Range+and+interquartile+range

Range, IQR

The lower quartile (Q1) is the 25th percentile and the upper quartile (Q3) is the 75th percentile. The middle quartile (Q2) is the 50th percentile and is also the median.
 * Range** -the difference between the largest and smallest numbers in a set of data
 * Interquartile Range (IQR)** -the difference between the upper and lower quartiles in a set of data



Outliers in a set of numbers could affect the range. Q1-1.5(IQR) =min. point (anything below this point is an outlier) Q3+1.5(IQR) =max. point (anything above this point is an outlier) Anything beyond the max. point or anything below the min. point are considered outliers.

The 5 number summary can be used to find the values of the range and IQR min Q1 Q2 Q3 max max-min =range Q3-Q1 =IQR

Range and IQR can be graphed using box plots In this example, the minimum is .05 and the maximum is 2.2. Since the range is max-min, the range is 2.2-.05 =2.15 Since IQR is Q3-Q1, the IQR is 1.15-.45 =.7

Another interesting piece of information that you can gather from this box plot is whether or not the data is skewed. Because the median line is not equidistant from the third and first quartile hinges, the data is skewed. Using a histogram to organize the data in correlation with a box plot will help to show what skewed means. In a skewed left graph the mean will be lower than the median. In a skewed right graph the median will be lower than the mean.

Another Example of finding range and iqr: Find the range and interqartile range of the following set of 21 numbers: 3, 5, 5, 6, 6, 8, 9, 9, 12, 13, 15, 15, 16, 17, 20, 22, 22, 23, 23, 26, 27. Repeat without the largest value (27). Repeat again without the largest two values (26 and 27).

5 5 6 6 Q1=7 8 9 9 12 13 15 median = 15 15 16 17 20 22 Q3=22 22 23 23 26 27 Maximum =27 || 3 Minimum =3 5 5 6 6 Q1=7 8 9 9 12 13 15 median = 14 15 16 17 20 22 Q3 =21 22 23 23 26 Maximum = 26 || 3 Minimum =3 5 5 6 6 Q1 =6 8 9 9 12 13 median = 13 15 15 16 17 20 Q3 =20 22 22 23 23 Maximum =23 || Set X contains 21 numbers. One number, 15, is in the middle; therefore the median is 15. When determining the values of Q1 and Q3, the median is excluded. Q1 is the median of the lower 10 numbers and Q3 is the median of the upper 10 numbers. Range of Set X = maximum – minimum = 27 – 3 = 24 Interquartile range of Set X = Q3 – Q1 = 22 – 7 = 15
 * Set X || Set Y || Set Z ||
 * 3 Minimum =3

Set Y contains 20 numbers. Two numbers are in the middle, so the median is their average, or 14. When determining the values of Q1 and Q3, the median is not excluded since an even number of values exists. Thus, Q1 is the median of the lower 10 numbers and Q3 is the median of the upper 10 numbers. Range of Set Y = maximum – minimum = 26 – 3 = 23 Interquartile range of Set Y = Q3 - Q1 = 21 – 7 = 14

Set Z contains 19 numbers. One number, 13, is in the middle; therefore the median is 13. When determining the values of Q1 and Q3, the median is excluded. Q1 is the median of the lower 9 numbers and Q3 is the median of the upper 9 numbers. Range of Set Z = maximum – minimum = 23 – 3 = 20 Interquartile range of Set Z = Q3 – Q1 = 20 – 6 = 14

Two sets of numbers that are very different can have the same mean and median. Range, interquartile range, and standard  __deviation__  are ways to measure the spread of a distribution. Example: Set A: 100, 90, 80, 70, and 60 Set B: 82, 81, 80, 79, and 78 Both sets of numbers have a mean and a median of 80 but Set A is more spread out.

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