# The range and interquartile range

The range and interquartile range 1 of 15 Boardworks 2012 Find the range Here are Laura and Jos duck herding times in minutes: Laura 6.59 6.45 9.41 5.10 7.30 Jo 8.52 7.41 8.35 6.20 5.15 In groups, discuss which girl is more consistent. How did you decide this? If the scores are spread out then the range will be higher and the scores are less consistent. If the scores are close together then the range will be lower and the scores are more consistent. The spread is called the range. range = highest score lowest score 2 of 15 Boardworks 2012 Find the range

Calculate the range for each girl to confirm the result of your discussion. Laura 6.59 6.45 9.41 5.10 7.30 Jo 8.52 7.41 8.35 6.20 5.15 Lauras range = 9.41 5.10 = 4.31 mins Jos range = 8.52 5.15 = 3.37 mins Jo is more consistent. Now calculate the median for each girl. Which girl would you enter into a competition? Why? Laura 4.31 mins range median 6.59 mins 3 of 15 Jo 3.37 mins 7.41 mins Remember, the range is not a measure of center. Boardworks 2012 Range Here is a summary of Chriss and Robs performances in 10

duck herding competitions. Chris Rob median 5.10 mins 4.26 mins range 2.93 mins 2.18 mins Discuss which of these conclusions are correct: Rob is more reliable. Chris is better because his median is higher. Chris is better because his range is higher. Rob must have gotten a better quickest time. On average, Rob is faster and more consistent. 4 of 15 Boardworks 2012 Range Chris Rob median 5.10 mins

4.26 mins range 2.93 mins 2.18 mins Here is the original data for Chris and Rob. Set A: 4.26 6.29 5.14 5.17 6.11 3.36 5.54 5.06 4.29 4.31 Set B: 3.52 4.14 4.03 5.09 3.37 4.46 5.55 4.41 3.51 5.42 Use the summary table above to decide which data set is Chriss and which is Robs. Who has the best time? Who has the worst time? 5 of 15 Boardworks 2012 Interquartile range When there are extreme values in the data, it is more appropriate to calculate the interquartile range (IQR). This is the range of the middle half of the data, when the values are written in order. interquartile range = upper quartile lower quartile The lower quartile is the value that is one quarter of the way along the list. The upper quartile is the

value that is three quarters of the way along the list. Extreme values in a data set are called outliers. 6 of 15 Boardworks 2012 What is an outlier? An outlier is defined as any data point that is more than 1.5 times the interquartile range (IQR) below the first quartile or above the third quartile. For the set of data values 18, 24, 26, 32, 45, 52, 96 we find the median, Q1, Q3, and the interquartile range in order to see if there are any outliers in the data set. 18, 24, 26, 32, 45, 52, 96 Q1 median Q3 IQR = 52 24 = 28 1.5 IQR = 1.5 28 = 42 Q1 42 = 24 42 = 18 and Q3 + 42 = 52 + 42 = 94. Are there any outliers in this data? What are they? 7 of 15 Boardworks 2012

Girls basketball statistics The number of free throw points made by each team member in the most recent season for a girls college basketball team is listed in the table on the left below. player free throw points Michala 3 Lauren 4 Heather 15 Caroline 19 Brianna 19 The IQR = 50 and 1.5 50 = 75

Kiah 32 Stefanie 51 Kelly 63 Bria 65 So, outliers are values less than (Q1 5) or greater than (Q3 + 75). i.e. 60 > outlier > 140 Kaleena 70 Tiffany 142 8 of 15 Determine the IQR then decide if there are any

outliers in the data. Q1 = 15 and Q3 = 65. The data point 142 is an outlier. Boardworks 2012 Using technology Using the STAT PLOT feature of your graphing calculator and choosing the modified box and whisker graph produces the plot below. It indicates that the number 142 is an outlier. outlier Use the trace feature to see the value of the outlier, which is displayed in the bottom corner. Notice that the plot fences in the expected data with the outlier lying outside of the fence. 9 of 15 Boardworks 2012 Interquartile range If there are lots of values in a data set, it is not always easy to find the interquartile range by inspection. In these cases, there are formulas we can use. When there are n values in an ordered data set: n+1 lower quartile =

median = n+1 upper quartile = 10 of 15 4 th 2 value value 3(n + 1) 4 th th value Boardworks 2012 Wife carrying race 2010 57.9 62.1

63.9 75.5 81.0 98.2 114.4 119.3 136.7 140.2 220.8 2011 61.5 62.2 64.9 74.2 87.3 95.8 115.8 125.5 139.1 148.5 149.2 The table shows the results (in seconds) of a 250 m Wife Carrying race held in Finland in 2010 and 2011. Compare the 2010 and 2011 results using the range and interquartile

range. Which gives a better summary of the data? Why? 2010 2011 mean 106.4 102.2 range 162.9 87.7 IQ Range 72.8 74.2 Interquartile range for 2010: 136.7 63.9 = 72.8 s Interquartile range for 2011: 139.1 64.9 = 74.2 s 11 of 15 Boardworks 2012

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