3.2 Measures of Central Tendency
MATH 166-506, Fall 2016, Jean Yeh
3.2 Measures of Central Tendency
Definition:
1. The average or mean of n numbers x1 , x2 , ..., xn , denoted by µ, is given by
µ=
x1 + x2 + · · · + xn
n
2. The median of n numbers x1 , x2 , ..., xn is the middle number when the numbers are arranged
in increasing (or decreasing) if n is odd. If n is the even, it is the average of the middle two
numbers.
3. The mode of n numbers x1 , x2 , ..., xn is the number that occurs the most frequently. If two
numbers occur the same number of times and more frequent than all other numbers, we say
the set is bimodal and has two modes. If no one or two numbers occur more frequently than
the rest, we say there is no mode.
Example: Find the mean, median and mode of the following sets of numbers
a. {4, 6, 2, 1, 2, 3, 2, 1}
{1, 1, 2, 2, 2, 3, 4, 6} There are total 8 many numbers.
The mean µ = 1+1+2+2+2+3+4+6
= 21
= 2.625. The mode is 2.
8
8
th
After arranged in increasing, the 4 number is 2 and the 5th number is 2. Therefore, the
median is 2+2
= 2.
2
b. {2, 3, 3, 5, 5, 6, 676}
There are total 7 many numbers.
The mean µ = 2+3+3+5+5+6+676
= 700
= 100. The mode is 3 and 5 ( bimodal ).
7
7
th
After arranged in increasing, the 4 number is 5, which is the median
c. The grade distribution in
Grade in Letter
A
Grade in Number
4
Number of Students 8
a certain
B C
3 2
20 12
math class is given in the following table.
D F
1 0
6 4
There are total 8+20+12+6+4=50 many students.
The mean µ = 8∗4+20∗3+12∗2+6∗1+4∗0
= 2.44. The mode is 3.
50
After arranged in increasing, the 25th number is 3 and the 26th number is 3. Therefore, the
median is 3+3
= 3.
2
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3.2 Measures of Central Tendency
MATH 166-506, Fall 2016, Jean Yeh
Calculate the mean and median for a set.(CANNOT FIND MODE)
1. Press STAT , and to select Edit , press Enter
2. Input the values in the list L1.
3. Press STAT again. Move the cursor to the right to the CALC menu and then select
1 - Var Stats .
4. Press 2ND and 1 to select list L1. The home screen should shows
“1 - Var Stats L1.”
5. Press Enter . x̄ is the mean and scroll down to see the median (Med).
To clear a list: Scroll up to the list name(L1), press CLEAR , and then press ENTER.
Make sure you do this every time before you start a new problem.
Calculate the mean and median for the data with frequencies.
1. Press STAT , and to select Edit , press Enter
2. Input the values in the list L1 Press the right arrow to input the frequencies
in the list L2.
3. Press STAT again. Move the cursor to the right to the CALC menu and then select
1 - Var Stats .
4. Press 2ND and 1 to select list L1. Press comma. And then press 2ND and 2
to selet list L2. This time, the home screen should shows
“1 - Var Stats L1, L2.”
5. Press Enter . x̄ is the mean and scroll down to see the median (Med).
Remind: Do not forget to clear the previous lists before you start a new problem.
Definition: Let X denote the random variable that has values x1 , x2 , ..., xn , and let the associated
(empirical) probabilities be p1 , p2 , ..., pn . Then expected value or mean of the random variable X,
denoted by E(X), is
E(X) = x1 p1 + x2 p2 + · · · + xn pn
Note: E(X) is what we “expect” over the long term, but E(X) need not be an actual outcome
(x1 , x2 , ..., xn ).
Example: A fair 6-sided die is rolled, find the expected value of the number observed.
The probabilities of observing each number are juse 1/6
E(X) = 1 ∗
1
1
1
1
1
1
1
+ 2 ∗ + 3 ∗ + 4 ∗ + 5 ∗ + 6 ∗ = 21 ∗ = 3.5
6
6
6
6
6
6
6
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3.2 Measures of Central Tendency
MATH 166-506, Fall 2016, Jean Yeh
Example: Two fair dice are rolled. If you roll a total of 7, you win $6; otherwise you lose $1.
What is the expected return of this game?
6
). The probability of lose $1 is
The probability of win $6 is the probability of roll a total of 7 ( 36
6
1 − 36 .
6
6
− 1(1 − ) = 0.3333
E(X) = 7 ∗
36
36
Example: Given the random variable X with the probability distribution shown in the following
X
12 24 36
table, find E(X).
P(X) 0.1 0.6 0.3
E(X) = 12 ∗ 0.1 + 24 ∗ 0.6 + 36 ∗ 0.3 = 26.4
Example: A hurricane and flood insurance policy costs 100 dollars a year. The company will pay
4000 dollars if there is hurricane and flood damage to the house. If the chance that a hurricane or
a flood will damage a house is 2% in this year, what is the expected profit or loss of this policy for
the insurance company?
E(X) = −4000 ∗ 2% + 100 ∗ (1 − 2%) = 18
Expected Value of Binomial Experiments: The expected value of the binomial distribution
with n trials and probability of success p in a single trial and q = 1 − p is
E(X) = np
.
Example: A unfair coin has the property that the probability of flipping a head is 40%. If we flip
the coin 15 times, what is the expected number of occurrences of a head?
p = 40%, n=15, E(X) = 15 ∗ 40% = 6.
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