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Descriptive Statistics A.A. Elimam College of Business San Francisco State University

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Statistics The Science of collecting, organizing, analyzing, interpreting and presenting data

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Topics Descriptive Statistics Frequency Distributions and Histograms Relative / Cumulative Frequency Measures of Central Tendency Mean, Median, Mode, Midrange

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Topics Measures of Dispersion (Variation) Range, Standard Deviation, Variance and Coefficient of variation Shape Symmetric, Skewed, using Box-and- Whisker Plots Quartile Statistical Relationships Correlation, Covariance

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A collection of quantitative measures and ways of describing data. This includes: Frequency distributions & histograms, measures of central tendency and measures of dispersion Descriptive Statistics

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Collect Data e.g. Survey Present Data e.g. Tables and Graphs Characterize Data e.g. Mean A Characteristic of a: Population is a Parameter Sample is a Statistic.

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Summary Measures Central Tendency Mean Median Mode Midrange Quartile Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range

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Measures of Central Tendency Central Tendency MeanMedianMode Midrange

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The Mean (Arithmetic Average) It is the Arithmetic Average of data values: The Most Common Measure of Central Tendency Affected by Extreme Values (Outliers) 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5Mean = 6 Sample Mean

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The Median 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Important Measure of Central Tendency In an ordered array, the median is the “middle” number. If n is odd, the median is the middle number. If n is even, the median is the average of the 2 middle numbers. Not Affected by Extreme Values

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The Mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 A Measure of Central Tendency Value that Occurs Most Often Not Affected by Extreme Values There May Not be a Mode There May be Several Modes Used for Either Numerical or Categorical Data 0 1 2 3 4 5 6 No Mode

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Midrange A Measure of Central Tendency Average of Smallest and Largest Observation: Affected by Extreme Value Midrange 0 1 2 3 4 5 6 7 8 9 10 Midrange = 5

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Quartiles Not a Measure of Central Tendency Split Ordered Data into 4 Quarters Position of i-th Quartile: position of point 25% Q1Q1 Q2Q2 Q3Q3 Q i(n+1) i 4 Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Position of Q 1 = 2.50 Q1Q1 =12.5 = 1(9 + 1) 4

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Quartiles Not a Measure of Central Tendency Split Ordered Data into 4 Quarters Position of i-th Quartile: position of point 25% Q1Q1 Q2Q2 Q3Q3 Q i(n+1) i 4 Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Position of Q 3 = 7.50 Q3Q3 =19.5 = 3(9 + 1) 4

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Summary Measures Central Tendency Mean Median Mode Midrange Quartile Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range

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Measures of Dispersion (Variation) Variation VarianceStandard DeviationCoefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range

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Understanding Variation The more Spread out or dispersed data the larger the measures of variation The more concentrated or homogenous the data the smaller the measures of variation If all observations are equal measures of variation = Zero All measures of variation are Nonnegative

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Measure of Variation Difference Between Largest & Smallest Observations: Range = Ignores How Data Are Distributed: The Range 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5

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Important Measure of Variation Shows Variation About the Mean: For the Population: For the Sample: Variance For the Population: use N in the denominator. For the Sample : use n - 1 in the denominator.

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Most Important Measure of Variation Shows Variation About the Mean: For the Population: For the Sample: Standard Deviation For the Population: use N in the denominator. For the Sample : use n - 1 in the denominator.

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Sample Standard Deviation For the Sample : use n - 1 in the denominator. Data: 10 12 14 15 17 18 18 24 s = n = 8 Mean =16 = 4.2426 s

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Comparing Standard Deviations s = = 4.2426 = 3.9686 Value for the Standard Deviation is larger for data considered as a Sample. Data : 10 12 14 15 17 18 18 24 N= 8 Mean =16

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Comparing Standard Deviations Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s =.9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C

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Coefficient of Variation Measure of Relative Variation Always a % Shows Variation Relative to Mean Used to Compare 2 or More Groups Formula ( for Sample):

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Comparing Coefficient of Variation Stock A: Average Price last year = $50 Standard Deviation = $5 Stock B: Average Price last year = $100 Standard Deviation = $5 Coefficient of Variation: Stock A: CV = 10% Stock B: CV = 5%

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Shape Describes How Data Are Distributed Measures of Shape: Symmetric or skewed

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Shape Describes How Data Are Distributed Measures of Shape: Symmetric or skewed Symmetric Mean =Median =Mode -0.5 <0 < 0.5

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Shape Describes How Data Are Distributed Measures of Shape: Symmetric or skewed Left-SkewedSymmetric Mean =Median =Mode Mean Median Mod e < -1-0.5 <0 < 0.5

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Shape Describes How Data Are Distributed Measures of Shape: Symmetric or skewed Right-Skewed Left-SkewedSymmetric Mean =Median =Mode Mean Median Mode Median Mean Mod e < -1 > 1 -0.5 <0 < 0.5

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Box-and-Whisker Plot Graphical Display of Data Using 5-Number Summary Median 4 6 8 10 12 Q 3 Q 1 X largest X smallest

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Distribution Shape & Box-and-Whisker Plots Right-SkewedLeft-SkewedSymmetric Q 1 Median Q 3 Q 1 Q 3 Q 1 Q 3

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A measure of the strength of linear relationship between two variables X and Y, and is measured by the (population) correlation coefficient: The numerator is the covariance Correlation

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The average of the products of the deviations of each observation from its respective mean: Covariance

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Sample Correlation Coefficient Correlation Coefficient ranges from –1 to +1 +1 perfect positive correlation 0 no linear correlation -1 perfect negative correlation

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Summary Discussed Measures of Central Tendency Mean, Median, Mode, Midrange Quartiles Addressed Measures of Variation The Range, Interquartile Range, Variance, Standard Deviation, Coefficient of Variation Determined Shape of Distributions Symmetric, Skewed, Box-and-Whisker Plot Mean =Median =ModeMean Median Mode Median Mean

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