4.4 The Normal Distribution: Introduction
Introduction
In many cases, measurements will tend to cluster near the mean so that many points lie
near the mean and fewer and fewer points lie further and further from the mean.
In these cases, the frequency distribution looks something like this:
The frequency is high near the mean,
And then frequencies taper off on both
sides.
Normal distribution: A distribution is a normal distribution if it is represented by an
ideal bell-shaped curve. (Frequencies are highest in the middle and taper symmetrically
as values move away from the center).
Other types of distributions:
Uniform
Left-tailed
Right-tailed
Normal Distribution
The normal distribution is an idealized bell-shaped curve which represents the frequency
of the data verses the datavalue. The center of the curve is the mean  and the std dev is
.
A normal distribution described an idealized case in which the mean, median and mode
are all equal and the frequencies on either side of the peak are perfectly symmetric. The
tails of the curve extend forever, from - to +.
68.3% of the data points lie within 1 standard deviation of the mean
95.44% of the data points lie within 2 standard deviations of the mean
99.74% of the data points lie within 3 standard deviations of the mean
Example: Book #10
A population is normally distributed with mean 18.9 and standard deviation 1.8.
a. Find the intervals representing 1, 2 and 3 standard deviations of the mean.
1 st dev: [18.9-1.8,18.9+1.8] = [17.1,20.7]
2 st dev: [18.9-2*1.8,18.9+2*1.8] = [15.3,22.5]
3 st dev: [18.9-3*1.8,18.9+3*1.8] = [13.5,24.3]
b. What percentage of the data lies in each of the intervals in part (a)?
68.3% 95.44% 99.74%
c. Draw a sketch of the bell curve.
The Shape of a Normal Distribution: Narrow or Spread-Out
The mean is the location of the
highest frequency.
The standard deviation determines
how narrow or spread out it is.
The larger the standard deviation,
the more spread out it is.
The Standard Normal Distribution (z-distribution)
Consider a normal distribution with mean  and std dev . If the mean is 0 and the
standard deviation is 1 we call it the standard normal curve, or the z-distribution.
Using the Z-Distribution To Determine Probabilities
The Area Under the Curve Represents The Probability of That Region
The area under the curve between points a and b corresponds to the probability that a
randomly selected datapoint z will be between a and b
(1) The total area under the curve is 1: prob that x is anywhere between - and + is 1.
(2) The total area of the entire right side is .5: prob that z>=.5
(3) The total area of the entire left side is .5: prob that z<=.5
For other regions, we need to use a body table to determine the probability.
Worksheet: Normal Distributions
Any Normal Distribution can be converted to a z-distribution
We convert to the standard normal curve by taking each datapoint x and normalizing it as
z = (x- μ) / σ.
Example:
D = {8 10 12}
mean = 30/3 = 10
st dev = 2
x=8: z=(x-10)/2 = -1
x=10: z=(x-10)/2 = 0
x=12: z=(x-10)/2 =1
D2 = {-1,0,1}
mean = 0
st dev = 1
Thus to study the normal distribution, you only have to study the standard normal
distribution in which the mean=0 and the standard dev=1.
Technical details:
To be perfect, the normal distribution must describe the frequency of a continuous rather
than discrete variable.
Continuous variable: measurements (heights of students in a class)
Discrete variable: whole numbers (# of times you flip heads when you flipping
coins)
Distributions for discrete variables can approximate normal distributions.
However, a normal curve is an ideal, perfectly smooth curve while discrete variables will
result in jagged edges.
Finding Probabilities of Regions under the Standard Normal Curve
1. Shade in the region on a graph of the z-distribution, and identify the sub-regions.
2. Calculate the probability of each sub-region and add the shaded probabilities
together.
The Probabilities of 3 Types of Regions: Bodies, Tails, Body And Tail Regions
Body Region of Length z: A body region is [0,z] (the interval from 0 to z under the bell
curve)

You will need to be able to use a body table to determine the probability that a
variable is between 0 and z.
Tail Region of Body Region of Length z: A tail region is the region not included in the
body region (the interval from z to ∞ under the bell curve).
Body and Tail Region: Including the body and tail together make half the area under the
bell curve, and has probability .5
The body table for the standard normal distribution provides the probability for body
regions of length z only. To calculate tail regions, you need to subtract the body region
from the body and tail region.

You need to be able to use a body table to determine the probability that a value will
lie in an interval [a,b].