Transcript Normal Distribution

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 “bellshaped”, 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.
2.
3.
4.
•
http://www.stat.psu.edu/~resources/ClassNotes/ljs_08/ind
ex.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…