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Chapter 6: Normal Distributions
6.1 Normal Distribution
Learning Activity 6.1-1 Plotting a normal curve
 Open z-plot.xls
 Click on the chart to see the source data
 Click in one of the cells in column C. See Excel
formula. f ( y )  1 exp(  1 ( y   )2 ) Z  (Y   ) /  f ( z)  1 exp(  z 2 )
2 
 2
2
Learning Activity 6.1-2 Practice with z-values
Open z-demo.xls!start.
 Calculate the mean and s.d of the X values
 In column C calculate the z-value for each of the X
values. Calculate the mean and s.d. of the z-values.
 Click on the z-demo worksheet tab. This worksheet
generates random numbers for the X values.
z-values: z= (X – mean)/s.d.
2
Example 1
IQ scores are close to normally distributed and standardized
to have mean  = 100 and a standard deviation of  = 16. If
you take an IQ test and get a score of 125, what is your
percentile, or what proportion of the population would be
below you?
Calculate z = (125 - 100)/16 = 1.56
Look up Tables.xls!Normal +z, and you find 0.94.
That is you are at the 94th percentile, or 94% of the population
would be at or below your IQ score of 125.
The normal probabilities are valid only to which population is
normal. Do a frequency distribution to see if it looks normal.
If a distribution has a peak in the middle and sort of tapers down
at both ends, it probably will be called approximately normal.
Benchmark z-values
The z-values corresponding to 1%, 5%, and 10%.
Probability one-tail
two-tail
0.1/.9
+ or -1.28
+ and -1.645
0.05/.95
+ or -1.645 + and -1.96
0.01/.99
+ or -2.33
+ and -2.58
(See Tables.xls!Benchmark_z_values.)
Learning Activity 6.1-5 Important z-values
Verify these values by
(1) Normal curve table
(2) Excel function NORMSINV(p) to find the z value
corresponding to P(Z < z) = p.
(3) MegaStat | Probability | Normal Distribution
Empirical Rule
What percent of a normal distribution falls within 1 s.d. of
the mean? That is the probability between z = -1 and z = +1.
From Tables.xls,
P(Z < 1) – P(Z < -1) = 0.8413-0.1587 = 0.6827 (68%)
Similarly, the probability within 2 s.d. of the mean is
P(Z < 2) – P(Z < -2) = 0.9772-0.0228 = 0.9544 (95%)
The probability with 3 s.d. of the mean is 0.9973.
Use MegaStat|Probability|Normal Distribution to show
the following figues.
(Mark Calcculate z given P, shading: upper/lower,
Color: Transparent, and Overlay)
Approximately 68% of the area is within +/-1
Approximately 95% of the area is within +/-2
Almost all of the area is within +/-3
 Calculate the values below using the three methods:
 Normal curve table
 Excel functions: NORMSDIST(z), or
NORMDIST(X,mean,stdev,1)
 MegaStat | Probability | Normal Distribution
Learning Activity 6.A-1 Practice with the normal distribution
Assume an examination has a mean of 84.2 and a standard
deviation of 10.9.
 What is the probability of being above 100?
 What is the probability of being between 77 and 97?
 If an examination score has to be at the 94th percentile to
be an A, what score would that be?
See Normal_practice.xls
Relationship between the normal and binomial distributions
As n approaches infinity, the binomial distributions are
mathematically equivalent.
Plot (1) the binomial probabilities with n = 1000 and p = 0.2
(2) the binomial probabilities with n = 1000 and p = 0.5
Binomial distribution (n = 1000, p = 0.2)
Binomial distribution (n = 1000, p = 0.5)
0.04
0.03
0.03
0.03
0.02
P(X)
0.02
0.02
0.02
X
X
571
562
553
544
535
526
517
508
499
490
481
472
463
454
445
436
262
255
248
241
234
227
220
213
206
199
192
185
178
171
0.00
164
0.00
157
0.01
150
0.01
427
0.01
0.01
143
P(X)
0.03
Example 2
If a university wanted to accept students at the 86th percentile
on an entrance examination that had a mean of 400 and a
standard deviation of 60, to what score would that
correspond?
From tables.xls!Normal +z, we see that
P(Z < 1.08) = 0.8599
Hence,
a z-value of 1.08 corresponds to
1.08*60 + 400 = 464.8
Using Excel Functions
To calculate normal probability
NORMSDIST(z)
NORMDIST(Y, mean, stdev, true/false)
To calculate z or Y given a probability
NORMSINV(p)
NORMINV(p, mean, stdev)
NORMINV(0.86, 400,60) = 464.8