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**Describing Data: Percentiles**

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

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

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**Quartiles, Deciles and Percentiles**

Learning Objective 2 Compute and understand quartiles, deciles, and percentiles. 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

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

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**Percentile Computation**

LO2 Percentile Computation To formalize the computational procedure, let Lp refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L33 and if we wanted the median, the 50th percentile, then L50. 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.

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LO2 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.

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**Percentiles – Example (cont.)**

LO2 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

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**Percentiles – Example (cont.)**

LO2 Percentiles – Example (cont.) Step 2: Compute the first and third quartiles. Locate L25 and L75 using: 4-8

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**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: the minimum value, Q1(the first quartile), the median, Q3 (the third quartile), and the maximum value.

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LO3 Boxplot - Example

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LO3 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).

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**Example: Draw a Box & Whisker for**

Sample Data in Ordered Array: (n = 9) Q1 (L25) is in the (9+1)*25/100 = 2.5 position of the ranked data so use the value half way between the 2nd and 3rd values, so Q1 = 12.5 Q1 and Q3 are measures of non-central location Q2 = median, is a measure of central tendency

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**Quartile Measures Calculating The Quartiles: Example**

Sample Data in Ordered Array: (n = 9) Q1 is in the (9+1)*25/100 = 2.5 position of the ranked data, so Q1 = (12+13)/2 = 12.5 Q2 is in the (9+1)*50/100 = 5th position of the ranked data, so Q2 = median = 16 Q3 is in the (9+1)*75/100 = 7.5 position of the ranked data, so Q3 = (18+21)/2 = 19.5 Q1 and Q3 are measures of non-central location Q2 = median, is a measure of central tendency

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**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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**Quartile Measures: The Interquartile Range (IQR)**

The IQR is Q3 – Q1 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 Q1, Q3, and IQR that are not influenced by outliers are called resistant measures

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**The Interquartile Range**

Example: Median (Q2) X X Q1 Q3 maximum minimum 25% % % % Interquartile range = 19.5 – 12.5 = 7

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**Distribution Shape and The Boxplot**

Negatively-Skewed Symmetrical Positively-Skewed Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..

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

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

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Stem and Leaf Diagrams 35 23 1 2 3 4 1 2 3 4 6 7 5 8 9 8 6 7 3 5 2 7 7 7 9 8 5 3 7 1 2 1 means 18

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Stem and Leaf Diagrams

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

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66 6 62 19 20 15 21 8 63 44 10 26 35 61 13 28 7 52 22 64 11 39 9 17 32 36 37 18 138 16 67 45 55 14 49 23 12 88 46 59 25 42 29 60 50 31 34 Total 92

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**Ages of cyclists in road accidents**

Always include a title Always include a Key Key 6|7 means 67 years

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Describing Data: Displaying and Exploring Data

Describing Data: Displaying and Exploring Data

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