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Business Statistics Spring 2005

Business Statistics Spring 2005. Summarizing and Describing Numerical Data. Topics. Measures of Central Tendency Mean, Median, Mode, Midrange, Midhinge Quartile Measures of Variation The Range, Interquartile Range, Variance and Standard Deviation, Coefficient of variation

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Business Statistics Spring 2005

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  1. Business Statistics Spring 2005 Summarizing and Describing Numerical Data

  2. Topics • Measures ofCentral Tendency • Mean, Median, Mode, Midrange, Midhinge • Quartile • Measures ofVariation • The Range, Interquartile Range, Variance and • Standard Deviation, Coefficient of variation • Shape • Symmetric, Skewed, using Box-and-Whisker • Plots

  3. Numerical Data Properties Central Tendency (Location) Variation (Dispersion) Shape

  4. Measures of Central Tendency forUngrouped Data Raw Data

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

  6. Measures of Central Tendency Central Tendency Mean Median Mode Midrange Midhinge

  7. 3-2 Population Mean • For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: • where µ stands for the population mean. • Nis the total number of observations in the population. • X stands for a particular value. • S indicates the operation of adding.

  8. Population Mean Example 3-3 • Parameter: a measurable characteristic of a population. • The Kane family owns four cars. The following is the mileage attained by each car: 56,000, 23,000, 42,000, and 73,000. Find the average miles covered by each car. • The mean is (56,000 + 23,000 + 42,000 + 73,000)/4 = 48,500

  9. 3-4 Sample Mean • For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: • where X stands for the sample mean • n is the total number of values in the sample

  10. Return on Stock Stock X Stock Y 1998 1997 1996 1995 1994 10% 8 12 2 8 17% -2 16 1 8 40% 40% Average Return on Stock = 40 / 5 = 8%

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

  12. 3-6 Properties of the Arithmetic Mean • Every set of interval-level and ratio-level data has a mean. • All the values are included in computing the mean. • A set of data has a unique mean. • The mean is affected by unusually large or small data values. • The mean is relatively reliable. • The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero.

  13. 3-7 EXAMPLE • Consider the set of values: 3, 8, and 4. • The mean is 5. • Illustrating the fifth property, (3-5) + (8-5) + (4-5) = -2 +3 -1 = 0. In other words,

  14. 3-10 The Median • Median: The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. There are as many values above the median as below it in the data array. • Note: For an even set of numbers, the median will be the arithmetic average of the two middle numbers.

  15. n + 1 Positionin g Point = 2 Median • Position of Median in Sequence

  16. The Median • 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 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5

  17. The Mode • 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 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9

  18. 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 0 1 2 3 4 5 6 7 8 9 10 Midrange = 5 Midrange = 5

  19. Quartiles • Not a Measure of Central Tendency • Split Ordered Data into 4 Quarters • Position of i-th Quartile: position of point 25% 25% 25% 25% Q1 Q2 Q3 i(n+1) Q  i 4 Data in Ordered Array: 11 12 13 16 16 17 18 21 22 1•(9 + 1) = Position of Q1 = 2.50 Q1 =12.5 4

  20. Quartiles • See text page 107 for “rounding rules” for position of the i-th quartile • Position (not value) of i-th Quartile: 25% 25% 25% 25% Q1 Q2 Q3 Q  i(n+1) i 4

  21. Midhinge • A Measure of Central Tendency • The Middle point of 1st and 3rd Quarters • Used to Overcome Extreme Values Midhinge = Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Midhinge =

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

  23. Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Range Population Variance Population Standard Deviation Sample Variance Sample Standard Deviation Interquartile Range

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

  25. Return on Stock Stock X Stock Y 1998 1997 1996 1995 1994 10% 8 12 2 8 17% -2 16 1 8 Range on Stock X = 12 - 2 = 10% Range on Stock Y = 17 - (-2) = 19%

  26. Interquartile Range • Measure of Variation • Also Known asMidspread: • Spread in the Middle 50% • Difference Between Third & First • Quartiles:Interquartile Range = Data in Ordered Array: 11 12 13 16 16 17 17 18 21 = 17.5 - 12.5 = 5

  27. Interquartile Range • IQR = 75th percentile - 25th percentile • The IQR is useful for checking for outliers • Not Affected by Extreme Values Data in Ordered Array: 11 12 13 16 16 17 17 18 21 = 17.5 - 12.5 = 5

  28. = 8.3 X Variance & Standard Deviation • Measures of Dispersion • Most Common Measures • Consider How Data Are Distributed • Show Variation About Mean (`X or m) 4 6 8 10 12

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

  30. 4-5 Population Variance • The population variance for ungrouped data is the arithmetic mean of the squared deviations from the population mean.

  31. Population Variance EXAMPLE • The ages of the Dunn family are 2, 18, 34, and 42 years. What is the population variance?

  32. PopulationStandard Deviation

  33. Population Standard Deviation EXAMPLE • The ages of the Dunn family are 2, 18, 34, and 42 years. What is the population variance?

  34. Standard Deviation • Most Important Measure of Variation • Shows Variation About the Mean: • For the Population: • For the Sample: For thePopulation:use Nin the denominator. For theSample : usen - 1in the denominator.

  35. The sample standard deviation = Sample Variance and Standard Deviation • The sample variance estimates the population • am variance. NOTE: important computation formriance estimates the population variance.

  36. = 129.71 = s = s = = Example of Standard Deviation 2 2

  37. Example of Standard Deviation(Computational Version) 2 = = 129.71 s =

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

  39. 4-14 Interpretation and Uses of the Standard Deviation • Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least 1 - 1/k2 where k is any constant greater than 1. • Multiply by 100% to get percentage of values within k standard deviations of the mean

  40. 4-15 Interpretation and Uses of the Standard Deviation • Empirical Rule: For any symmetrical, bell-shaped distribution, approximately 68% of the observations will lie within of the mean ();approximately 95% of the observations will lie within of the mean ( ); approximately 99.7% will lie within of the mean ( ).

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

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

  43. Coefficient of Variation • Measure ofRelative Variation • Always a% • Shows Variation Relative to Mean • Used toCompare 2 or More Groups • Formula ( for Sample):

  44. 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%

  45. Shape • Describes How Data Are Distributed • Measures of Shape: • Symmetric or skewed Right-Skewed Left-Skewed Symmetric Mean Median Mode Mean = Median = Mode Mode Median Mean

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

  47. Distribution Shape & Box-and-Whisker Plots Left-Skewed Symmetric Right-Skewed Q Median Q Q Median Q Q Median Q 1 3 1 3 3 1

  48. Summary • Discussed Measures ofCentral Tendency • Mean, Median, Mode, Midrange, Midhinge • Quartiles • Addressed Measures ofVariation • The Range, Interquartile Range, Variance, • Standard Deviation, Coefficient ofVariation • Determined Shape of Distributions • Symmetric, Skewed, Box-and-Whisker Plot Mean Median Mode Mean = Median = Mode Mode Median Mean

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