the normal distributions psls chapter 11 © 2009 w.h. freeman and company
TRANSCRIPT
The Normal distributions
PSLS chapter 11
© 2009 W.H. Freeman and Company
Objectives (PSLS 11)
The Normal distributions
Normal distributions
The 68-95-99.7 rule
The standard Normal distribution
Using the standard Normal table (Table B)
Inverse Normal calculations
Normal distributions
Normal curves are used to model many biological variables.
They can describe the population distribution or density curve.
Normal – or Gaussian – distributions are a family of symmetrical, bell
shaped density curves defined by a mean (mu) and a standard
deviation (sigma): N().
xx
2
2
1
2
1)(
x
exf
Human heights, by
gender, can be modeled
quite accurately by a
Normal distribution.
0
2
4
6
8
10
12
14
16
18
unde
r 56 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
72 o
r m
ore
Height (inches)
Per
cen
t
Guinea pigs survival times
after inoculation of a pathogen
are clearly not a good candidate
for a Normal model!
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
A family of density curves
Here means are different
( = 10, 15, and 20) while
standard deviations are the same
( = 3)
Here means are the same ( = 15)
while standard deviations are
different ( = 2, 4, and 6).
mean µ = 64.5 standard deviation = 2.5
N(µ, ) = N(64.5, 2.5)
The 68–95–99.7 rule for any N(μ,σ)
Reminder: µ (mu) is the mean of the idealized curve, while is the mean of a sample.
σ (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample.
About 68% of all observations
are within 1 standard deviation
(of the mean ().
About 95% of all observations
are within 2 of the mean .
Almost all (99.7%) observations
are within 3 of the mean.
Inflection point
x
Because all Normal distributions share the same properties, we can
standardize to transform any Normal curve N() into the standard
Normal curve N(0,1).
The standard Normal distribution
For each x we calculate a new value, z (called a z-score).
N(0,1)
=>
z
x
N(64.5, 2.5)
Standardized height, standard deviation units
z (x )
A z-score measures the number of standard deviations that a data
value x is from the mean .
Standardizing: calculating z-scores
When x is larger than the mean, z is positive.
When x is smaller than the mean, z is negative.
1 ,
zxfor
When x is 1 standard deviation larger
than the mean, then z = 1.
222
,2
zxfor
When x is 2 standard deviations larger
than the mean, then z = 2.
mean µ = 64.5"
standard deviation = 2.5"
height x = 67"
We calculate z, the standardized value of x:
( ) (67 64.5) 2.5
, 1 1 stand. dev. from mean2.5 2.5
xz z
Given the 68-95-99.7 rule, the percent of women shorter than 67” should be,
approximately, .68 + half of (1 - .68) = .84 or 84%. The probability of randomly
selecting a woman shorter than 67” is also ~84%.
Area= ???
Area = ???
N(µ, ) = N(64.5, 2.5)
= 64.5” x = 67”
z = 0 z = 1
X = Women heights follow the N(64.5”, 2.5”)
distribution. What percent of women are
shorter than 67 inches tall (that’s 5’7”)?
X
Z
Using Table B
(…)
Table B gives the area under the standard Normal curve to the left of, i.e., less
than, any z value.
.0062 is the area under
N(0,1) left of z = -
2.50
.0060 is the area under
N(0,1) left of z = -2.51
0.0052 is the area under
N(0,1) left of z = -2.56
P{X < 67} = P{Z < 1} = Area ≈ 0.84
P{X > 67} = P{Z > 1} = Area ≈ 0.16
N(µ, ) = N(64.5, 2.5)
= 64.5 x = 67 z = 1
84.13% of women are shorter than 67”.
The complementary, or 15.87% of women
are taller than 67" (5'6").
For z = 1.00, the area
under the curve to the
left of z is 0.8413, i.e.,
P{Z < 1.00} = 0.8413
Tips on using Table B
Because of the curve’s symmetry,
there are 2 ways of finding the
area under N(0,1) curve to the
right of a z value.
area right of z = 1 - area left of z
Area = 0.9901
Area = 0.0099
z = -2.33
area right of z = area left of -z
More tips on using Table B
To calculate the area between 2 z- values, first get the area under N(0,1)
to the left for each z-value from Table B.
area between z1 and z2 =
area left of z2 – area left of z1
Don’t subtract the z values!!!
Normal curves are not square!
Then subtract the
smaller area from the
larger area.
The area under N(0,1) for a single value of z is zero
Inverse Normal calculations
You may also seek the range of values that correspond to a given
proportion/ area under the curve. For that, use Table B backward:
first find the desired
area/ proportion in the
body of the table,
then read the
corresponding z-value
from the left column and
top row.For a left area of 1.25 % (0.0125),
the z-value is -2.24
25695.255
)15*67.0(266
)*()(
x
x
zxx
z
Vitamins and better food: The lengths of pregnancies when malnourished mothers
are given vitamins and better food is approximately N(266, 15). How long are the
75% longest pregnancies in this population?
?
upper 75%
The 75% longest pregnancies in this
population are about 256 days or longer.
We know μ, σ, and the area
under the curve; we want x.
Table B gives the area left of z
look for the lower 25%.
We find z ≈ -0.67
Checking your cholesterol
High levels of total serum cholesterol
increase the risk of cardiovascular disease.
Cholesterol levels above 240 mg/dl demand
medical attention because they place the
subject at high risk of CV disease.
In the hope of extending treatment benefits to patients with early disease,
various professional societies have recommended a lower threshold value
for diagnosis.
Levels above 200 mg/dl are considered elevated cholesterol and may place
the person at some risk of cardiovascular disease.
The cholesterol levels for women aged 20 to 34 follow an approximately Normal
distribution with mean 185 mg/dl and standard deviation 39 mg/dl.
What is the probability that a young woman has high cholesterol (> 240 mg/dl)?
What is the probability she has an elevated cholesterol (between 200 and 240)?
68 107 146 185 224 263 30229 68 107 146 185 224 263 302 341
39
x zarea left
area right
240 1.41 92% 8%
200 0.38 65% 35%
The blood cholesterol levels of men aged 55 to 64 are approximately Normal with
mean 222 mg/dl and standard deviation 37 mg/dl.
What percent of middle-age men have high cholesterol (> 240 mg/dl)?
What percent have elevated cholesterol (between 200 and 240 mg/dl)?
111 148 185 222 259 296 33329 68 107 146 185 224 263 302 341
37
x zarea left
area right
240 0.49 69% 31%
200 -0.59 28% 72%