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Descriptive (Univariate) Statistics Percentages (frequencies) Ratios and Rates Measures of Central Tendency Measures of Variability Descriptive statistics describes variables—We are not testing relationships between variables

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Central Tendency What is the average value of a variable in a range of values for a given population? MEASURES of Central Tendency MeanSum of all values / N MeanSum of all values / N MedianCenter of the distribution (case that cuts the MedianCenter of the distribution (case that cuts the sample into two) ModeMost frequently occurring value ModeMost frequently occurring value

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Calculating Each Measure Mean = ∑X / n –(Sum of all values divided by the sample size). Mode = Count the most frequently occurring value.

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Median Median = Odd # of cases (Md = middle value) Finding the Middle Position: (n + 1) / 2 (= position of the middle value) Example: 11, 12, 13, 16, 17, 20, 25 (N=7) Md = (7+1) / 2 = 4th value = 16 50% of cases lie above and below 16

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Median Continued Median = even # of cases There will be two middle cases Md. = the average of the scores of the two middle cases. Example: 11, 12, 13, 16, 17, 20, 25, 26 Position of the middle value = (8+1)/2 = 4.5 Md = 16 + 17 (two middle cases) / 2 = 16.5 NOTE = Need to sort your values before locating the Md

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Why might we use the median instead of the mean?

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Skewed Distributions See board Mean is most sensitive to outliers EXAMPLE: 5, 6, 6, 7, 8, 9, 10, 10 Md. 7.5 Mean 7.63 5, 6, 6, 7, 8, 9, 10, 95 Md. 7.5 Mean 18.25

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Measures of Variability Variability—scatter of scores around the mean. How do scores cluster around the mean? Example: Say the average price of a home in Bakersfield is (say 150,000). Can you buy a home in Hagen Oaks for 150,000? See bell curve (mean = 150K, Sd = 10K)

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Measures of Variability RangeThe distance between the highest and lowest score (subtract the lowest value from the highest value) A rough measure.

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Standard Deviation Deviation = The distance of a given raw score from the mean (X – Mean). We need a summary measure that accounts for all of the scores in a distribution. Variance and SD are summary measures Calculate the SD by taking the Square Root of Variance

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Variance Variance = ∑ (X-mean) squared/n Dividing by n controls for the number of scores involved. SD = Square root of variance We take the square root of variance b/c it is easier to interpret.

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Spread Around the Mean Theoretically: 34.13% of the cases fall 1 SD above & 1 SD below the mean. 47.72% fall 2 SDs above mean & 2 SDs below the mean. 49.87% of cases fall 3 SDs above & 3 SDs below the mean.

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Housing Cost Example Cont. If the mean is 150,000 & Sd is 10,000 then: 99.74% of the cases fall between 120,000 (3 SDs below the mean) & 180,000 (3 SDs above the mean)

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Levels of Measurement & Descriptive Statistics Nominal Frequency Distribution Modal Category Ordinal Frequency Dist. Modal Category Mean in some cases (i.e. a scale) Interval/Ratio Mean, Md., Mode Variance & Standard Deviation

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Practice Interpretation Descriptive Statistics Descriptive Statistics HIGHEST YEAR OF SCHOOL COMPLETED Minimum Maximum Mean SDVariance 02013.262.8698.232 N 2808

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