the point x - change the mean & sd as arguments…

• The 68-95-99.7 Rule describes the relationship between and . See Figure below -

• How many different normal curves are there? Ans: One for every combination of values of and …but they all are alike except for their and . So we take advantage of this and consider a process called standardization to reduce all normals to one we call the Standard Normal Distribution.

• Denote a normal distribution with mean and standard deviation by N(). Let X correspond to the variable whose distribution is N(). We may standardize any value of X by subtracting and dividing by - this re-writes any normal into a variable called Z whose values represent the number of standard deviations X is away from its mean. The standardized value is sometimes called a z-score.

• If X is N(), then Z is N(0,1), where Z=(X-)/.• We can find areas under Z from the Standard Normal Table

(Table A in the Book), and these areas equal the corresponding areas under X. See the next example . . .

• Consider this example: Let X=height (inches) of a young woman aged 18-24 years. Then X is ~N(64.5", 2.5"). – What proportion of these women's heights are between 62" and 67"?

– What proportion are above 67"? Below 72"?– What proportions of these women's heights are between 61" and 66"? NOTE: This cannot be solved by the 68-95-99.7 rule…

– What proportion are below 64.5"? Below 68"?– What proportion are between 58" and 60"?– Etc., etc., etc. ….– What height represents the 90th percentile of this aged woman?

• All problems of this type are solvable by sketching the picture, standardizing, and doing appropriate arithmetic to get the final answer…the last question above is what I call a "backwards problem", since you're solving for an X value while knowing an area…

• Do readings & practice before going on - now jump to the last slide!

• We’ve seen examples of data that seem to fit the normal model, and examples of data that don’t seem to fit … Because normality is an important property of data for specific types of analyses we’ll do later, it is important to be able to decide whether a dataset is normal or not. A histogram is one way but a better graphical method is through the normal quantile plot …

• We'll use R to draw a normal quantile plot it will allow us to assess the normality of our data in the following sense:– if the data points fall along the straight line (and within the bands on the plot) then the data can be treated as normal. Systematic deviations from the line indicate non-normal distributions - outliers often appear as points far away from the pattern of the points...

– the y-intercept of the line corresponds to the mean of the normal distribution and the slope of the line corresponds to the standard deviation of the normal distribution

Normal quantile plot of co2 data

Notice the systematic failure of the points to fall on the line, especially at the low end where the data is “piled up”. Also, note the outliers at the high end… Conclusion: Not normal

Normal quantile plot Hrs_Completed in our Stt Class data

Notice that the data points follow the line fairly well in the middle, but the high hours are too high and the low hours are too low for what would be expected of a normal distribution.

Conclusion: Not normal

Normal quantile plot of IQ data

These IQ scores follow the line fairly well, except for the lowest ones, which are lower than what we would expect. Conclusion: Normal; mean ~ 110, sd ~ 10

• Some on-line readings to help with the normal distribution:1. http://www.stat.psu.edu/~resources/ClassNote

s/ljs_08/index.htm (select "View Lecture Notes")

2. http://cnx.org/content/m16979/latest/ (start with Chapter 6, The Normal Distribution, and work through the Homework section.

3. http://www.stat.ucla.edu/textbook/ (check out the readings here in Chapter V - there are also several good examples of how to do these normal computations)

4. http://www-unix.oit.umass.edu/~biep540w/index.html (check out the link to the fifth chapter…)

• There are a total of 19 Homework problems given at the cnx.org site (#2) above. Make sure you can work all those problems. We'll have a quiz on the normal distribution soon…