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Describing Data: Percentiles Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

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4-2 LO1. Compute and understand quartiles, deciles, and percentiles. LO2. Construct and interpret box plots. LEARNING OBJECTIVES

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4-3 Quartiles, Deciles and Percentiles The median splits the data into equal sized halves Quartiles split the data into quarters Deciles into tenths And percentiles can be any split of our choosing These measures include quartiles, deciles, a Learning Objective 2 Compute and understand quartiles, deciles, and percentiles.

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4-4 Median 50% value - 50% 50% - 25% Q1 Q2 Q3 Quartiles 10% Deciles 1/10 10% Lowest Data Value Highest Data Value

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4-5 Percentile Computation To formalize the computational procedure, let L p refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L 33 and if we wanted the median, the 50th percentile, then L 50. The number of observations is n, so if we want to locate the median, its position is at (n + 1)/2, or we could write this as (n + 1)(P/100), where P is the desired percentile. LO2

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4-6 Percentiles - Example Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney’s Oakland, California, office. $2,038 $1,758 $1,721 $1,637 $2,097 $2,047 $2,205 $1,787 $2,287 $1,940 $2,311 $2,054 $2,406 $1,471 $1,460 Locate the median, the first quartile, and the third quartile for the commissions earned. LO2

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4-7 Percentiles – Example (cont.) Step 1: Organize the data from lowest to largest value $1,460 $1,471 $1,637 $1,721 $1,758 $1,787 $1,940 $2,038 $2,047 $2,054 $2,097 $2,205 $2,287 $2,311 $2,406 LO2

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Percentiles – Example (cont.) Step 2: Compute the first and third quartiles. Locate L 25 and L 75 using: LO2 4-8

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4-9 Boxplots Learning Objective 3 Construct and interpret box plots. A box plot is a graphical display, based on quartiles, that helps us picture a set of data. To construct a box plot, we need only five statistics: 1.the minimum value, 2.Q1(the first quartile), 3.the median, 4.Q3 (the third quartile), and 5.the maximum value.

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4-10 Boxplot - Example LO3

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4-11 Boxplot Example Step1: Create an appropriate scale along the horizontal axis. Step 2: Draw a box that starts at Q1 (15 minutes) and ends at Q3 (22 minutes). Inside the box we place a vertical line to represent the median (18 minutes). Step 3: Extend horizontal lines from the box out to the minimum value (13 minutes) and the maximum value (30 minutes). LO3

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4-12 Chap 3-12 (n = 9) Q 1 (L 25 ) is in the (9+1)*25/100 = 2.5 position of the ranked data so use the value half way between the 2 nd and 3 rd values, so Q 1 = 12.5 Example: Draw a Box & Whisker for Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Q 1 and Q 3 are measures of non-central location Q 2 = median, is a measure of central tendency

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4-13 Chap 3-13 (n = 9) Q 1 is in the (9+1)*25/100 = 2.5 position of the ranked data, so Q 1 = (12+13)/2 = 12.5 Q 2 is in the (9+1)*50/100 = 5 th position of the ranked data, so Q 2 = median = 16 Q 3 is in the (9+1)*75/100 = 7.5 position of the ranked data, so Q 3 = (18+21)/2 = 19.5 Quartile Measures Calculating The Quartiles: Example Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Q 1 and Q 3 are measures of non-central location Q 2 = median, is a measure of central tendency

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4-14 Chap 3-14 Quartile Measures- Calculation Rules When calculating the ranked position use the following rules ― If the result is a whole number then it is the ranked position to use ― If the result is a fractional half (e.g. 2.5, 7.5, 8.5, etc.) then average the two corresponding data values. ― If the result is not a whole number or a fractional half then interpolate between the data points.

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4-15 Chap 3-15 Quartile Measures: The Interquartile Range (IQR) ― The IQR is Q 3 – Q 1 and measures the spread in the middle 50% of the data ― The IQR is a measure of variability that is not influenced by outliers or extreme values ― Measures like Q 1, Q 3, and IQR that are not influenced by outliers are called resistant measures

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4-16 Chap 3-16 The Interquartile Range Median (Q 2 ) X maximum X minimum Q1Q1 Q3Q3 Example: 25% 25% 11 12.5 16 19.5 22 Interquartile range = 19.5 – 12.5 = 7

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4-17 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 3-17 Distribution Shape and The Boxplot Positively-SkewedNegatively-SkewedSymmetrical Q1Q1 Q2Q2 Q3Q3 Q1Q1 Q2Q2 Q3Q3 Q1Q1 Q2Q2 Q3Q3

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4-18 Interpolation If you found that the first quartile was the 13.75 th value then you interpolate like this: Take the 13 th and 14 th data values Find the difference |14 th -15 th | Multiply the difference by 0.75 Add the calculated value to the 13 th value

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4-19 Exercises – To Do Page 116 – Q4-21 Q4-23 Q4-25

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Stem and Leaf Diagrams 3523182520 1622273341 2737172527 2928313240 12341234 35 5 23 3 8 50 6 27 3 1 7 7 7 798 12 0 12341234 1 0 6 23 7 57 2 0 7 3 8 789 57 1 1 8 means 18

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Stem and Leaf Diagrams 5.26.64.38.35.1 7.58.67.17.82.2 6.65.83.57.56.1 3.82.52.78.84.8

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4-22 Raw data The following data were collected on the ages of cyclists involved in road accidents

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66662192015218 63441044 2635266113612821710521352 19226411392291391764328 622836371813816674510551466 4992312937736988461259 1820112574229660 165016 1815181731142214342096761 34 Total 92

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4-24 Ages of cyclists in road accidents Key 6|7 means 67 years Always include a Key Always include a title

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