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Measures of Variation

Measures of Variation. Prepared by: Josefina V. Almeda Professor and College Secretary School of Statistics University of the Philippines, Diliman August 2009. Learning Objectives. After the session, participants should be able to:

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Measures of Variation

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  1. Measures of Variation Prepared by:Josefina V. AlmedaProfessor and College SecretarySchool of StatisticsUniversity of the Philippines, DilimanAugust 2009

  2. Learning Objectives After the session, participants should be able to: • Gain skills in the computation of the different quantitative measures of dispersion; • Describe and compare groups and individuals within groups using the measures of dispersion; • Interpret results obtained from each measure

  3. Measures of Dispersion • indicate the extent to which individual items in a series arescattered about an average. 1. Measures of Absolute Dispersion  Use to compare two or more data sets with the same means and the same units of measurement. 2. Measures of Relative Dispersion Used to compare two or more data sets with different means and different units of measurement.

  4. Measures of Variation Variation Coefficient of Variation Variance Standard Deviation Population Standard Deviation Range Population Variance Sample Variance Sample Standard Deviation

  5. Measures of Absolute Dispersion: Range and Standard Deviation Range • highest observation – lowest observation Standard deviation • is the positive square root of the variance and measures on the average the dispersion of each observation from the mean.

  6. Range • Difference Between Largest & Smallest • Observations: • Range = X Largest - X Smallest • Simple to compute and easy to understand • Quick measure of spread • Ignores How Data Are Distributed

  7. Range = 12 - 7 = 5 Range = 12 - 7 = 5 7 8 9 10 11 12 7 8 9 10 11 12 Range Q: What is the range of the monthly salary of people working in Y company? 3050, 3273, 3552, 4009, 5118, 6370, 8950, 10835 R = 10835 – 3050 = 7785

  8. Variance • Important measure of variation • Shows variation about the mean • Sample variance: • Population variance:

  9. Standard Deviation • Most important measure of variation • Shows variation about the mean • Has the same units as the original data • It is always positive • Sample standard deviation: • Population standard deviation:

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

  11. Sample vs Pop’n SD Data :Xj 10 12 14 15 17 18 18 24 N= 8 Mean =16 s = = 4.2426 = 3.9686 Value for the Standard Deviation is larger for data considered as a Sample.

  12. Standard Deviation Remarks: 1. If there is a large amount of variation in the data set, then on the average, the data values will be far from the mean. Hence, the standard deviation will be large. 2. If there is only a small amount of variation in the data set, then on the average, the data values will be close to the mean. Hence, the standard deviation will be small.

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

  14. Comparing Standard Deviations Example: Team A - Heights of five marathon players in inches Mean = 65 s = 0 65 “ 65 “ 65 “ 65 “ 65 “

  15. Comparing Standard Deviations Example: Team B - Heights of five marathon players in inches Mean = 65” s = 3.6” 62 “ 67 “ 66 “ 70 “ 60 “

  16. Standard Deviation Advantages 11. It is the most widely used measure of dispersion. It is based on all the items and is rigidly defined.  2. It is of great significance for testing the reliability of measures calculated from samples, the difference between such measures, and in comparing the extent of fluctuation in two or more samples.

  17. Standard Deviation Disadvantages • D • The standard deviation is sensitive to the presence of extreme values. • It is not easy to calculate by hand.

  18. Measure of relative dispersion • are unitless and are used to compare the scatter of one distribution with the scatter of another distribution. Coefficient of Variation • utilizes two measures and these are the mean and the standard deviation.  is a percentage

  19. ComparingCV’s • Stock A:Average Price last year =P50 • Standard Deviation =P5 • Stock B:Average Price last year =P100 • Standard Deviation =P5 Coefficient of Variation: Stock A:CV = 10% Stock B:CV = 5%

  20. Thank you.

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