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Statistics: How We View the World…

Statistics: How We View the World…. Day 1: Center and Spread. One-Minute Question. Find the mean of the following grades: {70, 70, 80, 92, 98, 100}. One-Minute Question. Arithmetic Mean = average = (70+70+80+92+98+100)/6 = 85. Review.

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Statistics: How We View the World…

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  1. Statistics: How We View the World… Day 1: Center and Spread

  2. One-Minute Question • Find the meanof the following grades: • {70, 70, 80, 92, 98, 100}

  3. One-Minute Question • Arithmetic Mean = average = (70+70+80+92+98+100)/6 • = 85

  4. Review • Find all the other “Measures of Central Tendency” that you know… • {70, 70, 80, 92, 98, 100}

  5. Review • Did you name… • Median = 86 • Mode = 70 • {70, 70, 80, 92, 98, 100}

  6. Review • Besides “measures of Central Tendency”, what else do we care about?

  7. Review • Find all the “Measures of Dispersion” that you know… • {70, 70, 80, 92, 98, 100}

  8. Review • Range = 30 • Mean Absolute Deviation = 11.67 • Inner-Quartile Range (IQR) = 28 • {70, 70, 80, 92, 98, 100}

  9. New Concept • Mean Absolute Deviation is rarely used, but Standard Deviationis used quite often… • {70, 70, 80, 92, 98, 100}

  10. To find Standard Deviation • Find each number’s difference from the mean. 2. Square these differences. • Find the average of these squared differences. 4. Take the square root of your result. {70, 70, 80, 92, 98, 100}

  11. Standard Deviation • In other words: • Standard Deviation = σ • σ = • {70, 70, 80, 92, 98, 100}

  12. Standard Deviation • In other words: • Standard Deviation = • {70, 70, 80, 92, 98, 100}

  13. Variance = (Standard Deviation)2

  14. So, Who Cares?? • All of us who are “Normal”!

  15. So, Who Cares?? • Remember that histograms are graphs of the distribution of data??? • Well, think about other – more general distributions.

  16. So, Who Cares?? • Suppose we roll a single die 1,000,000 times. • What should the distribution of number of dots on the top face look like? • Uniform distribution • (Close to constant!) Frequency 1 2 3 4 5 6

  17. So, Who Cares?? • What should the distribution be for a VERY, VERY easy test? • It should be skewed to the left. • (There should be lots of data to the right of the graph.) Frequency F D C B A

  18. So, Who Cares?? • But the heights of 20 year old males is normal. • That means that the data forms a bell-shape that is symmetric about the mean. Frequency

  19. Furthermore, the percentage of the area covered by each standard deviation from the mean is shown by this graph.

  20. So, suppose the mean of a normal distribution is 100 with a standard deviation of 15. What percentage of the scores lie between 85 and 115?

  21. So, suppose the mean of a normal distribution is 100 with a standard deviation of 15. What percentage of the scores lie between 70 and 115?

  22. In order to find how many standard deviations away an x value is from the mean we use a z-score.

  23. If the mean of a set of test grades is 84 with a standard deviation of 6, what is the z-score for a test grade of 96? What percent of the other students scored lower that a 96?

  24. For homework,work all the odd problems on pages 262 and 266

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