3 Shape

3 Shape - Dr Harvey A Singer School of Management George...

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© 2002 by Harvey A. Singer 1 OM 210 Statistical Analysis for  Management Data Summarization: Part 4: Shape and Clustering Dr. Harvey A. Singer School of Management George Mason University

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© 2002 by Harvey A. Singer 2 Numerical Data Properties and Measures Coeff. of Variation Numerical Data Mean Median Mode Midrange (Quartiles) Midquartile Central Tendency Range Interquartile Range Variance Standard Deviation Variability Skew Shape Grouped Data (Quartiles) Symmetric Weighted mean Clustering
© 2002 by Harvey A. Singer 3 Shape Two issues: Shape. Skewed vs. symmetric. – Asymmetric vs. symmetric. – If skewed, which way? » Which side of the distribution (high end or low end) has the tail. If skewed, by how much? » Slight, mild, moderate, vs. severe skewing. Measurement of skew. » Skew factor. Data clustering. Wide vs. narrow data “peak.” – Wide vs. narrow “hump” in the data distribution. Measurement of data clustering. » IQR/R.

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© 2002 by Harvey A. Singer 4 Shape Right-Skewed Mode < Median < Mean Left-Skewed Mean < Median < Mode Symmetric Mean = Median = Mode (bell-shaped)
© 2002 by Harvey A. Singer 5 Symmetric Distribution Formally: All of the measures of central tendency coincide. All the central measures have the same identical value. Mean = Median = Mode (if there is a mode). Bounding curve of data distribution is symmetric about its center. “Tails” symmetric on both sides of center.

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© 2002 by Harvey A. Singer 6 Symmetric Distributions Symmetric if the measures of central tendency are the same value. Or are nearly the same value, practically. Operationally, symmetric if mean ~ median ~ midrange ~ midquartile Types: Uniform or even. Bell-shaped.
© 2002 by Harvey A. Singer 7 Symmetric Distribution Types If there is no mode, then the symmetric distribution is “uniform.” The data is evenly or uniformly distributed about the center of the distribution. If there is a mode and if the mode ~ mean, etc., then the distribution is “normal.” Has the shape of a bell-shaped curve. Normally, the data distribution will appear to have a bell shape. Normally, the value that is expected to occur is also the most likely value to occur.

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© 2002 by Harvey A. Singer 8 Uniform Distribution The data is uniformly or evenly spaced through the distribution if In general, mean = median = midrange for a uniform distribution. – mean = x min + ½ R = x max – ½ R No mode in a uniform distribution! midrange e midquartil x x x x = = = ˆ
9 Uniform Distribution Practically: Data are evenly or “uniformly” distributed over their range. In particular, all the data have the same frequency, so there is no mode. 7

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3 Shape - Dr Harvey A Singer School of Management George...

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