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DESCRIBING DATA: 2

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Numerical summaries of data using measures of central tendency and dispersion

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Central tendency--Mode Table 1. Undergraduate Majors

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Bimodal Distributions Table 1. Undergraduate Majors

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Mode for Grouped Frequency Distributions based on Interval Data Midpoint of the modal class interval

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Median The point in the distribution above which and below which exactly half the observations lie (50th percentile) Calculation depends on whether the no. of observations is odd or even.

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Median= 188

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MEDIAN for grouped frequency distributions based on interval data Median = 40 + ((20/30) * 10) = 40 + 6.67 = 46.67

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ARITHMETIC MEAN

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Mean for Grouped Data Mean = sum of weighted midpoints / n = 4650/100=46.5

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Mean is the balancing point of the distribution 0123456789 X X X X X X X MEAN

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Key Properties of the Mean Sum of the differences between the individual scores and the mean equals 0 sum of the squared differences between the individual scores and the mean equals a minimum value. The minimum value

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Weaknesses of each measure of central tendency MODE: ignores all other info. about values except the most frequent one MEDIAN: ignores the LOCATION of scores above or below the midpoint MEAN: is the most sensitive to extreme values

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Mode Mean Median Impacts of skewed distributions

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Measures of Dispersion Poverty Households (%) in 2 suburbs by tract Less dispersion more dispersion

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Range Highest value minus the lowest value problem: ignores all the other values between the two extreme values

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Interquartile range Based on the quartiles (25th percentile and 75th percentile of a distribution) Interquartile range = Q 3 -Q 1 Semi-interquartile range = (Q 3 -Q 1 )/2 eliminates the effect of extreme scores by excluding them

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Graphic representation: Box Plot Infant mortality rate AfricaAsia Latin America

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Variance A measure of dispersion based on the second property of the mean we discussed earlier: minimum

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Step 1: Calculate the total sum of squares around the mean

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Step 2: Take an average of this total variation Why n-1? Rather than simply n??? The normal procedure involves estimating variance for a population using data from a sample. Samples, especially small samples, are less likely to include extreme scores in the population. N-1 is used to compensate for this underestimate.

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Step 3: Take the square root of variance Purpose: expresses dispersion in the original units of measurement--not units of measurement squared Like variance: the larger the value the greater the variability

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Coefficient of Variation (V) V = (standard deviation / mean) Value : To allow you to make comparisons of dispersion across groups with very different mean values or across variables with very different measurement scales.

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