Normal Distribution

1
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 (gt 0) and whose total
  area under the curve is 1

2

• An important property of a density curve is that
  areas under the curve correspond to relative
  frequencies - see figures below.
• rel. freq287/947.303
  area .293
• Note the relative frequency of vocabulary scores
  lt 6 is roughly equal to the area under the
  density curve lt 6.

3

• 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.

4

• 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 (?)

xseq(-6,6,.01) plot(x,dnorm(x),col"red") lines(x
,dnorm(x,sd2.5),col"black")
dnorm(x) gives the value of the standard normal
density curve at the point x - change the mean
sd as arguments
5

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

6

• 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 . . .

7

• Consider this example Let Xheight (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
  answerthe 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!

8

• Weve seen examples of data that seem to fit the
  normal model, and examples of data that dont
  seem to fit Because normality is an important
  property of data for specific types of analyses
  well 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

9
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
10
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
11
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
12

• Some on-line readings to help with the normal
  distribution
• http//www.stat.psu.edu/resources/ClassNotes/ljs_
  08/index.htm (select "View Lecture Notes")
• http//cnx.org/content/m16979/latest/ (start
  with Chapter 6, The Normal Distribution, and work
  through the Homework section.
• 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)
• 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