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Math 166-copyright Joe Kahlig, 16A
Section 3.2: Measures of Central Tendency ( Part 2)
Section 3.3: Measures of Spread
Definition: The mean of a data set is the traditional average.
The median of a data set is the middle of the data when the data is ordered.
The mode of a data set are the values that have the largest frequencies.
The standard deviation of a data set is the measure of how the data is spread about its mean.
The variance of a data set is the average of the square of the distance from the data value to the mean.
Variance =
Notation:
(x1 − µ)2 + (x2 − µ)2 + ... + (xn − µ)2
n
Sample
Sx
x
st. deviation
mean
Sample Variance formula =
Population
σ
µ
(x1 − x)2 + (x2 − x)2 + ... + (xn − x)2
n−1
Example: What is the difference between a population and a sample?
Example: Find the mean, median, and mode of these data sets.
A) 40, 45, 50, 90, 95, 100
B) 60, 65, 65, 70, 75, 75, 80
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Math 166-copyright Joe Kahlig, 16A
Example: Compute the mean, median, mode, standard deviation, and variance of this data.
data
freq.
7
9
9
9
12
7
15
10
Example: Compute the mean, standard deviation and variance. Let X = the number of Dr. Peppers
purchased during a semester.
X
freq.
10
50
23
30
43
49
26
73
250
1
Example: Compute the mean, median, and mode of this probability distribution.
data
1
3
8
10
prob. 0.3 0.2 0.4 0.1
Math 166-copyright Joe Kahlig, 16A
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Example: Five people take an exam. The grades are
70, 73, 81, and 92
A) If the median is 81, what is the missing grade?
B) If the median is 78, what is the missing grade?
Example: Let X be a discrete random variable with integer values such that 0 ≤ X ≤ 90. The random
variable has an expected value of 15.3 and standard deviation of 3.5.
A) What values of X are within 1.25 standard deviations of the mean?
B) What values of X are upto 2 standard deviations above the mean?
Definition: The standard deviation of a binomial distribution with n trials and probability of success
√
p on a single trial is σ = npq
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Math 166-copyright Joe Kahlig, 16A
Example: The probability of an adverse reaction to a flu shot is 0.17. Flu shots are given to a group
of 90 people. Let X represent the number of people with an adverse reaction. Find the probability
that the number of people with an adverse reaction is within 2 standard deviations of the mean.
Chebyshev’s Inequality: Let X be a random variable with mean µ and standard deviation σ. Then
the probability that X will be within k standard deviations of the mean is
P (µ − kσ ≤ X ≤ µ + kσ) ≥ 1 −
1
k2
Example: The random variable X has a mean of 50 and a standard deviation of 14. Find
P (30 ≤ X ≤ 70).
Math 166-copyright Joe Kahlig, 16A
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Example: The expected lifetime of a product is 2 years with a standard deviation of 3.5 months.
For a shipment of 5000 items, estimate the number of item that will last between 17.7 months and
30.3 months.