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Understanding Measures of Variation in Statistics

Learn about quartiles, range, standard deviation, and variance in statistics to analyze data variability and make informed decisions. Explore practical examples and calculations to enhance your statistical analysis skills.

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Understanding Measures of Variation in Statistics

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  1. SP 225Lecture 8 Measures of Variation

  2. Challenge Question • A randomized, double-blind study of 50 subjects shows daily administration of Echinacea supplements shortens the average duration of an Upper Respiratory Infection (URI) from 14 to 13 days. • Based on this study, is Echinacea an effective treatment for URI’s?

  3. Roll of the Dice • All outcomes are equally likely • The probability of any outcome is 1/6 or 16.7%

  4. Casinos Patrons: Risky Fun Red, White and Blue Slots • 82% chance of loss on any spin • Prizes for a dollar bet range from $2400 to $1 • Patrons are expected to lose $0.10 for each dollar bet

  5. Casinos: False Risk Soaring Eagle • 4300 slot machines • 25 spins per hour • Open 24/7/365 • 94,170,000 possible spins

  6. Statistics vs. Parameters • Statistics: numerical description of a sample • Parameter: numerical description of a population • Statistics are calculated randomly selected members of a population

  7. Differences Between Statistics and Parameters Sample: 3 Randomly Selected People Statistic: 0 of 3 or 0% wear glasses Population: All People Parameter: 5 of 15 or 33% wear glasses

  8. Random Sampling Activity • Number of siblings of each student in the freshman class of Powers Catholic High school • Take 3 samples, with replacement, of sizes 1, 5 and 10 • Calculate the sample mean • Record results in class data chart

  9. Challenge Question • A randomized, double-blind study of 50 subjects shows daily administration of Echinacea supplements shortens the average duration of an Upper Respiratory Infection (URI) from 14 to 13 days. • Based on this study, is Echinacea an effective treatment for URI’s?

  10. Why Do We Need Measures of Variation? • What is the average height of a male child? • How many children are that tall? • When is a child unusually tall or short?

  11. Range • Difference between the maximum and minimum value • Quick to Compute • Not Comprehensive Range = (maximum value) – (minimum value)

  12. Quartiles • Often used in the education field • Can be used with any data distribution • Measures distance in relation to the MEDIAN not MEAN

  13. Quartiles • Q1 (First Quartile)separates the bottom 25% of sorted values from the top 75%. • Q2 (Second Quartile)same as the median; separates the bottom 50% of sorted values from the top 50%. • Q3 (Third Quartile)separates the bottom 75% of sorted values from the top 25%.

  14. 25% 25% 25% 25% Q1 Q2 Q3 (minimum) (maximum) (median) Quartiles (2) Q1, Q2, Q3 dividerankedscores into four equal parts

  15. Quartile Statistics • Interquartile Range (or IQR): Q3- Q1

  16. Example • Given the following data calculate Q1, Q2 and Q3 • 4.2, 4.4, 5.1, 5.6, 6.0, 6.4, 6.8, 7.1, 7.4, 7.4, 7.9, 8.2, 8.2, 8.7, 9.1, 9.6, 9.6, 10.0, 10.5, 11.6

  17. Example Continued http://www.maths.murdoch.edu.au/units/statsnotes/samplestats/boxplot.html

  18. (x - x)2 s= n -1 Standard Deviation for a Population • Calculated by the following formula: • Used to show distance from the mean • Tells how usual, or unusual a measurement is  =

  19. Standard Deviation for a Sample (x - x)2 s= n -1

  20. Standard Deviation - Important Properties • Standard Deviation is always positive • Increases dramatically with outliers • The units of standard deviation s are the same as the units of the mean

  21. Calculating the Standard Deviation of a SAMPLE • Data points 1, 3, 5, 7, 9

  22. Variance • A measure of variation equal to the square of the standard deviation • Sample Variance = s • Population Variance = 2 2

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