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Math 141-copyright Joe Kahlig, 015C
Section 8.2: Expected Value (Part II)
Section 8.3: Variance and Standard Deviation
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 median is sometime
called the second quartile which means that at least 50% of the data is that number or greater.
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 141-copyright Joe Kahlig, 015C
Example: Compute the mean, median, mode, standard deviation, and variance of this data.
X
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 and the standard deviation.
data
prob.
1
0.3
3
0.2
8
0.4
10
0.1
Example: The distribution of grades on an exam are given in the table. Discuss the values for the
mean, median, and mode.
grades
90 ≤ x ≤ 99
80 ≤ x ≤ 89
70 ≤ x ≤ 79
60 ≤ x ≤ 69
50 ≤ x ≤ 59
40 ≤ x ≤ 49
Freq.
8
15
18
11
5
2
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Math 141-copyright Joe Kahlig, 015C
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?
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. Estimate
P (30 ≤ X ≤ 70).
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.