1 / 36

Business and Finance College Principles of Statistics Lecture 5 Eng. Heba Hamad week 3- 2008

Business and Finance College Principles of Statistics Lecture 5 Eng. Heba Hamad week 3- 2008. Slides Prepared by JOHN S. LOUCKS St. Edward’s University. Chapter 3 Descriptive Statistics: Numerical Measures Part B. Measures of Distribution Shape, Relative Location, and Detecting Outliers.

penha
Télécharger la présentation

Business and Finance College Principles of Statistics Lecture 5 Eng. Heba Hamad week 3- 2008

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Business and Finance College Principles of StatisticsLecture 5Eng. Heba Hamadweek 3- 2008

  2. Slides Prepared by JOHN S. LOUCKS St. Edward’s University

  3. Chapter 3 Descriptive Statistics: Numerical MeasuresPart B • Measures of Distribution Shape, Relative Location, and Detecting Outliers • Exploratory Data Analysis • Measures of Association Between Two Variables • The Weighted Mean and Working with Grouped Data

  4. Measures of Distribution Shape,Relative Location, and Detecting Outliers • Distribution Shape • z-Scores • Empirical Rule • Detecting Outliers

  5. Distribution Shape: Skewness • An important measure of the shape of a distribution is called skewness. • The formula for computing skewness for a data set is somewhat complex. • Skewness can be easily computed using statistical software.

  6. .35 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness • Symmetric (not skewed) • Skewness is zero. • Mean and median are equal. Skewness = 0 Relative Frequency

  7. .35 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness • Moderately Skewed Left • Skewness is negative. • Mean will usually be less than the median. Skewness = - .31 Relative Frequency

  8. .35 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness • Moderately Skewed Right • Skewness is positive. • Mean will usually be more than the median. Skewness = .31 Relative Frequency

  9. .35 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness • Highly Skewed Right • Skewness is positive (often above 1.0). • Mean will usually be more than the median. Skewness = 1.25 Relative Frequency

  10. Distribution Shape: Skewness • Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed in ascending order on the next slide.

  11. Distribution Shape: Skewness

  12. .35 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness Skewness = .92 Relative Frequency

  13. z-Scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value xi is from the mean.

  14. z-Scores • An observation’s z-score is a measure of the relative location of the observation in a data set. • A data value less than the sample mean will have a • z-score less than zero. • A data value greater than the sample mean will have • a z-score greater than zero. • A data value equal to the sample mean will have a • z-score of zero.

  15. z-Scores • z-Score of Smallest Value (425) Standardized Values for Apartment Rents

  16. of the values of a normal random variable are within of its mean. 68.26% +/- 1 standard deviation of the values of a normal random variable are within of its mean. 95.44% +/- 2 standard deviations of the values of a normal random variable are within of its mean. 99.72% +/- 3 standard deviations Empirical Rule For data having a bell-shaped distribution:

  17. 99.72% 95.44% 68.26% Empirical Rule x m m + 3s m – 3s m – 1s m + 1s m – 2s m + 2s

  18. Detecting Outliers • An outlier is an unusually small or unusually large • value in a data set. • A data value with a z-score less than -3 or greater • than +3 might be considered an outlier. • It might be: • an incorrectly recorded data value • a data value that was incorrectly included in the • data set • a correctly recorded data value that belongs in • the data set

  19. Detecting Outliers • The most extreme z-scores are -1.20 and 2.27 • Using |z| > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents

  20. Exploratory Data Analysis • Five-Number Summary • Box Plot

  21. Five-Number Summary 1 Smallest Value 2 First Quartile 3 Median 4 Third Quartile 5 Largest Value

  22. Five-Number Summary Lowest Value = 425 First Quartile = 445 Median = 475 Largest Value = 615 Third Quartile = 525

  23. Five-Number Summary 1 Smallest Value = 425 2 First Quartile = 445 3 Median = 475 4 Third Quartile = 525 5 Largest Value = 615

  24. 625 450 375 400 500 525 550 575 600 425 475 Box Plot • A box is drawn with its ends located at the first and third quartiles. • A vertical line is drawn in the box at the location of • the median (second quartile). Q1 = 445 Q3 = 525 Q2 = 475

  25. Box Plot • The lower limit is located 1.5(IQR) below Q1. Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(80) = 325 • The upper limit is located 1.5(IQR) above Q3. Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(80) = 645 • Data outside these limits are considered outliers. • The locations of each outlier is shown with the symbol * . • There are no outliers (values less than 325 or • greater than 645) in the apartment rent data.

  26. 625 450 375 400 500 525 550 575 600 425 475 Box Plot • Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. Smallest value inside limits = 425 Largest value inside limits = 615

  27. Weighted Mean • When the mean is computed by giving each data • value a weight that reflects its importance, it is • referred to as a weighted mean. • In the computation of a grade point average (GPA), • the weights are the number of credit hours earned for • each grade. • When data values vary in importance, the analyst • must choose the weight that best reflects the • importance of each value.

  28. Weighted Mean where: xi= value of observation i wi = weight for observation i

  29. GPA Example Grade#Credits (Weight)Product A 4 4 16 B 3 3 9 B 3 2 6 C 2 12 10 33 (sum of above) A = 4, B = 3, C = 2 GPA = 33/10 = 3.3

  30. Grouped Data • The weighted mean computation can be used to • obtain approximations of the mean, variance, and • standard deviation for the grouped data. • To compute the weighted mean, we treat the • midpoint of each class as though it were the mean • of all items in the class. • We compute a weighted mean of the class midpoints • using the class frequencies as weights. • Similarly, in computing the variance and standard • deviation, the class frequencies are used as weights.

  31. Mean for Grouped Data • Sample Data • Population Data where: fi = frequency of class i Mi = midpoint of class i

  32. Sample Mean for Grouped Data Given below is the previous sample of monthly rents for 70 efficiency apartments, presented here as grouped data in the form of a frequency distribution.

  33. Sample Mean for Grouped Data This approximation differs by $2.41 from the actual sample mean of $490.80.

  34. Variance for Grouped Data • For sample data • For population data

  35. continued Sample Variance for Grouped Data

  36. End of Chapter 3, Part B

More Related