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20, 22, 23, 24, 24, 25, 25, 27, 35

20, 22, 23, 24, 24, 25, 25, 27, 35. Are there any outliers? Draw a skeleton boxplot. Draw a modified boxplot. Chebyshev’s & The Empirical Rule. Describing Data in terms of the Standard Deviation. Test Mean = 80 St. Dev. = 5. Chebyshev’s Rule.

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20, 22, 23, 24, 24, 25, 25, 27, 35

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  1. 20, 22, 23, 24, 24, 25, 25, 27, 35 • Are there any outliers? • Draw a skeleton boxplot. • Draw a modified boxplot.

  2. Chebyshev’s & The Empirical Rule

  3. Describing Data in terms of the Standard Deviation. Test Mean = 80 St. Dev. = 5

  4. Chebyshev’s Rule The percent of observations that are within k standard deviations of the mean is at least

  5. Facts about Chebyshev • Applicable to any data set – whether it is symmetric or skewed. • Many times there are more than 75% - this is a very conservative estimation.

  6. # St. Dev. % w/in k st. dev. of mean 2 3 4 4.472 5 10

  7. Interpret using Chebyshev Test Mean = 80 St. Dev. = 5 • What percent are between 75 and 85? • What percent are between 60 and 100?

  8. Collect wrist measurements (in) • Create distribution • Find st. dev & mean. • What percent is within 1 deviation of mean

  9. Practice Problems • Using Chebyshev, solve the following problem for a distribution with a mean of 80 and a st. dev. Of 10. • a. At least what percentage of values will fall between 60 and 100? • b. At least what percentage of values will fall between 65 and 95?

  10. Normal Distributions • These are special density curves. • They have the same overall shape • Symmetric • Single-Peaked • Bell-Shaped • They are completely described by giving its mean () and its standard deviation (). • We abbreviate it N(,)

  11. Normal Curves…. • Changing the mean without changing the standard deviation simply moves the curve horizontally. • The Standard deviation controls the spread of a Normal Curve.

  12. Standard Deviation • It’s the natural measure of spread for Normal distributions. • It can be located by eye on a Normal curve. • It’s the point at which the curve changes from concave down to concave up.

  13. Why is the Normal Curve Important? • They are good descriptions for some real data such as • Test scores like SAT, IQ • Repeated careful measurements of the same quantity • Characteristics of biological populations (height) • They are good approximations to the results of many kinds of chance outcomes • They are used in many statistical inference procedures.

  14. Empirical Rule • Can only be used if the data can be reasonably described by a normal curve. • Approximately • 68% of the data is within 1 st. dev. of mean • 95% of the data is within 2 st. dev. of mean • 99.7% of data is within 3 st. dev. of mean

  15. Empirical Rule • What percent do you think…… • www.whfreeman.com/tps4e

  16. Empirical Rule (68-95-99.7 Rule) • In the Normal distribution with mean () and standard deviation (): • Within 1 of  ≈ 68% of the observations • Within 2 of  ≈ 95% of the observations • Within 3 of  ≈ 99.7% of the observations

  17. The distribution of batting average (proportion of hits) for the 432 Major League Baseball players with at least 100 plate appearances in the 2009 season is normally distributed defined N(0.261, 0.034). • Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard deviations from the mean. • What percent of the batting averages are above 0.329? • What percent are between 0.227 and .295?

  18. Scores on the Wechsler adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed. N(110, 25). • What percent are between 85 and 135? • What percent are below 185? • What percent are below 60?

  19. A sample of the hourly wages of employees who work in restaurants in a large city has a mean of $5.02 and a st. dev. of $0.09. • a. Using Chebyshev’s, find the range in which at least 75% of the data will fall. • b. Using the Empirical rule, find the range in which at least 68% of the data will fall.

  20. The mean of a distribution is 50 and the standard deviation is 6. Using the empirical rule, find the percentage that will fall between 38 and 62.

  21. A sample of the labor costs per hour to assemble a certain product has a mean of $2.60 and a standard deviation of $0.15, using Chebyshev’s, find the values in which at least 88.89% of the data will lie.

  22. Homework • Worksheet • Quiz Monday on Boxplots & outliers

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