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Q1 = 16 Q3 = 44 IQR = 26 * 1.5 = 42 Prob and Stats, Sep 16 Measures of Variation II – Understanding Measures of Variation and Other Relationships Book Sections: 2.4 Essential Questions: How do I compute and use statistical values? How are measures of variation used and how do relationships between categories manifest themselves? Standards: S.ID.1, .2, .3, .4 Back to the Future • We start this lesson at the doorstep of the basic data representation, the frequency histogram. Manifestation of a Distribution • A distribution (which is a sample set of data depicted graphically) will appear today as a frequency histogram with a curve imposed to illustrate symmetry. President’s Ages at First Inauguration Distribution Shapes There are four basic shapes that the frequency histogram takes on: • Symmetric • Uniform • Skewed Left • Skewed Right The Picture of Data Examples • Name the approximate distribution that each graph represents. 1) 2) 3) 4) More Examples (on the edge) • Name the approximate distribution that each graph represents. 5) 6) 7) 8) What Causes a Distribution to Be Skewed • If the mean of a distribution is right of the median, the set will be skewed right (the ‘bump’ is on the left) • If the mean of a distribution is left of the median, the set will be skewed left (the ‘bump’ is on the right) • In a perfectly symmetric data set, the mean and median are equal Threshold of Skewness • A distribution does not have to be perfect to be called symmetric (and thus for the Empirical Rule to apply). We will use an eyeball amount of tolerance on a not perfectly symmetrical data set. Mean and Standard Deviation • Mean and standard deviation are related statistics Remember that the mean was a component of the variance formula • Standard deviation is a measure of how close (on average) that the data is to the mean • If a data set is fairly symmetrical, there is a special rule that predicts how much data is within a multiple of s to the mean – It is called the Empirical Rule Understanding Measures of Variation What is the Relationship Between Mean & Standard Deviation The Empirical Rule Normally Distributed Data When the Empirical Rule Applies • Data that has a symmetric distribution (generally) is one in which the empirical rule applies. Empirical Rule Words and Graph Together • 68% of the data lies within one standard deviation of the mean • 95% of the data lies between two standard deviations of the mean • 99.7% of the data lies between two standard deviations of the mean Example 1 • A data set has a mean of 58 and a standard deviation of 19. What is the range of 68%, 95%, and 99.7% of the data? Example 2 Of a sample of 808 men measured, the mean was 69.9 in and the standard deviation was 3 in. • Roughly, which two heights contain the middle 95% of this data? Example 3 • A symmetric data set has a mean of 50 and a standard deviation of 10. What percent of the data is between 40 and 60? Class work: Classwork Handout 1-9 Homework: None