section 3.1 measures of central tendency: mode, median, and mean
TRANSCRIPT
Section 3.1
Measures of Central Tendency: Mode, Median, and Mean
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Measures of Central Tendency: Mode, Median, and Mean
Usually, one number is used to describe the entire sample or population – the average.
three of the major ways to measure center of data:
1.Mode
2.Median
3.Mean
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Mode
- Data value that occurs the most- Not every data set has a mode
(Ex: professor assigns equal # of A’s, B’s, C’s, D’s, F’s)- Mode is not stable- Think tallest bar on a histogram- Most in a class (ex: bimodal means 2 modes)
- Relevant in cases like most frequently requested shoe size
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- Order data from smallest to largest- 50% of the data below and 50% above the median
Ex:
Data on price per ounce in cents of chips:
19 19 27 28 18 35
a) Mode?
b) Median?
c) Average?
d) What if you add 80 to the data set?
Median
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If we take out 35 from the data.
Median = 19
e) Is $10.45 reasonable to serve an ounce of chips to 55 people?
Yes, the median price of the chips is 19 cents per ounce.
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NOTE #1: The median uses the position - extreme values usually does not change it much.
Ex: the median is often used as the average for house prices.
NOTE #2: Extreme values inflate or deflate the average (mean)
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Mean
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A resistant measure is one that is not influenced by extremely high or low data values.
***The mean is not a resistant measure of center
***The median is more resistant measure of center
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Trimmed Mean
***More resistant than the regular mean
-- trim the lowest 5% of the data and highest 5% of the data (works the same for a 10% trimmed mean)
Procedure:
1.Order data
2.Multiply 5% by n and round to the nearest integer
3.that value is how many data points you trim from each end
4.Take the average of the remaining values
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Measures of Central Tendency: Mode, Median, and Mean
Symmetrical data: mean, median, and mode are the same or almost the same.
Left-Skewed data: mean < median and
median < mode
Right-Skewed data: mean > median and
Median > mode
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Relationship: Mode, Median, and Mean
Figure, shows the general relationships among the mean, median, and mode for different types of distributions.
(a) Mound-shaped symmetrical (b) Skewed left (c) Skewed right
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Weighted Mean
Suppose your midterm test score is 83 and your final exam score is 95.
Using weights of 40% for the midterm and 60% for the final exam, compute the weighted average of your scores.
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Solution
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Harmonic Mean
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Geometric Mean
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Example 1 – In the calculatorBelleview College must make a report to the budget committee about the average credit hour load a full-time students carries. (A 12-credit-hour load is minimum requirement for full-time status. For the same tuition, students may take up to 20 credit hours.) A random sample of 40 students yielded the following information (in credit hours):
17 12 17 12 18 19 12
12 14 15 15 12 13 14
15 14 16 16 20 13 15
18 16 17 12 12 17 15
12 15 13 18 20 12
17 16 14 19 13 12
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Example 2Barron’s Profiles of American Colleges, 19th edition, lists average class size for introductory lecture courses at each of the profiled institutions. A sample of 20 colleges and universities in California showed class sizes for introductory lecture courses to be:
14 20 20 20 20 23 25 30 30
30 35 35 35 40 40 42 50 50
80 80
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