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Chapter 3, Numerical Descriptive Measures Data analysis is objective –Should report the summary measures that best meet the assumptions about the data set Data interpretation is subjective –Should be done in fair, neutral and clear manner

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Summary Measures Arithmetic Mean Median Mode Describing Data Numerically Variance Standard Deviation Coefficient of Variation Range Interquartile Range Geometric Mean Skewness Central TendencyVariationShape Quartiles

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Arithmetic Mean The arithmetic mean (mean) is the most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) Sample size Observed values

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Geometric Mean Geometric mean –Used to measure the rate of change of a variable over time Geometric mean rate of return –Measures the status of an investment over time –Where R i is the rate of return in time period I

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Median: Position and Value In an ordered array, the median is the “middle” number (50% above, 50% below) The location (position) of the median: The value of median is NOT affected by extreme values

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Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may may be no mode There may be several modes

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Quartiles Quartiles split the ranked data into 4 segments with an equal number of values per segment Find a quartile by determining the value in the appropriate position in the ranked data, where First quartile position: Q 1 = (n+1)/4 Second quartile position: Q 2 =2 (n+1)/4 (the median position) Third quartile position: Q 3 = 3(n+1)/4 where n is the number of observed values

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Same center, different variation Measures of Variation Variation Variance Standard Deviation Coefficient of Variation RangeInterquartile Range Measures of variation give information on the spread or variability of the data values.

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Range and Interquartile Rage Range –Simplest measure of variation –Difference between the largest and the smallest observations: Range = X largest – X smallest –Ignores the way in which data are distributed –Sensitive to outliers Interquartile Range –Eliminate some high- and low-valued observations and calculate the range from the remaining values –Interquartile range = 3 rd quartile – 1 st quartile = Q 3 – Q 1

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Average (approximately) of squared deviations of values from the mean –Sample variance: Variance Where = arithmetic mean n = sample size X i = i th value of the variable X

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Standard Deviation Most commonly used measure of variation Shows variation about the mean Has the same units as the original data It is a measure of the “average” spread around the mean Sample standard deviation:

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Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare two or more sets of data measured in different units

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Shape of a Distribution Describes how data are distributed Measures of shape –Symmetric or skewed Mean = Median Mean < Median Median < Mean Right-Skewed Left-SkewedSymmetric

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Using the Five-Number Summary to Explore the Shape Box-and-Whisker Plot: A Graphical display of data using 5-number summary: The Box and central line are centered between the endpoints if data are symmetric around the median Minimum, Q1, Median, Q3, Maximum Min Q 1 Median Q 3 Max

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Distribution Shape and Box-and-Whisker Plot Right-SkewedLeft-SkewedSymmetric Q1Q2Q3Q1Q2Q3 Q1Q2Q3

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If the data distribution is bell-shaped, then the interval: – contains about 68% of the values in the population or the sample – 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 Relationship between Std. Dev. And Shape: The Empirical Rule

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Population Mean and Variance Population variance Population Mean

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Covariance and Coefficient of Correlation The sample covariance measures the strength of the linear relationship between two variables (called bivariate data) The sample covariance: Only concerned with the strength of the relationship No causal effect is implied

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Covariance between two random variables: cov(X,Y) > 0 X and Y tend to move in the same direction cov(X,Y) < 0 X and Y tend to move in opposite directions cov(X,Y) = 0 X and Y are independent Covariance does not say anything about the relative strength of the relationship. Coefficient of Correlation measures the relative strength of the linear relationship between two variables

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Coefficient of Correlation: –Is unit free –Ranges between –1 (perfect negative) and 1(perfect positive) –The closer to –1, the stronger the negative linear relationship –The closer to 1, the stronger the positive linear relationship –The closer to 0, the weaker any positive linear relationship –At 0 there is no relationship at all

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Correlation vs. Regression A scatter plot (or scatter diagram) can be used to show the relationship between two variables Correlation analysis is used to measure strength of the association (linear relationship) between two variables –Correlation is only concerned with strength of the relationship –No causal effect is implied with correlation

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