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Math 166, Fall 2015, Robert
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3.2 & 3.3- Measures of Central Tendency & Spread
There are three different measures we use when we are interested in what the
“center” of our data looks like:
The mean or average of a set of data, often denoted µ or x̄, of the n numbers
x1 , x2 , . . . , xn is given by
µ=
The median of a set of data is the number in the middle when the data is
arranged from largest to smallest number. If there is not an exact middle (i.e.
there are an even number of numbers so two numbers are “in the middle”), then
the median is the mean of those two numbers.
The mode of a set of observations is the number that occurs most frequently.
If two numbers occur equally often and those two numbers occur more often
than any other number, the mode is the mean of those two numbers. If there is
not one or two numbers that occur more frequently than every other number,
we say that the data set has no mode.
Example 1 Find the mean, median, and mode of the following data sets:
• {4, 7, 18, 23, 23, 27, 32, 45}
• {64, 73, 78, 82, 85, 89, 95, 98}
• {1, 1, 1, 5, 5, 1, 6, 1, 4, 4, 8}
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Math 166, Fall 2015, Robert
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Example 2 A small town of 300 residents has the income distribution given
below
Annual Income in Thousands of Dollars
25
30
35
40
Number of Residents
50
100
130
20
Answer the following questions:
1. Find the mean, median, and mode of the town’s income
2. An economic crisis hits the town and everyone’s annual income is lowered
by $5, 000. At the same time, a wealthy person who makes $15 million
a year off of previous investments decides to move to the small town to
retire on his investment profits. What is the mean, median, and mode of
the town’s income?
If X is a random variable that has values x1 , x2 , . . . , xn with probabilities
p1 , p2 , . . . , pn , then the expected value or mean of the random variable X,
denoted by E(X), is
E(X) = x1 p1 + x2 p2 + · · · + xn pn
If X has a binomial distribution with n trials, probabilty of success p, and
probability of failure q = 1 − p, then this formula is simplified to
E(X) = np
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Math 166, Fall 2015, Robert
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Example 3 Find the expected value of
1. the outcome of rolling a six-sided die
2. the number of sales a store will make in a day if they expect 1, 243 customers and each customer will buy something with probability 0.7
3. the number of pens in a case of 500 if each pen has a 0.5% chance of being
defective
Example 4 On a roulette table, there are 38 possible outcomes for the ball to
land: 0, 00, and each of the numbers 1 − 36. The numbers 0 and 00 are green,
while the rest of the numbers are split evenly between red and black. A bet of
$100 is made. Find the expected outcome if
1. the player bet on a single number (this pays out 36 times the original bet
if correctly predicted)
2. the player bets on red (this pays out double the original bet if correctly
predicted)
Example 5 A flood insurance policy charges a customer $35 for a one year
policy that will pay $25, 000 in damages if the customer’s house is in a flood in
the next year. If the company calculates the probability of the customer’s house
being damaged by a flood to be .0012, how much can the company expect to make
from his deal?
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Math 166, Fall 2015, Robert
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Let X be a random variable that takes on the values x1 , x2 , . . . , xn with probabilities p1 , p2 , . . . , pn , and let µ = E(X). Then the variance of the random
variable X, denoted Var(X), is
Var(X) = (x1 − µ)2 p1 + (x2 − µ)2 p2 + · · · + (xn − µ)2 pn
If X is a binomial distribution with n trials, probability of success p, and probability of failure q = 1 − p, then this simplifies to
Var(X) = npq
The standard deviation of a random variable X, denoted by σ(X), is
σ(X) =
p
Var(X)
Example 6 After several years of attending college, a student who has only
taken 3 hour credit classes compiles a list of grades recieved in class thus far:
Grade
A
B
C
D
F
Grade Points
4
3
2
1
0
Frequency
10
7
4
2
1
Let the random variable X denote the grade points recieved. Find the student’s
grade point average (i.e. the expected value of X), as well as the variance and
standard deviation of this student’s performance.
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Math 166, Fall 2015, Robert
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Example 7 A three question true-false quiz is given and the number of incorrect
answers is for each student is documented. The (empirical) probability of each
student’s performance is then recorded in the following histogram:
Find the mean, median, variance, and standard deviation of the data.
Example 8 Without calculating anything, circle the histogram with the largest
variance:
Example 9 When modeling the spread of an infectous disease, the CDC estimates that everyone in a specific city will catch the disease with probability .003.
If the city has a population of 20, 000, what is the expected number of people who
will catch the disease? What is the standard deviation?
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Math 166, Fall 2015, Robert
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Example 10 Consider the probability distribution below:
x
10
11
12
13
P (X = x)
0.15
0.3
0.4
0.15
Find the following:
• Var(X)
• Var(X − 8)
• Var(3X)
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