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Warm Up Find the variance for the given data. Round your answer to one more decimal place than the original data. 77251 Find the standard deviation for the given sample data. Round your answer to one more decimal place than is present inthe original data. 18 18 18 9 15 5 10 5 15 RECALL:Sample Standard Deviation Formula ( x x ) s n 1 x bar is the sample mean n is the number or sample values Sigma means add s is the standard deviation 2 RECALL: Variance is the square of standard deviation. s ( x x ) n 1 2 ( x x ) s n 1 2 To find variance 1.Find the mean=xbar 2. Find n and n-1 3. Find each x-xbar (subtract xbar from each data value.) 4. Square each x-xbar 5. Add these and divide by n1 To find standard deviation 1. Find the mean=xbar 2. Find n and n-1 3. Find each x-xbar (subtract xbar from each data value.) 4. Square each x-xbar 5. Add these and divide by n1 6. Square root Solution#1 Find the variance for the given data. Round your answer to one more decimal place than the original data. 77251 To find the variance: Find the mean = 4.4 Find n = 5 so n-1 = 4 Find each x-xbar = 2.6, 2.6,-2.4, .6, -3 Square each x-xbar = 6.76,6.76,5.76,.36,11.56 Add these and divide by n-1 = 7.8 The variance is 7.8. Solution#2 Find the standard deviation for the given sample data. Round your answer to one more decimal place than is present in the original data. 18 18 18 9 15 5 10 5 15 To find the standard deviation: Find the mean = 12.56 Find n = 9 so n-1 = 8 Find each x-xbar = 5.44, 5.44,5.44, -3.56, 2.44,-7.56, -2.56, -7.56, 2.44 Square each x-xbar =29.64, 29.64, 29.64, 12.64, 5.98, 57.09, 6.53, 57.09, 5.98 Add these and divide by n-1 = 29.29 6. Square root = 5.41 (Note: we do not square root for variance.) The standard deviation is 5.41 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots FIRST-Empirical (or 68-95-99.7) Ru For data sets having a distribution that is approximately bell shaped, the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean. The Empirical Rule The Empirical Rule (In English) Assuming the values are normally distributed, Most (95%) values are within 2 standard deviations of the mean. A value within 2 sd’s of the mean is considered USUAL A value outside of 2 sd’s of the mean is considered UNUSUAL Example The author of the text measured his pulse rate to be 48 beats per minute. Is that pulse rate unusual if the mean adult male pulse rate is 67.3 beats per minute with a standard deviation of 10.3? SOLUTION: Find the interval (48-2*10.3,48+2*10.3) = (27.4, 68.6). Is the data value in that interval? Yes. It is usual. Finding the interval is a long way to test usual. Today we learn a faster way called z scores. Objective We will use z scores to test whether values are usual. Measures of relative standing Numbers showing the location of data values relative to the other values within a data set. These are used to -compare values from different data sets -compare values within the same data set. The most important measure of relative standing is the z score. Also important are percentiles and quartiles, as well as a new statistical graph called the boxplot. Part 1 Basics of z Scores, Percentiles, Quartiles, and Boxplots z score z Score (or standardized value) The z score of a data value is the number of standard deviations that the data value is away from the mean Measures of Position z Score Sample xx z s Population z x Round z scores to 2 decimal places Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: 2 z score 2 Unusual Values: z score 2 or z score 2 Example using z scores The author of the text measured his pulse rate to be 48 beats per minute. Is that pulse rate unusual if the mean adult male pulse rate is 67.3 beats per minute with a standard deviation of 10.3? x x 48 67.3 z 1.87 s 10.3 Answer: Since the z score is between – 2 and +2, his pulse rate is not unusual. Percentiles are measures of location. Percentiles are denoted P1, P2, . . ., P99, which divide a set of data into 100 groups with about 1% of the values in each group. Given the Data Value Find the Percentile To find the Percentile of a data value x the number of values less than x total number of values Count the number of values less than x Divide by total number of values Convert to a percent EXAMPLE For the 40 Chips Ahoy cookies, find the percentile for a cookie with 23 chips. Answer: We see there are 10 cookies with fewer than 23 chips, so 10 Percentile of 23 100 25 40 A cookie with 23 chips is in the 25th percentile. Given the Percentile Find the Data Value Notation k L n 100 1. Multiply the number of data values*the percentile/100 2. Count up that many from the bottom percentile being used locator that gives the position of a value Pk kth percentile k L Converting from the kth Percentile to the Corresponding Data Value Quartiles Are measures of location, denoted Q1, Q2, and Q3, which divide a set of data into four groups with about 25% of the values in each group. 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%. Quartiles Q1, Q2, Q3 divide sorted data values into four equal parts 25% (minimum) 25% 25% 25% Q1 Q2 Q3 (median) (maximum) Other Statistics Interquartile Range (or IQR): Semi-interquartile Range: Midquartile: Q3 Q1 Q3 Q1 2 Q3 Q1 2 10 - 90 Percentile Range: P90 P10 5-Number Summary For a set of data, the 5-number summary consists of these five values: 1. Minimum value 2. First quartile Q1 3. Second quartile Q2 (same as median) 4. Third quartile, Q3 5. Maximum value Boxplot A boxplot (or box-and-whisker-diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1, the median, and the third quartile, Q3. Boxplot - Construction Find the 5-number summary. Construct a scale with values that include the minimum and maximum data values. Construct a box (rectangle) extending from Q1 to Q3 and draw a line in the box at the value of Q2 (median). Draw lines extending outward from the box to the minimum and maximum values. Boxplots Boxplots - Normal Distribution Normal Distribution: Heights from a Simple Random Sample of Women Boxplots - Skewed Distribution Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches Part 2 Outliers and Modified Boxplots Outliers An outlier is a value that lies very far away from the vast majority of the other values in a data set. Important Principles An outlier can have a dramatic effect on the mean and the standard deviation. An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured. Putting It All Together So far, we have discussed several basic tools commonly used in statistics – Context of data Source of data Sampling method Measures of center and variation Distribution and outliers Changing patterns over time Conclusions and practical implications This is an excellent checklist, but it should not replace thinking about any other relevant factors. Quiz Solve the problem. Round results to the nearest hundredth. 1) Scores on a test have a mean of 66 and a standard deviation of 9. Michelle has a score of 57. Convert Michelle's score to a z-score. 2) Find the standard deviation and variance for the data 26 32 29 16 45 19. Find the indicated measure. 3) The test scores of 40 students are listed below. Find P85. 30 35 43 44 47 48 54 55 56 57 59 62 63 65 66 68 69 69 71 72 72 73 74 76 77 77 78 79 80 81 81 82 83 85 89 92 93 94 97 98 4) Find the data value that corresponds to P30. HW 123/1-31 odds 127,128 all problems