13Bspreads

= 13B Measures of Spread =


 * When comparing two sets of data, it is not enough to simple compare their centres
 * Two sets of data can have the same centre but be quite different because one is more spread out than the other


 * Range **


 * Range is the simplest way to measure the spread of the data

... ... Range = Maximum Value minus Minimum Value

... ... Range = max – min


 * Example 1 **

Find the range of the following data: 10, 11, 14, 17, 19, 21, 24


 * Solution:**

... ... max = 24 ... ... min = 10

... ... range = 24 – 10 = 14


 * Note **
 * The range is badly affected by any outliers (extreme values)
 * It does not reflect how the data is clumped between the two end values
 * So the range is a poor measure of the spread of the data


 * Interquartile Range (IQR) **


 * The Interquartile Range is a better measure of the spread of the data than the range
 * It is not affected by outliers and does a better job of reacting to how the data clumps
 * The interquartile range (IQR) is the range of the central 50% of the data


 * Quartiles **


 * Quartiles are at the quarter points through the data
 * Q 1 (first quartile) is one quarter of the way through the data
 * Q 2 (second quartile) is two quarters of the way through the data (ie the __**median**__)
 * Q 3 (third quartile) is three quarters of the way through the data


 * The Interquartile Range = third quartile minus first quartile
 * IQR = Q 3 – Q 1


 * Finding the IQR **


 * Put the data in ascending order (smallest to largest)
 * Find the median (middle value) and
 * circle it (if there is one middle value because n is odd)
 * put a short vertical line between the two middle values (if n is even)
 * Consider only the first half of the data NOT INCLUDING THE MEDIAN
 * Find the median of the first half of the data, that will be Q 1
 * Now consider only the second half of the data NOT INCLUDING THE MEDIAN
 * Find the median of the second half of the data, that will be Q 3
 * IQR = Q 3 – Q 1


 * Example 2 **

Find the Interquartile Range of the following data ... ... 25, 18, 15, 16. 13, 27, 21, 23, 12, 20, 15


 * Solution:**


 * Put data in ascending order:

... ... 12, 13, 15, 15, 16. 18, 20, 21, 23, 25, 27


 * Find median

math \\ . \qquad \text{number of values } n = 11 \\. \\ . \qquad \text{median is } \dfrac{11+1}{2} = 6 \text{th value} \\. \\ . \qquad \text{median is } 18 math


 * Consider only the first half of the data (not counting the median)
 * Q 1 is the middle of the first half of the data
 * Q 1 = 15


 * Consider only the second half of the data (not counting the median)
 * Q 3 is the middle of the second half of the data
 * Q 3 = 23
 * IQR = Q 3 – Q 1
 * IQR = 23 – 15
 * IQR = 8


 * Example 3 **

Find the Interquartile Range of the following data ... ... 20, 22, 24, 24, 27, 28, 30, 31, 33, 36, 38, 40


 * Solution:**


 * The data is already in ascending order
 * Find the median

math \\ . \qquad \text{number of values } n = 12 \\. \\ . \qquad \text{median is } \dfrac{12+1}{2} = 6.5 \text{th value} \\. \\ . \qquad \text{median is between the 6th and 7th value} \\. \\ . \qquad \text{median is } \dfrac{28+30}{2} = 29 math


 * Consider only the first half of the data (not counting the median)
 * Q 1 is the middle of the first half of the data
 * Q 1 is midway between 24 and 24
 * Q 1 = 24




 * Consider only the second half of the data (not counting the median)
 * Q 3 is the middle of the second half of the data
 * Q3 is midway between 33 and 36
 * Q 3 = 34.5

math . \qquad \qquad \text{Q}_3 = \dfrac{33+36}{2} = 34.5 math




 * IQR = Q 3 – Q 1


 * IQR = 34.5 – 24
 * IQR = 10.5

.
 * The data in Example 2 had an IQR of 8
 * So the data in Eg 3 (IQR = 10.5) is a little more spread out than that in Eg 2