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Plan for Today: Chapter 11: Displaying Distributions with Graphs Chapter 12: Describing Distributions with Numbers
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Histograms Histogram is the most common graph of the distribution of a quantitative variable. Pie chart and bar graph are the common graphs of the distribution of a categorical variable.
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Histograms Note: There is no space between bars. the obs from 85 to 95 number /percentage at this range implies number
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Overall Pattern of a Distribution See if the distribution has a simple shape that you can describe in a few words. The center and the spread.
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Histograms: center and the spread Histogram B Histogram A
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Histograms: shape Symmetric: if the right and left sides of the histogram are approximately mirror images of each other.
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Histograms: shape Skewed to the right: if the right side of the histogram extends much farther out than left side.
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Histograms: shape Skewed to the left: if the left side of the histogram extends much farther out than right side.
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Stemplot A stemplot (a.k.a. stem-and-leaf plot) is quicker to make and presents more detailed information.
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Stemplot The max temperatures for the first 11 days this February at West Lafayette (I faked the number 19). 56 49 55 42 48 36 36 35 33 38 19 1 9 2 3 35668 4 289 5 56 Largest place value Next place to the right Keep this row even you don’t have any 20s Duplicates have to be labeled separately.
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Boxplots: The median M is the midpoint of a distribution. Half the observation are smaller that M and the other half are larger. How to find the median: 1) Arrange all observations in order of size, from smallest to largest. 2) If the number of observations n is odd, the median M is the center observation in the ordered list. 3) If the number of observations n is even, the median M is the average of the two center observations in the ordered list.
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Boxplots: The first quartile Q 1 is the median of the left subgroup. The third quartile Q 3 is the median of the right. The median divided the sequence into left/right subgroups.
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Boxplots: 7 9 10 11 14 17 19 20 21 25 27 29 30 median [ ] Q 1 = 10.5Q 3 = 26
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Q1Q1 Boxplots (without Outliers): median Q3Q3 Minimum Maximum Without outliers 25% of the data
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Outliers: The interquartile range (IQR) is the distance between first quartile Q 1 and third quartile Q 3. IQR = Q 3 – Q 1 Any data observation which lies more than 1.5*IQR lower than the first quartile or 1.5*IQR higher than the third quartile is considered an outlier. Median Q1Q1 Q3Q3 IQR 1.5*IQR
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Modified Boxplots (with Outliers) With outliers Largest non-outlier point Minimum(since we don’t have any outliers
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Center and Spread : We often use two indexes to measure the central tendency: 2) Mean/ average: sample mean: 1) Median
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Center and Spread : We often use two indexes to measure the variability or “spread” : 1) Interquartile range (IQR) 2) Standard deviation (std dev): sample variance: sample std dev:
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Center and Spread : The median, Q1, Q3 suffer less impact at the present of outliers. Mean and standard deviation have better numerical properties.
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Center and Spread : The max temperatures for the first 10 days this February at West Lafayette. The researcher made a typo when he recorded the value 49. Before: 56 49 55 42 48 36 36 35 33 38 After: 56 149 55 42 48 36 36 35 33 38 BeforeAfter Median40 Q136 Q34955 BeforeAfter Mean42.852.8 Std Dev 8.5734.8
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