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MAT 135 Introductory Statistics and Data Analysis Adjunct Instructor Kenneth R. Martin Lecture 8 October 19, 2016 Confidential - Kenneth R. Martin Agenda • Housekeeping – Readings – HW #5 – Quiz #2 • Chapter 1, 14, 10, 2, & 3 Confidential - Kenneth R. Martin Housekeeping • • • • • • Read, Chapter 1.1 – 1.4 Read, Chapter 14.1 – 14.2 Read, Chapter 10.1 Read, Chapter 2 Read, Chapter 3 Read, Chapter 4 Confidential - Kenneth R. Martin Housekeeping • HW #5 issued Confidential - Kenneth R. Martin Housekeeping • Quiz #2 – Wednesday, October 26th – Chapter 3 material – Same format as Quiz #1 Confidential - Kenneth R. Martin Review • What have we learned so far ? Confidential - Kenneth R. Martin Statistics Standard Deviation - Example B = {50, 150, 300, 450, 550} Confidential - Kenneth R. Martin Statistics Normal Curve • AKA, Gaussian distribution • Mean, Median, and Mode have the approx. same value. – Associated with mean () at center and dispersion () X N(,) [when a random variable x is distributed normally] – Observations have equal likelihood on both sides of mean *** When data is normally distributed, Mean is used to describe Central Tendency • The graph of the associated distribution is called “Bell Shaped” Confidential - Kenneth R. Martin Statistics Standard Normal Curve - Distribution of Data Confidential - Kenneth R. Martin Statistics Standard Normal Curve - Distribution of Data Confidential - Kenneth R. Martin Statistics Measures of Dispersion – Chebyshev’s Theorem • Specifies the minimum proportions of the spread of data in terms of SD • Applies to any distribution, regardless of shape. Confidential - Kenneth R. Martin Statistics Measures of Dispersion – Chebyshev’s Theorem Confidential - Kenneth R. Martin Statistics Measures of Dispersion – Chebyshev’s Theorem μ σ Confidential - Kenneth R. Martin Statistics Chebyshev’s Theorem - Example Confidential - Kenneth R. Martin Statistics Chebyshev’s Theorem - Example Confidential - Kenneth R. Martin Statistics Chebyshev’s Theorem - Example Confidential - Kenneth R. Martin Statistics Measures of Position – Percentiles • Percentiles are divisions of data in 100 equal groups. • Percentiles are measures to indicate the position of an individual data point within a group. – • Thus, a percentile ranking of a data point indicates its position in the data set, with respect to the other data points. Percentile graphs are the same as Cum. Freq. graphs using percentages. Confidential - Kenneth R. Martin Statistics Measures of Position – Percentiles Confidential - Kenneth R. Martin Statistics Measures of Position – Percentiles Confidential - Kenneth R. Martin Statistics Percentiles - Example Confidential - Kenneth R. Martin Statistics Percentiles - Example Confidential - Kenneth R. Martin Statistics Percentiles - Example Confidential - Kenneth R. Martin Statistics Percentiles - Example Confidential - Kenneth R. Martin Statistics Percentiles - Example Confidential - Kenneth R. Martin Statistics Box and Whisker Plot – Boxplot Simple graphical tool to summarize data. Need to determine 5 values (five-number summary) from data, to generate a boxplot: 1. 2. 3. 4. 5. Median (2nd Quartile) Maximum data value Minimum data value 1st Quartile (value above 1/4 observations) [whisker] 3rd Quartile (value above 3/4 observations) [whisker] Confidential - Kenneth R. Martin Statistics Box and Whisker Plot – Boxplot Example • Process aim = 9.0 minutes • Spec = + / - 1.5 minutes • n = 125 • R = 1.7 Confidential - Kenneth R. Martin Statistics Box and Whisker Plot - Boxplot Example Inside box is the median value, and approximately 50% of observations Whiskers extend from the box to extreme values • Example: 1. 2. 3. 4. 5. Median; n=125: Median = 63rd value = 9.8 Max = 10.7 Min = 9.0 1st Quartile = X 125 * 0.25 ~ X Avg 31 & 32 value = 9.6 3rd Quartile = X 125 * 0.75 ~ X Avg 94 & 95 value = 10.0 Confidential - Kenneth R. Martin Statistics Box and Whisker Plot - Boxplot Example • Long Whiskers denote the existence of values much larger than other values. • • For this example, mean median. Variants exist, i.e. + / - 1.5*IQR [whisker ends], all other points are “outliers” as depicted as asterisks • IQR = Inner Quartile Range Confidential - Kenneth R. Martin Statistics Box and Whisker Plot - Boxplot Example Confidential - Kenneth R. Martin Statistics Measures of Variability (Dispersion) - IQR • IQR – Interquartile Range IQR = Q3 – Q1 Confidential - Kenneth R. Martin Statistics Box and Whisker Plot - Boxplot Example • For this example, IQR = ? Confidential - Kenneth R. Martin