2.1 visualizing distributions: shape, center, and spread
TRANSCRIPT
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2.1 Visualizing Distributions:Shape, Center, and Spread
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The student will be able:
• To identify and sketch the basic shapes of distributions of data – uniform, normal, skewed
• To describe the characteristics of the shape of a distribution, including symmetry, skewness, modes, outliers, gaps, and clusters.
• To describe a uniform distribution using the range and the frequency.
• To estimate graphically the mean and standard deviation of a normal distribution and use them to describe the distribution.
• The estimate graphically the median and quartiles and use them to describe a skewed distribution.
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Important Terms and Concepts• Basic shape of a distribution (listed on next slide)• Measure of center– Mean– Median
• Measures of spread– Standard deviation– Quartiles
• Other features– Outliers– Gaps– Clusters
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Important Terms and Concepts
Shape, Center, and Spread – Always giveAlways label graphs!
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Four Most Common Shapes of Distribution
• Uniform (Rectangular) Distributions• Normal Distributions• Skewed Distributions• Bimodal Distributions
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Uniform Distributions
• All values occur equally often or nearly equally often
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Normal Distribution
Bell shaped Single peak
Mode At line of
symmetry
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Normal Distribution
• The curve drops off smoothly on both sides, flattening towards the x-axis but never quite reaching it and stretching infinitely far in both directions
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Normal Distribution
• On either side of the mode are inflection points – where the curve changes from concave down to concave up
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Normal Distribution
• You should use the mean to describe the center• You should use the Standard Deviation, SD, to describe
spread.• The Standard Deviation – the horizontal distance from
the mean to an inflection point.• Use area to estimate the standard deviation.• Roughly 68% of the total area under the curve is
between the vertical lines through the two inflection points. – In other words, the interval between one standard deviation
on either side of the mean accounts for roughly68% of the area under the normal curve
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Normal Distribution
• Discuss measure the diameter of a tennis ball• Discuss Page 31, weight of pennies
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Graph a Normal Distribution
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Skewed Distributions
• Distribution with bunching a one end and a long tail stretching out in the other direction.
• The direction of the tail tells whether the distribution is skewed right or skewed left.
Left Skewed Right Skewed
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Skewed Distributions
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Skewed
• Since there is no symmetry the ideas of center and spread are not as clear-cut as they are for a normal distribution.
• Typically you should use median to describe center.
• You should use the lower and upper quartile to indicate spread.
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What is Lower and Upper Quartile?
• The lower quartile is the value that divides the lower half of the distribution into two halves, with equal numbers of dots on either side.
• The upper quartile is the value that divides the upper half of the distribution into two halves, with equal numbers of dots on either side.
• The three values—lower quartile, median, and upper quartile—divide the distribution into quarters. This allows you to describe a distribution as in the introduction to this chapter: “The middle 50% of the SAT math scores were between 630 and 720, with half above 680 and half below.”
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Skewed Distributions
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Bimodal Distributions
• Bimodal Distributions have two or more obvious peaks.
• It is worth asking whether your cases represent two or more groups.
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Bimodal Distributions
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Bimodal Distributions
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Other Features
• Outliers – a value that stand apart from the bulk of the data. An unusual value.
• Gaps – a separation – there is no formal definition
• Clusters – a grouping of values – there is no formal definition.
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Other Features
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Practice
• P1 – P5
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Entertainment
• E1 – E8, E11, E14