roughly describes where the center of the data is in the set. 2 important measures of center: a)...
TRANSCRIPT
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Central Tendency
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Roughly describes where the center of the data is in the set.
2 Important measures of center: a) Mean b) Median
Center
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Mean
Sample Population
Mean =
n
xx
N
x
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Example: Find the mean of the following sample.
23, 25, 26, 29, 39, 42, 50
4.337
2347
50423929262523
x
n
xx
Show Your Work – Including the Formula that you used!
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The 50 states plus the District of Columbia have a total of 3137 counties. There are a total of 248,709,873 people in each of these counties. Find the average population per county.
Example:
countyperresidents
N
x
7.282,79
3137
873,709,248
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What if I used the Almanac Book of Facts and chose a few samples? Find the means.
Sample 1 Sample 2 Sample 3
20,095 28,895 16,934
108,978 10,032 519
15,384 16,174 73,478
13,931 959,275 14,798
24,960 30,797 13,859
6.669,36x 6.034,209x 6.917,23x
Do you suppose these are close to the values we’d get if we could use the population?
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We will study this further later on to see how to be able to use samples to predict the populations better.
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What Exactly is the Mean?
The mean tells us how large each observation in the data set would be if the total were split equally among all the observations.
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A group of elementary school children was asked how many pets they have. Here are there responses. Find the mean and explain what it means.
1 3 4 4 4 5 7 8 9
Example
petsx 5 We can look at it this way:If every child in the group had the same number of pets, each would have 5 pets.
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In the last unit, we introduced the median as an informal measure of center that described the “midpoint” of a distribution.
Now, it is time to offer an official “rule” for calculating the median.
Median
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The median is the value in the middle!
50% of the data is above and below this value.
Steps: 1. Put numbers in order. 2. If the # of numbers is odd – median is the middle number. 3. If the # of numbers is even – median is the average of the two #’s in the middle.
Median
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Find the median.
17,14,12,8,6
Median = 12
28,27,23,22,15,7
Median=
Median = 22.5
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Example. The stemplot shows travel times to work for New York workers. Find and interpret the median.
58
7
5006
5
5004
003
50002
5555001
50 .min4554: Key
min5.222
2520 M
“In the sample of New York workers, about half of the people reported traveling less than 22.5 minutes to work, and about half reported traveling more.”
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88M
Example: Find the mean and median of the following test grades.
20 97 93 84 71
85 87 94 88 76
92 88 98 89 60
Stemplot:
847329
9887548
617
06
5
4
3
02
5.8115
1222
N
x
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Comparing the Mean and Median
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The median travel time is 20 minutes. The mean travel time is higher, 22.5
minutes. The mean is pulled toward the right
tail of this right-skewed distribution. The median, unlike the mean, is
resistant. If the longest travel time were 600
minutes rather than 60 minutes, the mean would get higher, but the median would not change at all!
Take a Look at the Stemplot and let’s discuss the mean and the median.
06
5
004
003
5002
5200001
50
Mean = 22.5Median = 20
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The mean and median of a roughly symmetric distribution are close together.
If the distribution is exactly symmetric, the mean and median are exactly the same.
In a skewed distribution, the mean is usually farther out in the long tail than is the median.
In General
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The mean is greatly affected by outliers – it is very sensitive to them – which means it is pulled towards the outlier.
The median is insensitive to outliers. It is often used more because it is more stable.
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Salaries for MLB players or NFL players
Scores on a test when there’s one that has NOT been made up yet.
Home prices
Personal Incomes
College tuitions
Examples of where median is the best choice
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Graphical Representations
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Based only on the stemplot, would you expect the mean travel time to be less than, about the same as, or larger than the median? Why?
Use the stemplot to answer the following questions.
58
7
5006
5
5004
003
50002
5555001
50
Since the distribution is skewed to the right, we would expect the mean to be larger than the median.
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Use your calculator to find the mean and median travel time. Was your answer to Question 1 correct?
Use the stemplot to answer the following questions.
58
7
5006
5
5004
003
50002
5555001
50
The mean is 31.25 minutes, which is bigger than the median of 22.5 minutes.
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Interpret your result from Question 2 in context without using the words “mean” or “average.”
Use the stemplot to answer the following questions.
58
7
5006
5
5004
003
50002
5555001
50
If we divided the travel time up evenly among all 20 people, each would have a 31.25 minute travel time.
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Would the mean or median be a more appropriate summary of the center of this distribution of drive times? Justify your answer.
Use the stemplot to answer the following questions.
58
7
5006
5
5004
003
50002
5555001
50
Since the distribution is skewed, the median would be a better measure of the center.
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M F F F M F M F M M F F F M
Proportion of Success:
14
8
#
p
n
successesofp
For a Population use “P”
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Trimmed Mean A disadvantage of the mean is that it can be affected by extremely high or low values.
One way to make the mean more resistant to exceptional values and still sensitive to specific data values is to do a trimmed mean.
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Order the data – delete a selected number of values from each end of the list then average the remaining values.
Trimming Percentage: The percent of values trimmed from the list.
Trimmed Mean
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Example80805050424040353535
30303025232020202014
a) Compute the mean for the entire sample.
b) Compute a 5% trimmed mean.
0.3620
719
n
xx
.8014
.
1
105.020:%5
ANDREMOVE
SETTHEOFBOTTOMANDTOP
THEFROMVALUEREMOVE
Trim
7.3418
625
n
xx
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Example80805050424040353535
30303025232020202014
c) Compute the median for the entire sample.
d) Compute a 5% trimmed median.
5.322
3530 Median
The median is still 32.5.
e) Is the trimmed mean or the original mean closer to the median?
Trimmed Mean
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Ed took 5 tests and his average was 85. If his average after the first three tests was 83, what’s the average of the last two tests?
TestsAllofSumTotalx
x
x
425
5855
85
TestsstofSumTotalx
x
x
31249
3833
83
TestslastofSumx
x
2176
249425
882
176
x
x
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On Thursday, 20 out of 25 students took a test and their average was 80. On Friday, the other 5 students took it and their average was 90. What was the class average?
82
25
9058020
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The first 3 hours of a trip, Susan drove 50 mph. Due to delays, she drove 40 mph for the next 2 hours. What was her average speed?
mph
hrs
mphhrsmphhrs46
.5
40.250.3
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Ed’s average on 4 tests is 80. What does he need to get on the 5th test to raise his average to an 84?
320
4
480
x
scoresofx
100
4203205
32084
x
x
x