Download Measuring Center Sections 2.1, 2.2, 2.3 Lecture 5 Robb T. Koether

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Transcript 
Measuring Center
Sections 2.1, 2.2, 2.3
Lecture 5
Robb T. Koether
Hampden-Sydney College
Wed, Jan 20, 2016
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
1 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
2 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
3 / 23
Measuring Center
We expect the “middle” or “center” of a distribution to contain the
“typical” or “representative” values in the data set.
However, these are all vague concepts.
We need to give them a precise meaning.
And there is more than one way to define the middle of a
distribution.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
4 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
5 / 23
The Mean
Definition (Mean)
The mean of a data set is the average, that is, the sum of all the values
divided by the number of values. The symbol for the mean is x,
pronounced “x bar.”
The mean is the most common measure of center.
If there are n values and we label them x1 , x2 , . . . , xn , then the
mean is
P
xi
x=
.
n
The mean is the “balancing point” of the data.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
6 / 23
The Mean
The Balancing Point
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
7 / 23
The Mean
The Balancing Point
Average
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
7 / 23
The Mean
The Balancing Point
Average
-5
-2
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
7 / 23
The Mean
The Balancing Point
Average
+4
-5
+2
-2
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
+1
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
7 / 23
Example
Example (The Mean)
Compute the average height of the students in this class.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
8 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
9 / 23
The Mean on the TI-83
The Mean on the TI-83
Enter the data into a list, say L1 .
Press STAT > CALC > 1-Var Stats.
Press ENTER. “1-Var-Stats” appears in the display.
Type L1 and press ENTER.
A list of statistics appears. The first one is the mean.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
10 / 23
Example
Example (Rainfall Data)
Rainfall data for August in Richmond, VA (1986 - 2015).
6.74 1.24 4.04 4.90 5.72 2.88
6.91 5.58 2.52 8.42 4.44 1.41
1.84 2.00 2.79 2.30 3.15 3.59
16.02 2.56 5.99 6.81 5.73 4.04
3.92 7.10 3.50 7.64 3.61 2.77
Use the TI-83 to find the mean of the rainfall data.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
11 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
12 / 23
The Median
Definition (Median)
The median of a data set is the number in the middle of the set. If
there are n numbers in the set, then the middle number is in position
n+1
.
2
If n is even, then we get a half integer which we interpret as indicating
the number halfway between the middle two numbers.
The median is a better measure of center than the mean when the
data are strongly skewed.
The median divides the data set into the lower half and the upper
half; it is the halfway point.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
13 / 23
The Median
The Halfway Point
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
14 / 23
The Median
The Halfway Point
Median
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
14 / 23
The Median
The Halfway Point
Median
-6
-3
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
14 / 23
The Median
The Halfway Point
Median
-6
+3
-3
1
2
3
Robb T. Koether (Hampden-Sydney College)
4
5
+1
6
7
8
Measuring CenterSections 2.1, 2.2, 2.3
9
10
Wed, Jan 20, 2016
14 / 23
Example
Example (The Median)
Compute the median height of the students in this class.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
15 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
16 / 23
The Median on the TI-83
The Median on the TI-83
Follow the same procedure that was used to find the mean.
When the list of statistics appears, scroll down to the one labeled
“Med." It is the median.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
17 / 23
Example
Example (Rainfall Data)
Rainfall data for August in Richmond, VA (1986 - 2015).
6.74 1.24 4.04 4.90 5.72 2.88
6.91 5.58 2.52 8.42 4.44 1.41
1.84 2.00 2.79 2.30 3.15 3.59
16.02 2.56 5.99 6.81 5.73 4.04
3.92 7.10 3.50 7.64 3.61 2.77
Use the TI-83 to find the median of the rainfall data.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
18 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
19 / 23
If the distribution is symmetric, then the mean and the median
have the same value.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
20 / 23
If the distribution is symmetric, then the mean and the median
have the same value.
If the data are skewed in one direction, then the mean and the
median are pulled in that direction, but the mean is pulled further.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
20 / 23
If the distribution is symmetric, then the mean and the median
have the same value.
If the data are skewed in one direction, then the mean and the
median are pulled in that direction, but the mean is pulled further.
For that reason, if the data are strongly skewed, then the median
is more representative than the mean.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
20 / 23
The Mean vs. the Median
The Mean vs. the Median
Mean
Median
Peak
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
21 / 23
The Mean vs. the Median
The Mean vs. the Median
Peak
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
21 / 23
The Mean vs. the Median
The Mean vs. the Median
Peak
Robb T. Koether (Hampden-Sydney College)
Mean
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
21 / 23
The Mean vs. the Median
The Mean vs. the Median
Peak
Mean
Median
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
21 / 23
Outline
1
Measuring Center
2
The Mean
Using the TI-83
3
The Median
Using the TI-83
4
Comparing the Mean and the Median
5
Assignment
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
22 / 23
Assignment
Assignment
Read Section 2.1: Measuring Center: The Mean.
Read Section 2.2: Measuring Center: The Median.
Read Section 2.3: Comparing the Mean and the Median.
Apply Your Knowledge: 2.1, 2.3, 2.4.
Check Your Skills: –.
Exercises: 2.25, 2.26.
Robb T. Koether (Hampden-Sydney College)
Measuring CenterSections 2.1, 2.2, 2.3
Wed, Jan 20, 2016
23 / 23
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