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Name: _______________________
Class: __________
Date:_____________
Math SL: 24B Normal Distribution
Review:
25
1. The mean of the population x1, x2, ........ , x25 is m. Given that
x
i
= 300 and
i 1
25
 ( x – m)
i
2
= 625, find
i 1
(a)
the value of m;
(b)
the standard deviation of the population.
2. A bag contains 10 red balls, 10 green balls and 6 white balls. Two balls are drawn at
random from the bag without replacement. What is the probability that they are of different
colors?
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24 B Normal Distributions
Today’s Objectives:
(1) to understand the nature of a continuous probability density function
(2) to define the mean m or expected value E(X)
(3) to understand how the normal distribution can be used to determine
probabilities
(4) to understand the significance of mean and standard deviation for a normal
distribution
Background ONLY: [24A] Continuous Probability Density Function
For a continuous random variable X, x can take any value. Consequently, we use a
function to describe the probability distribution and that function is called the
probability density function. We also need to have an understanding of calculus.
You are not required to use calculus this year.
What is the symbol for an integral? _____________
An integral is used to determine the _________ under a curve.
A continuous probability density function (pdf) is a function f (x) such that:
f (x) ³ 0 on a given interval a £ x £ b and
ò
b
a
f (x) dx = 1
Similarly, the sum of all probabilities of a discrete random variable always equals __.
Probabilities are areas under the curve over a certain interval: i.e.
d
P(c £ X £ d) = ò f (x) dx
c
Since the variable is continuous, it is important to realize that the probability of X
taking any exact value is zero. Why?? (Hint: What is the area?)
Instead, we can only measure the probability of X being in a particular interval.
Therefore, it doesn’t make any difference whether we use the equality sign with our
inequality signs.
This means P(c £ X £ d) = P(c £ X < d) = P(c < X £ d) = P(c < X < d)
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These probabilities can be calculated quickly on the calculator or by hand using
calculus. Both methods will be revisited at a later date.
The mean m or expected value E(X) of a continuous pdf is defined as:
b
m = ò x f (x) dx where a £ x £ b is the domain of f (x)
a
Unfortunately, we are currently not able to calculate any E(X) values when X is
continuous, but we should note the similarity with E(X) when X is discrete. These
examples will also have to be revisited after our calculus unit.
[24B] Normal Distribution
The normal distribution is the most important distribution in statistics. It models
many naturally occurring variables. For example,
 Student exam scores
 Time to complete an assignment
 Reaction times
 IQ scores
 Different times when flowers were fertilized
Most of these measurements are close to the mean, but there are always a few much
higher or much lower. This results in the bell-shaped normal distribution.
Rather than a single normal curve, there is a family of curves each defined by the
parameters m and s 2 :
m = mean and s 2 = variance.
These three curves have the same
s but different means.
These three curves have the same
mean but different s .
While these curve have different means and standard deviations they all have the
same characteristics.
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Characteristics of the Normal Probability Density Function
 These curves are all symmetrical about the mean.
 More scores are distributed closer to the mean than further away. This results in
the typical bell-shaped curve.
 As x ® ±¥ the normal curve approaches the x-axis.
 f (x) > 0 for all x.
¥
 The total area under the curve is always one, so
ò
Review: The Geometrical Significance of m and
s
-¥
f (x) dx = 1.
The line x = m marks the line of symmetry of the normal probability distribution
function. For a given normal curve, the points of inflection are one standard
deviation either side of the mean, i.e. at x = m + s and x = m - s .
For any normal distribution with mean m and standard deviation s , the
proportional breakdown of where the random variable could lie is constant. In
particular:
» 68.26% of values lie between m - s and m + s
» 95.44% of values lie between m - 2s and m + 2s
» 99.74% of values lie between m - 3s and m + 3s
**Note** If a normal distribution question appeared on a non-calculator test, then it
would have to refer to the above percentages.
Finding Probabilities by Graphics Calculator
We can use a graphics calculator to quickly find probabilities for a normal
distribution. Remember that probabilities for continuous variables can only be
found for a particular interval.
normalcdf(lower, upper, mean, standard deviation)
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Example 1: X is a random variable that is normally distributed with mean 70 and
standard deviation 4. Draw a normal curve to represent each probability. Then, if
possible, solve without a calculator. Lastly, use a calculator to find the probabilities.
a) P(70 £ X £ 74)
b) P(68 < X £ 72)
c)
P( X < 65)
d) P( X ³ 60)
Hmwk#55:
24B Normal distributions
pg. 651 # 4 – 9
pg. 653 # 2, 3(a,b)
USE GRAPHING CALCULATOR and always draw a picture where possible.
By now, most of you should have an idea of what you want
to explore. Take this opportunity to look for relevant
math. .
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