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Statistics 528 - Lecture 3 Prof. Kate Calder
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Section 1.1/1.
Graphical and Numerical Summaries of Data
- Shape of a Distribution
- Modes
- Symmetric vs. Skewed
- Outliers
- Measures of the Center
- Measures of Spread
- Choosing Summaries of Distributions
- Changing the Units of Measurement
Statistics 528 - Lecture 3 Prof. Kate Calder
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Modes
- Question: Does the distribution have one or several major peaks? Look at histograms and stemplots.
- A distribution with one major peak is called unimodal. A distribution with two major peaks is called bimodal.
- Example of a bimodal distribution: scores on an exam
Statistics 528 - Lecture 3 Prof. Kate Calder
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Symmetric vs. Skewed
- A distribution is symmetric if the values larger or smaller than the midpoint are mirror images of each other.
- A distribution is skewed to the right if the right tail (larger values) is much longer than the left tail (smaller values).
- A distribution is skewed to the left if the left tail (smaller values) is much longer than the right tail (larger values).
Statistics 528 - Lecture 3 Prof. Kate Calder
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Left Skewed Symmetric Right Skewed
Statistics 528 - Lecture 3 Prof. Kate Calder
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Outliers
Outliers – values that fall outside the overall pattern and are far from the bulk of the data
- Can be a result of natural variation.
- Or, can be evidence of a mistake (equipment failure, incorrect recording of an observation, etc.).
Removing an outlier? Big Decision
Statistics 528 - Lecture 3 Prof. Kate Calder
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Measures of the Center
Two different ideas for the “center” of a distribution - can be very different.
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a) sort observations from smallest to largest b) if n is odd ( n = number of observations) median = middle value of the sorted list = (n+1)/2th^ observation up from the bottom of the list c) if n is even median = mean of the middle two observations
Statistics 528 - Lecture 3 Prof. Kate Calder
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Mean vs. Median
- The median is a more resistant measure of the center of a distribution,
i.e., the median is not as affected by extreme observations (long tails, outliers)
Mean vs. Median Applet - example of a dot plot (http://bcs.whfreeman.com/ips4e/default.asp)
Statistics 528 - Lecture 3 Prof. Kate Calder
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Left Skewed Symmetric Right Skewed
Mean < Median Mean = Median Mean > Median
Statistics 528 - Lecture 3 Prof. Kate Calder
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Example: Phyllis received 6 HW grades in her statistics class:
86 88 92 44 89 90
Her mean grade is:
Her median grade is:
44 86 88 89 90 92
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Question: Does the mean, 81.5, give a good idea of her “typical” grade?
No, it is lower than all but one of her grades.
Question: What about the median, 88.5?
88.5 is more typical.
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Choosing a Summary
- The median, IQR, or five-number summary are better than the mean and the standard deviation for describing a skewed distribution or a distribution with outliers.
- The mean and standard deviation should only be used for describing symmetric distributions with no outliers.
- Why should we ever used the mean and standard deviation?
Answer: They completely specify a normal distribution which allows us to easily perform statistical inference.
Statistics 528 - Lecture 3 Prof. Kate Calder
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Changing the Unit of Measurement
Linear Transformations : xnew = a + b x
- a (constant) shifts all of the values of x up or down by the same amount
- b (positive constant) changes the size of the unit of measurement
- A linear transformation will not change the shape of a distribution.
- Multiplying each observation by a positive constant b multiplies both measures of the center (mean and median) and measures of spread (IQR and standard deviation) by b.
- Adding the same number a (either positive or negative) to each observation adds a to the measures of the center (mean and median) and to the quartiles (and other percentiles) but does not change measures of spread.
