Statistical Reasoning for everyday life

Statistical Reasoningfor everyday life Intro to Probability and Statistics Mr. Spering – Room 113

4.2 Shapes of Distribution • CLASS WORK: • Worksheet REVIEW • ACTIONS are REMEMBERED, WORDS can be FORGOTTEN! • MAKE an EFFORT, NOT an EXCUSE

4.2 Shapes of Distribution • Variation: • Describes how widely data are spread out about the center of a distribution. • ????How would you expect the variation to differ between the heights of NCAA Division 1A Men’s College Basketball Centers and the heights of all High School Boy Basketball Players???? • NCAA Division 1A Centers less variation • High School Boy Basketball Players more variation

4.3 Measures of Variation • How do we investigate variation? • Study all of the raw data… • Range… • Quartiles… • Five-number summary (BOXPLOT or BOX-and-WHISKER)… • Interquartile range… • Semi-quartile range… • Percentiles… • MAD… • Variance & Standard Deviation…

4.3 Measures of Variation • RANGE: • The range of a distribution is the difference between the highest and lowest data values.

4.3 Measures of Variation • Find the range of the data. • 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3, 11.0 • Range = 11.0 – 4.1 = 6.9

4.3 Measures of Variation • Misleading range: • Which Quiz Set has greater variation? Quiz Set 1: 1, 10, 10, 10, 10, 10, 10, 10, 10, 10 Quiz Set 2: 2, 3, 4, 5, 6, 7, 8, 9, 10, 8, 9, 10, 6, 5 ** Even though Set 1 has a greater range than Set 2 (9 > 8). Set 2 has a greater variation because Set 1 contains an outlier. Therefore, we use quartiles.**

4.3 Measures of Variation • Quartiles: • Quartiles divide the data into four quarters. • Lower Quartile (1st Quartile): is the median of the data values in the lower half of a data set. Exclude the middle value in the data set if the number of data points is odd. • Middle Quartile (2nd Quartile): is the overall median • Upper Quartile (3rd Quartile): is the median of the data values in the upper half of a data set. Exclude the middle value in the data set if the number of data points is odd.

4.3 Measures of Variation • Find quartiles… Example 1 – Upper and lower quartiles Lower Quartile Q1 Median Q2 Upper Quartile Q3

4.3 Measures of Variation • Example 2 – Range and quartiles • A year ago, Angela began working at a computer store. Her supervisor asked her to keep a record of the number of sales she made each month. • The following data set is a list of her sales for the last 12 months: • 34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37 • Use Angela's sales records to find: • the median • b) the range • c) the upper and lower quartiles • Find quartiles…

4.3 Measures of Variation • Answers • The values in ascending order are: • 1, 11, 15, 19, 20, 24, 28, 34, 37, 47, 50, 57. a) Median = (6th + 7th observations) ÷ 2 = (24 + 28) ÷ 2 = 26 b) Range = difference between the highest and lowest values = 57 - 1 = 56

4.3 Measures of Variation c) Lower quartile = value of middle of first half of data Q1 = the median of 1, 11, 15, 19, 20, 24 = (3rd + 4th observations) ÷ 2 = (15 + 19) ÷ 2 = 17 d) Upper quartile = value of middle of second half of data Q3 = the median of 28, 34, 37, 47, 50, 57 = (3rd + 4th observations) ÷ 2 = (37 + 47) ÷ 2 = 42 These results can be summarized as follows:

4.3 Measures of Variation • Five-number summary: • Consists of the following… • Low Value • Q1 (lower quartile) • Q2 (median) • Q3 (upper quartile) • High Value # Summary

4.3 Measures of Variation Vertical box plot showing “normal” distribution “FORESHADOWING” • BOXPLOT or BOX-and-WHISKER: • Box plots show variation along the number line. Steps for creating a box plot: • Draw a number line that spans the entire data set. • Above the number line, enclose the values from the lower to the upper quartile in a box. • Draw a line through the box at the value corresponding to the median. • Add “whiskers” extending to the low and high values.

4.3 Measures of Variation • Example of 5 number summary and box plot. So for the data set 1, 4, 9, 12, 12, 16, 23, 24 here is our box plot:

4.3 Measures of Variation Digest of BOXPLOTS and SKEWNESS Symmetric Right-Skewed Left-Skewed Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3

4.3 Measures of Variation • Below is a Box-and-Whisker plot for the following data: 0 2 2 2 3 3 4 5 5 10 27 • The data are right skewed, as the plot depicts Min Q1 Q2 Q3 Max 0 2 3 5 27

4.3 Measures of Variation • Interquartile range: i.e. If the five number summary is low: 3, high: 23, Q1: 4, Q2: 12, Q3: 19. Then the interquartile range is IQR: (Q3-Q1) = (19 – 4) =15. Interquartile range The interquartile range is another range used as a measure of the variation. The difference between upper and lower quartiles (Q3–Q1), which is called the interquartile range, also indicates the dispersion of a data set. The inter-quartile range spans 50% of a data set, and eliminates the influence of outliers because, in effect, the highest and lowest quarters are removed.

4.3 Measures of Variation • Next Time: • Semi-quartile range… • Percentiles… • MAD… • Variance & Standard Deviation… • According to the box-n-whisker above what are the values for the 5 number summary: • Low: 12 • Q1: 22 • Q2: 31 • Q3: 45 • High: 50

4.3 Measures of Variation • Classwork: • PRACTICE MAKES PERMANENT • Pg 174 # 2-6 even and # 25-27 (Letters a, b only)

Statistical Reasoning for everyday life