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04 Describing Data Graphically and Numerically Part 3

1σ μ μ 68 1σ μ the empirical rule 36 contains

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μ ± μ 68% μ ±
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The Empirical Rule 36 contains about 95% of the values in the population or the sample contains about 99.7% of the values in the population or the sample μ ± μ ± μ ± 99.7% 95% μ ±
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Regardless of how the data are distributed, at least (1 - 1/ k 2 ) of the values will fall within k standard deviations of the mean Examples: (1 - 1/1 2 ) = 0% …… ..... k=1 (μ ± 1σ) (1 - 1/2 2 ) = 75% … .......... k=2 (μ ± 2σ) (1 - 1/3 2 ) = 89% ………. k=3 (μ ± 3σ) Tchebysheff’s (Chebyshev’s) Theorem (Aside) 37 within At least
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A standardized data value refers to the number of standard deviations a value is from the mean Standardized data values are sometimes referred to as z-scores Can be used to compare data sets Will be addressed in more detail later in the course Standardized Data Values 38
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Standardized population values: Standardized sample values: Standardized Data Values 39 σ μ x z - = x = original data value μ = population mean σ = population standard deviation z = standard score (number of standard deviations x is from μ) x = original data value x = sample mean s = sample standard deviation z = standard score (number of standard deviations x is from x) s x x z - =
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Measures of Variation 40 Variation Variance Standard Deviation Coefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartil e Range
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