03Dispersion

03Dispersion - Statistical Techniques I EXST7005 Measures...

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Statistical Techniques I EXST7005 Start here Measures of Dispersion
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Objective - Hypothesis testing Background We will test primarily means, but also variances Testing means requires a measure of the variability in the data set Course Progression
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MEASURES OF DISPERSION These are measures of variation or variability among the elements (observations) of a data se RANGE - difference between the largest and smallest observation This is a rough estimator which does not use all the information in the data set.
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MEASURES OF DISPERSION (continued) Interquartile range - Q1 to Q3 (25% to 75%) better than range What are Quartiles? The first quartile (Q1) is the value that has one quarter of the values below (smaller than Q1) i and three quarters above it (larger than Q1) The second quartile has half the values smalle and half the values larger The third quartile has 3/4 smaller and 1/4 large
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MEASURES OF DISPERSION (continued) Percentile - a given percentile has that percent o the values below it and the remaining values above it. e.g. The 40th percentile has 40% of the values smaller and 60% of the values larger
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MEASURES OF DISPERSION (continued) VARIANCE the "average" squared deviation from the mean. the POPULATION VARIANCE (called sigma squared) σ 2 This is a parameter, and therefore a constant where N is the size of the population σ μ 2 2 1 = - F H G I K J = Y N i i N
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S2 is the SAMPLE VARIANCE (called s- squared). This is a statistic, and therefore a variable where n is the size of the sample NOTE that the divisor is n-1 rather than n. If n is used then the calculation is a biased estimator of σ 2. MEASURES OF DISPERSION (continued) S Y Y n i i n 2 2 1 1 = - - F H G I K J =
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MEASURES OF DISPERSION (continued) STANDARD DEVIATION a standard measure the deviation of observations from the mean. it is calculated as the square root of the variance σ = √σ 2 this is a parameter S = S2 this is a statistic
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the VARIANCE is the average squared deviation so we take the square root of this to get back to the same units. Absolute Mean Deviation the "average deviation from the mean, but using absolute values. This i another possibility, but is not used much because the Variance is more flexible. MEASURES OF DISPERSION (continued)
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A valid, useful measure of dispersion should use all of the available information be independent of other parameters (and statistics) for large data sets be capable of being expressed in the same units as the variables be small when the spread among the points in th data set is small, and large when the spread is wider. The Standard deviation fills these criteria.
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03Dispersion - Statistical Techniques I EXST7005 Measures...

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