Download point data each indicate to indices uses variable the x set. data in

Section 8.3 Measures of Center
Mean: is the most widely known measure of center of a
data set. It is also known as the average. It is calculated
by summing all of the observations and then by dividing
the number of observations.
Example 1 The average length of stay by patients in a
short term hospital
A random sample of nine patients yielded the following
data on the length of stay in days 4, 12, 18, 9, 12, 6, 7, 3,
55
What is the mean of these data points?
Summation Notation
xi the variable uses indices to indicate each data point
x1 is the first data point, x2 is the second data point,
and x n is the last data point in the data set.
 is the Greek letter Sigma which represents the letter
S for sum (to add)
n
 xi is read as the summation of x - values from
i 1
where i  1 to n where n is the sample size.
These symbols are used to define the equation in finding
the sample mean.
Sample Mean
For a variable X, the mean of the observations for a
sample is called a sample mean and is denoted by
Symbolically,
x
 xi
n
Where n is the sample size.
Hospital Stay Data
i
x
i
1
2
3
4
5
6
7
8
9
4
12
18
9
12
6
7
3
55
9
 xi  126
i 1
x.
 xi
n
126
x
 14 days
9
x
Deviation: the difference between a data point in a data
set and the mean of the data set.
deviation  xi  x
The mean is the center of the data set and has the
property that the sum of the deviations of the data set is 0
 xi  x   0
Median: is the value in an ordered data set that lies most
nearly in the middle of the sample. It is the number that
divides the bottom 50% of the data from the top 50%.
Procedure 1 Median of a Data Set
Arrange the data in increasing order
 If the number of observations is odd, then the median
is the observation exactly in the middle of the ordered
list.
 If the number of observations is even, then the
median is the mean of the two middle observations in
the ordered list.
In both cases, if we let n denote the number of
observations, then the median is at the position
n  1
2
the ordered list.
Example 2 Length of stay in a short term hospital
Find the median of the length of stay in a short term
hospital
i
yi
1
2
3
4
5
6
7
8
9
3
4
6
7
9
12
12
18
55
in
Sample mode: is a value that appears most often in a
collection. (There may be more than one mode in a
sample.)
Example 3
Length of stay in a short term hospital
Find the mode of the length of stay in a short term hospital
i
yi
1
2
3
4
5
6
7
8
9
3
4
6
7
9
12
12
18
55
Example 4
Compute the mean, median and mode of the data
samples
2. 2, 6, 6, 7, -1
8. 4.2, -3,2, 0, 1.7, 0
Random variables: is simply a variable that takes on
numerical values that depend on the outcomes of a
chance operation.
A Math class was surveyed on how many siblings they
had
Expected Value of a Finite Random Variable
Number of
Siblings
Frequency
Relative
Frequency
Pr( X i )
Xi
0
7
1
18
2
11
3
3
4
1
Total =40
7
 0.175
40
18
 0.450
40
11
 0.275
40
3
 0.075
40
1
 0.025
40
Total =1
Mean of a discrete random variable
 Mean of a discrete random variable Y is denoted by
    xi  Pr( X  xi )
This mean is also called the expected value
Example 5
For the STP 231 sibling data problem find the expected
number of siblings for that class.
Example 6
Calculate the Expected value of X for the given probability
distribution.
12.
1
2
3
4
x
P( X  x) 0.1
0.2
0.5
0.2
16.
x
P( X  x)
-20
0.2
-10
0.4
0
0.2
10
0.1
20
0
30
0.1
34. Your company Sonic Video, Inc., has conducted
research that shows the following probability distribution
where X is the number of video arcades in a randomly
chosen city with more than 500,000 inhabitants.
0
1
2
3
4
5
6
7
8
9
x
P( X  x) 0.07 0.09 0.35 0.25 0.15 0.03 0.02 0.02 0.01 0.01
a. Compute   E (X ) and interpret the result.
b. Which is larger, P( X   ) or P( X   ) ? Interpret the
result.
We can also find the mean and standard deviation of a
Binomial Random Variable
If X is the binomial random variable associated with
n independent Bernoulli trials, each with probability p
of success, then the expected value of X is
  E( X )  n  p
Calculate the expected value of the given random variable
X.
17. X is the number that faces up when a fair die is rolled.
24. X is the number of green marbles that Suzan has in
her hand after she selects four marbles from a bag
containing three red marbles and two green ones.