mod 6
To create the boxplot for each distribution,
Draw a box from Q1 to Q3. Draw a vertical line in the box at the median. Extend a tail from Q1 to the smallest value that is not an outlier and from Q3 to the largest value that is not an outlier. Indicate outliers with asterisks (*) which makes it a MODIFIED Boxplots
the 5-number summary of the age distribution for actors winning the Oscar for Best Actor between 1970 and 2001. n = 32; Min = 31; Q1 = 37.5; Median = 42.5; Q3 = 49.5; Max = 76 How large is the sample? At least half of the actors won the Oscar before what age? What is the range of the actors' ages? What is the interquartile range?
32 42.5 45 12
A class of 37 students took a test worth 100 points. The five-number summary of their scores is 55, 67, 73, 81, 98. Give an interval of typical test scores that represents the middle 50% of grade distribution.
67 and 81 Q1 = 67 and Q3 = 81. So the middle 50% of the scores fell between these two values.
A class of 37 students took a test worth 100 points. The five-number summary of their scores is 55, 67, 73, 81, 98. Identify the FALSE statement: At least 25% of the students scored a 67 or below. At least 18 students scored below a 67. At least 9 students scored between 67 and 73. At least 25% of the students scored an 81 or above.
At least 18 students scored below a 67. This is a false statement. There are 37 students. Half (18) will have a score at or below the median. The median is not 67. The median is 73.
summary
The range measures the variability of a distribution by looking at the interval covered by all the data. The IQR measures the variability of a distribution by giving us the interval covered by the middle 50% of the data. The five-number summary of a distribution consists of the minimum, quartile 1, median, quartile 3, and maximum. The IQR is the measure of spread we should use when using the median to measure center. When using the median and IQR to measure center and spread, a data point is considered an outlier if it satisfies one of the following conditions. The data value is more than 1.5 IQRs greater than Q3 (i.e., the value is greater than Q3 + 1.5 * IQR) The data value is more than 1.5 IQRs less than Q1 (i.e., the value is less than Q1 − 1.5 * IQR)
A survey taken in a large statistics class contained the question: "What's the fastest you have driven a car (in miles per hour)?" The five-number summary for the 87 males surveyed is: min = 55, Q1 = 95, Median = 110, Q3 = 120, Max = 155 Should the largest observation in this data set be classified as an outlier?
The IQR in this case is 120 - 95 = 25. Applying the 1.5(IQR) rule, we find: Q3 + 1.5(IQR) = 120 + 1.5(25) = 157.5, and therefore the largest observation, 155, should NOT be classified as an outlier. Note, however, that in this case: Q1 - 1.5(IQR) = 95 - 1.5(25) = 57.5, and therefore the smallest observation, 55, should be classified as an outlier.
A survey taken of 140 sports fans asked the question: "What is the most you have ever spent for a ticket to a sporting event?" The five-number summary for the data collected is: min = 85, Q1 = 130, Median = 145, Q3 = 150, Max = 250 Should the smallest observation be classified as an outlier?
The IQR is 150 - 130 = 20. Using the 1.5(IQR) criterion we get 130 - 1.5(20) = 100. Since the smallest observation of 85 is smaller than 100, it should be considered an outlier.
The five-number summary
The five-number summary uses quartiles to identify the center and spread of a distribution. The median (which is Q2) is a measure of center. We also view the median as a typical value that represents the distribution. The values between Q1 and Q3 give a typical range of values. The IQR is a way to measure the variability about the median.
Wrap Up "Measures of Spread about the Median"
The range measures the variability of a distribution by looking at the interval covered by all the data. The IQR measures the variability of a distribution by giving us the interval covered by the middle 50% of the data. The five-number summary of a distribution consists of the minimum, quartile 1, median, quartile 3, and maximum. The IQR is the measure of spread we should use when using the median to measure center. When using the median and IQR to measure center and spread, a data point is considered an outlier if it satisfies one of the following conditions.More than 1.5 IQRs greater than Q3 (i.e., the value is greater than Q3 + 1.5 * IQR).More than 1.5 IQRs less than Q1 (i.e., the value is less than Q1 - 1.5 * IQR). The boxplot is a graphical representation of a data set. It displays the five-number summary and highlights any points that are considered outliers (using the 1.5 * IQR rule described in the previous bullet). Side-by-side boxplots are commonly used to compare two data sets.
