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Descriptive Statistics

Chapter Contents 4.1 Numerical Description 4.2 Measures of Center 4.3 Measures of Variability 4.4 Standardized Data 4.5 Percentiles, Quartiles, and Box Plots 4.6 Correlation and Covariance 4.7 Grouped Data 4.8 Skewness and Kurtosis. Descriptive Statistics. Chapter 4.

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Descriptive Statistics

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  1. Chapter Contents 4.1 Numerical Description 4.2 Measures of Center 4.3 Measures of Variability 4.4 Standardized Data 4.5 Percentiles, Quartiles, and Box Plots 4.6 Correlation and Covariance 4.7 Grouped Data 4.8 Skewness and Kurtosis Descriptive Statistics Chapter 4

  2. Chapter Learning Objectives LO4-1:Explain the concepts of center, variability, and shape. LO4-2:Use Excel to obtain descriptive statistics and visual displays. LO4-3:Calculate and interpret common measures of center. LO4-4:Calculate and interpret common measures of variability. LO4-5: Transform a data set into standardized values. LO4-6:Apply the Empirical Rule and recognize outliers. LO4-7:Calculate quartiles and other percentiles. LO4-8:Make and interpret box plots. LO4-9:Calculate and interpret a correlation coefficient and covariance. LO4-10:Calculate the mean and standard deviation from grouped data. LO4-11:Assess skewness and kurtosis in a sample Descriptive Statistics Chapter 4

  3. 4.1 Numerical Description LO4-1 Chapter 4 LO4-1: Explain the concepts of center, variability, and shape. Three key characteristics of numerical data:

  4. 4.1 Numerical Description LO4-2 Chapter 4 LO4-2: Use Excel to obtain descriptive statistics and visual displays. EXCEL Displays for Table 4.3

  5. 4.2 Measures of Center LO4-3 Chapter 4 LO4-3: Calculate and interpret common measures of center.

  6. 4.2 Measures of Center LO4-1 Chapter 4 LO4-1: Explain the concepts of center, variability, and shape. Shape • Compare mean and median or look at histogram to determine degree of skewness.

  7. 4.3 Measures of Variability LO4-4 Chapter 4 LO4-4: Calculate and interpret common measures of variability. • Variation is the “spread” of data points about the center of the distribution in a sample. Consider the following measures of variability:

  8. 4.4 Standardized Data Chapter 4 Chebyshev’s Theorem • For any population with mean m and standard deviation s, the percentage of observations that lie within k standard deviations of the mean must be at least 100[1 – 1/k2]. • For k = 2 standard deviations, 100[1 – 1/22] = 75%. So, at least 75.0% will lie within m+ 2s. • For k = 3 standard deviations, 100[1 – 1/32] = 88.9% • So, at least 88.9% will lie within m+ 3s • Although applicable to any data set, these limits tend to be too wide to be useful.

  9. 4.4 Standardized Data LO4-6 Chapter 4 LO4-6: Apply the Empirical Rule and recognize outliers. Note: No upper bound is given. Data values outside m+ 3sare rare. Unusual observations are those that lie beyond m+ 2s. Outliers are observations that lie beyond m+ 3s. The Empirical Rule

  10. 4.4 Standardized Data LO4-5 Chapter 4 LO4-5: Transform a data set into standardized values. Defining a Standardized Variable • A standardized variable(Z) redefines each observation in terms the number of standard deviations from the mean. A negative z value means the observation is below the mean. Standardization formula for a population: Standardization formula for a sample: Positive z means the observation is above the mean.

  11. 4.5 Percentiles, Quartiles, and Box-Plots LO4-7 Chapter 4 LO4-7: Calculate quartiles and other percentiles. Percentiles • Percentilesare data that have been divided into 100 groups. • For example, you score in the 83rd percentile on a standardized test. That means that 83% of the test-takers scored below you. • Deciles are data that have been divided into 10 groups. • Quintiles are data that have been divided into 5 groups. • Quartiles are data that have been divided into 4 groups.

  12. 4.5 Percentiles, Quartiles, and Box-Plots LO4-8 Chapter 4 LO4-8: Make and interpret box plots. • A useful tool of exploratory data analysis(EDA). • Also called a box-and-whisker plot. • Based on a five-number summary: Xmin, Q1, Q2, Q3, Xmax • A box plot shows central tendency, dispersion, and shape. Fences and Unusual Data Values Values outside the inner fences are unusualwhile those outside the outer fences are outliers

  13. 4.6 Correlation and Covariance LO4-9 Chapter 4 LO4-9: Calculate and interpret a correlation coefficient and covariance. Correlation Coefficient • The sample correlation coefficient r is a statistic that describes the degree of linearity between paired observations on two quantitative variables X and Y. Note: -1 ≤ r ≤ +1. The covariance of two random variables X and Y (denoted σXY )measures the degree to which the values of X and Y change together. Population Sample

  14. 4.6 Correlation and Covariance LO4-9 Chapter 4 LO4-9: Calculate and interpret a correlation coefficient and covariance. Covariance A correlation coefficient is the covariance divided by the product of the standard deviations of X and Y.

  15. 4.7 Grouped Data LO4-10 Chapter 4 LO4-10: Calculate the mean and standard deviation from grouped data. Group Mean and Standard Deviation

  16. 4.8 Skewness and Kurtosis LO4-11 Chapter 4 LO4-11: Assess skewness and kurtosis in a sample. Skewness

  17. 4.8 Skewness and Kurtosis LO4-11 Chapter 4 LO4-11: Assess skewness and kurtosis in a sample. Kurtosis

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