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Math 166-copyright Joe Kahlig, 12C
Section 3.3: Measures of Spread
Example: Find the mean and median of these data sets.
A) 40, 40, 40, 100, 100, 100
B) 60, 65, 65, 70, 75, 75, 80
Definition: 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 =
(x1 − µ)2 + (x2 − µ)2 + ... + (xn − µ)2
n
Sample
Sx
x
Notation:
st. deviation
mean
Sample Variance formula =
Population
σ
µ
(x1 − x)2 + (x2 − x)2 + ... + (xn − x)2
n−1
Example: Compute the median, standard deviation and the variance.
data
freq.
7
9
9
9
12
7
15
10
Example: Compute the median, standard deviation and the variance. Let X = the number of Dr.
Peppers drank during a semester.
10 23 43 26 250
X
freq. 50 30 49 73
1
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Math 166-copyright Joe Kahlig, 12C
Example: Compute the standard deviation.
data
prob.
1
0.3
3
0.2
8
0.4
10
0.1
Definition: The standard deviation of a binomial distribution with n trials and probability of success
√
p on a single trial is σ = npq
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.
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?
C) Find the probability that the number of people with an adverse reaction is within 2 standard
deviations of the mean.
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Math 166-copyright Joe Kahlig, 12C
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).
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.