04 Describing Data Graphically and Numerically Part 3

# If i is an integer the median is the average of the

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If i is an integer , the median is the average of the values in position i and i + 1.

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Weighted Mean The mean of data values that have been weighted according to their relative importance Weighted Mean for a population: Weighted Mean for a sample: Example Application: Computing GPA 10 μ W = w i x i w i x W = w i x i w i x i = value of observation i w i = weight for observation i
Percentiles 11 A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. The p th percentile is a value such that at least p percent of the observations are less than or equal to this value and, at least (100 - p ) percent of the observations are greater than or equal to this value. Examples: Colleges and universities frequently report admission test scores in terms of percentiles.

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Percentiles 12 Calculating the p th percentile: Arrange the data in ascending order (smallest value to largest value). Compute the percentile index: If i is not an integer , round up. The next integer greater than i denotes the position of the p th percentile. If i is an integer , the p th percentile is the average of the values in positions i and i +1. i = p 100 (n) p = percentile of interest n = number of observations
Percentiles 13 Example: 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 3925 85 th percentile: i is not an integer , so we round up. Data value in the 11 th position will be the 85 th percentile, i.e. the 85 th percentile is 3730. 50 th percentile: i is an integer , 50 th percentile is the average of the sixth and seventh data values, i.e. the 50 th percentile is (3490+3520)/2 = 3505. Note, the 50 th percentile is also the median. i = 85 100 (12) = 10.2 85 = percentile of interest 12 = number of observations i = 50 100 (12) = 6

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Quartiles 14 Quartiles split the data into four parts, with each part containing approximately one- fourth, or 25% of the observations. Q 1 : first quartile, or 25 th percentile Q 2 : second quartile, or 50 th percentile (also the median) Q 3 : third quartile, or 75 th percentile Interquartile Range (IQR) = Q 3 – Q 1 25% Q 1 Q 2 Q 3 25% 25% 25%
Quartiles 15 Example: 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 3925 We already identified Q 2 (the median) as 3505. Q 1 : i is an integer , 25 th percentile is the average of the 3 rd and 4 th data values, thus Q 1 =(3450+3480)/2 = 3465. Q 3 : i is an integer , 75 th percentile is the average of the 9 th and 10 th data values, thus Q 3 =(3550+3650)/2 = 3600. 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 3925 i = 25 100 (12) = 3 i = 75 100 (12) = 9 Q 1 =3465 Q 2 =3505 Q 3 =3600

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Box and Whisker Plot 16 A Box and Whisker Plot is a graphical display of data using a central “box” and extended “whiskers” Example: 25% 25% 25% 25% Outliers Lower 1st Median 3rd Upper Limit Quartile Quartile Limit * * The lower limit is Q 1 – 1.5 (Q 3 Q 1 ) The upper limit is Q 3 + 1.5 (Q 3 – Q 1 )
The center box extends from Q 1 to Q 3 The line within the box is the median The whiskers extend to the smallest and largest values within the calculated limits Outliers are plotted outside the calculated limits Box and Whisker Plot 17 25% 25% 25% 25%

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