Section 2.3
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Transcript of Section 2.3
![Page 1: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/1.jpg)
Section 2.3
Measures of Central Tendency
Larson/Farber 4th ed.
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Section 2.3 Objectives
• Determine the mean, median, and mode of a population and of a sample
• Determine the weighted mean of a data set and the mean of a frequency distribution
• Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each
Larson/Farber 4th ed.
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Measures of Central Tendency
Measure of central tendency• A value that represents a typical, or central, entry of a
data set.• Most common measures of central tendency:
Mean Median Mode
Larson/Farber 4th ed.
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Measure of Central Tendency: Mean
Mean (average)• The sum of all the data entries divided by the number
of entries.• Sigma notation: Σx = add all of the data entries (x)
in the data set.• Population mean:
• Sample mean:
Larson/Farber 4th ed.
![Page 5: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/5.jpg)
Example: Finding a Sample Mean
The education cost per student (in thousands of dollars) from a sample of 10 liberal arts colleges?
30 35 19 22 22 20 23 21 35 25
Larson/Farber 4th ed.
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Solution: Finding a Sample Mean
30 35 19 22 22 20 23 21 35 25
Larson/Farber 4th ed.
• The sum of the educational costs
Σx = 30+35+19+22+22+20+23+21+35+25 = 252
• To find the mean price, divide the sum of the prices by the number of prices in the sample
The mean price of the educational costs is about $25.2 (in thousands of dollars).
![Page 7: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/7.jpg)
Measure of Central Tendency: Median
Median• The value that lies in the middle of the data when the
data set is ordered.• Measures the center of an ordered data set by
dividing it into two equal parts.• If the data set has an
odd number of entries: median is the middle data entry.
even number of entries: median is the mean of the two middle data entries.
Larson/Farber 4th ed.
![Page 8: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/8.jpg)
Example: Finding the Median
Larson/Farber 4th ed.
The education cost per student (in thousands of dollars) from a sample of 10 liberal arts colleges?
30 35 19 22 22 20 23 21 35 25
![Page 9: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/9.jpg)
Solution: Finding the Median
Larson/Farber 4th ed.
• First order the data.
19 20 21 22 22 23 25 30 35 35
• There are ten entries (an even number), the median is the average of the middle two entries.
• Median = (23 + 22) ÷ 2 = 22.5
The median cost of education is $22.5 (in thousands of
dollars).
30 35 19 22 22 20 23 21 35 25
![Page 10: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/10.jpg)
Example: Finding the Median
If there were only 9 colleges, say we remove one of the 35k colleges, then what is the median?
30 19 22 22 20 23 21 35 25
Larson/Farber 4th ed.
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Solution: Finding the Median
30 19 22 22 20 23 21 35 25
Larson/Farber 4th ed.
• First order the data.
19 20 21 22 22 23 25 30 35
• There are 9 entries entries (an odd number), the median is the middle entry.
The median cost is $22 (in thousands of dollars).
median = 22
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Measure of Central Tendency: Mode
Mode• The data entry that occurs with the greatest
frequency.• If no entry is repeated the data set has no mode.• If two entries occur with the same greatest frequency,
each entry is a mode (bimodal).
Larson/Farber 4th ed.
![Page 13: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/13.jpg)
Example: Finding the Mode
The education cost per student (in thousands of dollars) from a sample of 9 liberal arts colleges?
30 35 19 22 22 20 23 21 25
Larson/Farber 4th ed.
![Page 14: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/14.jpg)
Solution: Finding the Mode
30 35 19 22 22 20 23 21 25
Larson/Farber 4th ed.
• Ordering the data helps to find the mode.
19 20 21 22 22 23 25 30 35
• The entry of 22 occurs twice, whereas the other data entries occur only once.
The mode of educational cost is $22, (in thousands of
dollars).
![Page 15: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/15.jpg)
Example: Finding the Mode
At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses?
Larson/Farber 4th ed.
Political Party Frequency, f
Democrat 34
Republican 56
Other 21
Did not respond 9
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Solution: Finding the Mode
Larson/Farber 4th ed.
Political Party Frequency, f
Democrat 34
Republican 56
Other 21
Did not respond 9
The mode is Republican (the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single affiliation.
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Comparing the Mean, Median, and Mode
• All three measures describe a typical entry of a data set.
• Advantage of using the mean: The mean is a reliable measure because it takes
into account every entry of a data set.• Disadvantage of using the mean:
Greatly affected by outliers (a data entry that is far removed from the other entries in the data set).
Larson/Farber 4th ed.
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Example: Comparing the Mean, Median, and Mode
Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers?
Larson/Farber 4th ed. 80
Ages in a class
20 20 20 20 20 20 21
21 21 21 22 22 22 23
23 23 23 24 24 65
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Solution: Comparing the Mean, Median, and Mode
Larson/Farber 4th ed.
Mean:
Median:
20 years (the entry occurring with thegreatest frequency)
Ages in a class
20 20 20 20 20 20 21
21 21 21 22 22 22 23
23 23 23 24 24 65
Mode:
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Solution: Comparing the Mean, Median, and Mode
Larson/Farber 4th ed.
Mean ≈ 23.8 years Median = 21.5 years Mode = 20 years
• The mean takes every entry into account, but is influenced by the outlier of 65.
• The median also takes every entry into account, and it is not affected by the outlier.
• In this case the mode exists, but it doesn't appear to represent a typical entry.
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Solution: Comparing the Mean, Median, and Mode
Larson/Farber 4th ed.
Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set.
In this case, it appears that the median best describes the data set.
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Weighted Mean
Weighted Mean• The mean of a data set whose entries have varying
weights.
• where w is the weight of each entry x
Larson/Farber 4th ed.
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Solution: Finding a Weighted Mean
Larson/Farber 4th ed.
Source Score, x Weight, w x∙w
Quiz Mean 79 0.10 79(0.10)= 7.9
Exam 1 85 0.20 85(0.20) = 17.0
Exam 2 93 0.20 93(0.20) = 18.6
Final Exam 89 0.40 89(0.40) = 35.6
Extra Credit 45 0.10 45(0.10) = 4.5
Homework 100 0.05 100(0.05) = 5.0
Σw = 1 Σ(x∙w) = 88.6
Your weighted mean for the course is 88.6. You did not get an A.
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Mean of Grouped Data
Mean of a Frequency Distribution• Approximated by
where x and f are the midpoints and frequencies of a class, respectively
Larson/Farber 4th ed.
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Finding the Mean of a Frequency Distribution
In Words In Symbols
Larson/Farber 4th ed.
1. Find the midpoint of each class.
2. Find the sum of the products of the midpoints and the frequencies.
3. Find the sum of the frequencies.
4. Find the mean of the frequency distribution.
![Page 26: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/26.jpg)
The Shape of Distributions
Larson/Farber 4th ed.
Symmetric Distribution• A vertical line can be drawn through the middle of
a graph of the distribution and the resulting halves are approximately mirror images.
![Page 27: Section 2.3](https://reader030.fdocuments.in/reader030/viewer/2022020417/5681365b550346895d9de68e/html5/thumbnails/27.jpg)
The Shape of Distributions
Larson/Farber 4th ed.
Uniform Distribution (rectangular)• All entries or classes in the distribution have equal
or approximately equal frequencies.• Symmetric.
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The Shape of Distributions
Larson/Farber 4th ed.
Skewed Left Distribution (negatively skewed)• The “tail” of the graph elongates more to the left.• The mean is to the left of the median.
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The Shape of Distributions
Larson/Farber 4th ed.
Skewed Right Distribution (positively skewed)• The “tail” of the graph elongates more to the right.• The mean is to the right of the median.
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Section 2.3 Summary
• Determined the mean, median, and mode of a population and of a sample
• Determined the weighted mean of a data set and the mean of a frequency distribution
• Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each
• Homework 2.3 1 - 24 EO, 25 - 55 EOO
Larson/Farber 4th ed.