MBA7020_04.ppt/June 120, 2005/Page 1 Georgia State University - Confidential MBA 7020 Business Analysis Foundations Descriptive Statistics June 20, 2005

MBA7020_04.ppt/June 120, 2005/Page 2 Georgia State University - Confidential Agenda Confidence Interval Descriptive Summary Measures 1. Measures of Central Location Mean, Median, Mode 2. Measures of Variation The Range, Variance and Standard Deviation 3. Measures of Association Covariance and Correlation

MBA7020_04.ppt/June 120, 2005/Page 3 Georgia State University - Confidential 1. Measures of Central Location Mean, Median, Mode 2. Measures of Variation The Range, Variance and Standard Deviation 3. Measures of Association Covariance and Correlation Describing Data: Summary Measures

MBA7020_04.ppt/June 120, 2005/Page 4 Georgia State University - Confidential 1. It is the Arithmetic Average of data values: 2. The Most Common Measure of Central Tendency 3. Affected by Extreme Values (Outliers) Mean = 5Mean = 6 Sample Mean Mean

MBA7020_04.ppt/June 120, 2005/Page 5 Georgia State University - Confidential Median = 5 1.Important Measure of Central Tendency 2.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. 3.Not Affected by Extreme Values Median

MBA7020_04.ppt/June 120, 2005/Page 6 Georgia State University - Confidential Mode = 9 1.A Measure of Central Tendency 2.Value that Occurs Most Often 3.Not Affected by Extreme Values 4.There May Not be a Mode 5.There May be Several Modes 6.Used for Either Numerical or Categorical Data No Mode Mode

MBA7020_04.ppt/June 120, 2005/Page 7 Georgia State University - Confidential Agenda Confidence Interval Descriptive Summary Measures 1. Measures of Central Location Mean, Median, Mode 2. Measures of Variation The Range, Variance and Standard Deviation 3. Measures of Association Covariance and Correlation

MBA7020_04.ppt/June 120, 2005/Page 8 Georgia State University - Confidential Variation Variance / Standard Deviation Coefficient of Variation Population Sample Range / Percentiles Measures of Variability

MBA7020_04.ppt/June 120, 2005/Page 9 Georgia State University - Confidential Measure of Variation Difference Between Largest & Smallest Observations: Range = Ignores How Data Are Distributed: The Range Range = = Range = = 5

MBA7020_04.ppt/June 120, 2005/Page 10 Georgia State University - Confidential Percentile Scores 1.Arrange data in ascending order. 2.The middle number is the median. 3.The number halfway to the median is the first quartile. 4.The number halfway past the median is the 3 rd quartile. 5.A number with (no more than) 66% of the values less than it is the 66 th percentile, and so forth.

MBA7020_04.ppt/June 120, 2005/Page 11 Georgia State University - Confidential Box Plot Median Q1Q3 SmallestLargest

MBA7020_04.ppt/June 120, 2005/Page 12 Georgia State University - Confidential 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.

MBA7020_04.ppt/June 120, 2005/Page 13 Georgia State University - Confidential 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.

MBA7020_04.ppt/June 120, 2005/Page 14 Georgia State University - Confidential Sample Standard Deviation For the Sample : use n - 1 in the denominator. Data: s = n = 8 Mean =16 = s

MBA7020_04.ppt/June 120, 2005/Page 15 Georgia State University - Confidential Comparing Standard Deviations Mean = 15.5 s = Data B Data A Mean = 15.5 s = Mean = 15.5 s = 4.57 Data C

MBA7020_04.ppt/June 120, 2005/Page 16 Georgia State University - Confidential Agenda Confidence Interval Descriptive Summary Measures 1. Measures of Central Location Mean, Median, Mode 2. Measures of Variation The Range, Variance and Standard Deviation 3. Measures of Association Covariance and Correlation

MBA7020_04.ppt/June 120, 2005/Page 17 Georgia State University - Confidential Measure of Relative Variation Always a % Shows Variation Relative to Mean Used to Compare 2 or More Groups Formula (for Sample): Coefficient of Variation

MBA7020_04.ppt/June 120, 2005/Page 18 Georgia State University - Confidential 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% Comparing Coefficient of Variation

MBA7020_04.ppt/June 120, 2005/Page 19 Georgia State University - Confidential Describes How Data Are Distributed Measures of Shape: Symmetric or skewed Right-Skewed Left-SkewedSymmetric Mean =Median =Mode Mean Median Mode Median Mean Mode Shape

MBA7020_04.ppt/June 120, 2005/Page 20 Georgia State University - Confidential Agenda Confidence Interval Descriptive Summary Measures

MBA7020_04.ppt/June 120, 2005/Page 21 Georgia State University - Confidential Confidence Interval Sample Mean + Margin of Error (MOE) Called a Confidence Interval To Compute Margin of Error, One of Two Conditions Must Be True: The Distribution of the Population of Incomes Must Be Normal, or The Distribution of Sample Means Must Be Normal.

MBA7020_04.ppt/June 120, 2005/Page 22 Georgia State University - Confidential A Side-Trip Before Constructing Confidence Intervals 1.What is a Population Distribution? 2.What is a Distribution of the Sample Mean? 3.How Does Distribution of Sample Mean Differ From a Population Distribution? 4.What is the Central Limit Theorem?

MBA7020_04.ppt/June 120, 2005/Page 23 Georgia State University - Confidential Assume Small Population of Lexus Owners’ Incomes (N = 200) Estimating the Population Mean Income of Lexus Owners

MBA7020_04.ppt/June 120, 2005/Page 24 Georgia State University - Confidential Distribution of N = 200 Incomes (Population Mean ) Mean

MBA7020_04.ppt/June 120, 2005/Page 25 Georgia State University - Confidential Obs 1Obs 2Obs 3Obs 4Obs 5 Mean Constructing a Distribution of Samples of Size 5 from N = 200 Owners

MBA7020_04.ppt/June 120, 2005/Page 26 Georgia State University - Confidential Distribution of Sample Mean Incomes (Column #7) Distribution of Sample Means Near Normal! Estimated Std. Error

MBA7020_04.ppt/June 120, 2005/Page 27 Georgia State University - Confidential Central Limit Theorem Even if Distribution of Population is Not Normal, Distribution of Sample Mean Will Be Near Normal Provided You Select Sample of Five or Ten or Greater From the Population. For a Sample Sizes of 30 or More, Distribution of the Sample Mean Will Be Normal, with –mean of sample means = population mean, and –standard error = [population deviation] / [sqrt(n)] Thus Can Use Expression:

MBA7020_04.ppt/June 120, 2005/Page 28 Georgia State University - Confidential Why Does Central Limit Theorem Work? As Sample Size Increases: 1.Most Sample Means will be Close to Population Mean, 2.Some Sample Means will be Either Relatively Far Above or Below Population Mean. 3.A Few Sample Means will be Either Very Far Above or Below Population Mean.

MBA7020_04.ppt/June 120, 2005/Page 29 Georgia State University - Confidential Impact of Side-Trip on MOE 1.Determine Confidence, or Reliability, Factor. 2.Distribution of Sample Mean Normal from Central Limit Theorem. 3.Use a “Normal-Like Table” to Obtain Confidence Factor. Without Taking Repeated Samples) 4.Determine Spread in Sample Means (Without Taking Repeated Samples)

MBA7020_04.ppt/June 120, 2005/Page 30 Georgia State University - Confidential Drawing Conclusions about a Population Mean Using a Sample Mean Select Simple Random Sample Compute Sample Mean and Std. Dev. For n < 10, Sample Bell-Shaped? For n >10 CLT Ensures Dist of Normal Draw Conclusion about Population Mean

MBA7020_04.ppt/June 120, 2005/Page 31 Georgia State University - Confidential Federal Aid Problem Suppose a census tract with 5000 families is eligible for aid under program HR- 247 if average income of families of 4 is between $7500 and $8500 (those lower than 7500 are eligible in a different program). A random sample of 12 families yields data on the next page.

MBA7020_04.ppt/June 120, 2005/Page 32 Georgia State University - Confidential Federal Aid Study Calculations 7,300 7,700 8,100 8,400 7,800 8,300 8,500 7,600 7,400 7,800 8,300 8,600 Representative Sample

MBA7020_04.ppt/June 120, 2005/Page 33 Georgia State University - Confidential Estimated Standard Error If We Took Repeated SamplesMeasures Variation Among the Sample Means If We Took Repeated Samples. But We Only Have One Sample! How Can We Compute Estimated Standard Error? Based on Constructing Distribution of Sample Mean Slide, Will Estimated Standard Error Be Smaller or Larger Than Sample Standard Deviation (s)? Estimated Std. Error ______ than s. Estimated Standard Error Expression: For Federal Aid Study

MBA7020_04.ppt/June 120, 2005/Page 34 Georgia State University - Confidential Confidence Factor for MOE: Can Use t-Table Provided Distribution of Sample Mean is Normal

MBA7020_04.ppt/June 120, 2005/Page 35 Georgia State University - Confidential 95% Confidence Interval Interpretation of Confidence Interval PopulationNot Sample) 95% Confident that Interval $7,983 + $280 Contains Unknown Population (Not Sample) Mean Income. If We Selected 1,000 Samples of Size 12 and Constructed 1,000 Confidence Intervals, about 950 Would Contain Unknown Population Mean and 50 Would Not. So Is Tract Eligible for Aid???

MBA7020_04.ppt/June 120, 2005/Page 36 Georgia State University - Confidential Sample Means versus Sample Proportion Income/Loss Time to Complete Loan Papers Number of Fat Calories in Burger Breaking Strength of Cellular Phone Housing Americans Who Believe that Japan is #1 Economic Power Circuit Boards with One or More Failed Solder Connections African-Americans Who Pass CPA MeanProportion of Means and Proportions Not the Same!!!!

MBA7020_04.ppt/June 120, 2005/Page 37 Georgia State University - Confidential Similarities and Differences Between Sample Means and Proportions Sample Means Measured tComputed from Data that Are Measured. tEstimate Population Means. Sample Proportions Counted tComputed from Data that Are Counted. tEstimate Population Proportions.

MBA7020_04.ppt/June 120, 2005/Page 38 Georgia State University - Confidential Drawing Conclusions about a Population Proportion From a Sample Proportion Select Simple Random Sample Compute Sample Proportion Check for Normality - Table 7.8 Draw Conclusion About Population Proportion, p

MBA7020_04.ppt/June 120, 2005/Page 39 Georgia State University - Confidential Japan Business Survey N =200 Californians Yes = 116 No = 84 Is Japan the Foremost Economic Power Today?

MBA7020_04.ppt/June 120, 2005/Page 40 Georgia State University - Confidential 90% Confidence Interval on P 90% Confident that Between 52.3%and 63.7% of Californians Believe Japan is Leading Economic Power. After Calculations Done. Common Practice to Report Proportions as Percentages. After Calculations Done.