2LEARNING OBJECTIVESLO1. Compute and understand quartiles, deciles, and percentiles. LO2. Construct and interpret box plots.
3Quartiles, Deciles and Percentiles Learning Objective 2 Compute and understand quartiles, deciles, and percentiles.Quartiles, Deciles and PercentilesThe median splits the data into equal sized halvesQuartiles split the data into quartersDeciles into tenthsAnd percentiles can be any split of our choosingThese measures include quartiles, deciles, a
5Percentile Computation LO2Percentile ComputationTo 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.
6LO2Percentiles - ExampleListed 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 $1,460Locate the median, the first quartile, and the third quartile for the commissions earned.
7Percentiles – Example (cont.) LO2Percentiles – 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
8Percentiles – Example (cont.) LO2Percentiles – Example (cont.)Step 2: Compute the first and third quartiles. Locate L25 and L75 using:4-8
9Learning 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), andthe maximum value.
11LO3Boxplot ExampleStep1: Create an appropriate scale along the horizontal axis.Step 2: Draw a box that starts at Q1 (15 minutes) and ends at Q3 (22minutes). 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 (13minutes) and the maximum value (30 minutes).
12Example: 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 dataso use the value half way between the 2nd and 3rd values,so Q1 = 12.5Q1 and Q3 are measures of non-central locationQ2 = median, is a measure of central tendency
13Quartile 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 = 12.5Q2 is in the (9+1)*50/100 = 5th position of the ranked data,so Q2 = median = 16Q3 is in the (9+1)*75/100 = 7.5 position of the ranked data,so Q3 = (18+21)/2 = 19.5Q1 and Q3 are measures of non-central locationQ2 = median, is a measure of central tendency
14Quartile Measures- Calculation Rules When calculating the ranked position use the following rulesIf the result is a whole number then it is the ranked position to useIf 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.
15Quartile Measures: The Interquartile Range (IQR) The IQR is Q3 – Q1 and measures the spread in the middle 50% of the dataThe IQR is a measure of variability that is not influenced by outliers or extreme valuesMeasures like Q1, Q3, and IQR that are not influenced by outliers are called resistant measures
18InterpolationIf 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