Lesson 75 Objectives Calculate quartile values for a data set Create a box-and-whisker plot Quartiles and Box-and-Whisker Plots Richard recorded the number of people that attended his team s minor league hockey games this year Identify the first, second, and third quartiles of his data Game Attendance 211 246 242 305 195 272 296 140 279 250 246 265 242 209 311 Quartiles Quartiles are boundaries that break the data into fourths You already know how to find the second quartile, Q 2, because the second quartile is better known as the median The first quartile, Q 1, is the median of the lower half of the data The third quartile, Q 3, is the median of the upper half of the data One-fourth of the data in a set falls at or below Q 1 One-fourth of the data falls between Q 1 and Q 2 One-fourth of the data falls between Q 2 and Q 3 One-fourth of the data falls at or above Q 3 Example 1 Finding Quartiles Use Richard s data from the opening paragraph of this lesson Identify the first, second, and third quartiles Solution First sort the data from least to greatest Then locate the median The set has 15 numbers so the median is one of the numbers in the set The median, or second quartile, is 246 Because there are seven numbers in the lower half of the data, the first quartile is 211 Because there are seven numbers in the upper half of the data, the third quartile is 279 140 195 209 211 242 242 246 246 250 265 272 279 296 305 311 Q 1 median, or Q 2 Q 3 416 Chapter 7 Statistics
Box-and-Whisker Plots A box-and-whisker plot is a visual presentation of the data that divides the data into quartiles, or fourths This plot also shows the quartile values, and the least and greatest numbers in the set One advantage of displaying data in a box-andwhisker plot is that it does not become more difficult to interpret the plot as the number of data values increase One disadvantage of displaying data in a box-and-whisker plot occurs when there are not many data values Example 2 Creating a Box-and-Whisker Plot Use Richard s data from the opening paragraph of this lesson to create a box-and-whisker plot Solution Draw a number line where you can easily locate the least value, the greatest value, and each of the quartile values A box is drawn below the number line that is the length from Q 1 to Q 3 The box has a vertical line drawn to separate it into two parts at Q 2 Points are made at the least value and greatest value Then lines, known as whiskers, are drawn to connect these points to the ends of the box Box & Whiskers If an outlier is part of the set, it is shown but noted as an outlier An outlier is a data value that is clearly not near (or close in range) to the rest of the data Outliers are unusually small or unusually large values The interquartile range is Q 3 Q 1 You can usually conclude that a data value is an outlier if it lies more than 15 times the interquartile range from Q 1 or Q 3 75 Quartiles and Box-and-Whisker Plots 417
Activity Making a Box-and-Whisker Plot see margin Make a box-and-whisker plot using the weekly-hours data from an earlier Activity Weekly Hours Worked Office Temp, Inc 43 26 27 33 18 27 30 31 31 33 19 41 40 28 6 29 40 19 40 39 1 First, sort the data from least to greatest 2 Find the median value, Q 2 3 Find the first-quartile value, Q 1 4 Find the third-quartile value, Q 3 5 Identify the least and greatest values the whisker values 6 Draw a scale for your plot that will include the whiskers 7 Locate the values for Q 1, Q 2, and Q 3 on the scale Draw the box spanning these three values 8 Locate the least and greatest values on the scale Draw the whiskers from the box ends to the least and greatest values Note that when calculating the medians of the lower or upper quartiles of a data set that has an odd number of data values does not include the overall median in its calculation Example 3 Using Technology a Use the data from the Activity and your graphing calculator to verify the box-and-whisker plot you created above b Determine how to adjust the list in your graphing calculator to reflect that the data set has an outlier 418 Chapter 7 Statistics
Solution a To enter the data into a list, press S Select Edit Enter the data into L1 Press h s and select Plot1 Turn on Plot 1 and select the box-and-whisker plot for type Be sure that the Xlist named is L1 Set your viewing window to show the first quadrant of the coordinate plane Because the Yscl is not used for the box-and-whisker plot set the minimum at 1 and maximum at 2 To view your graph, press G b Because you want to use 18 as one whisker, delete 6 from the L1 list The outlier can be plotted using Plot2 and the L2 and L3 lists Cultural Connection The definition of quartile is as follows: quartile n [ISV, fr L quartus] (1879): any one of three values that divides the items of a frequency distribution into four classes with each containing one fourth of the total population: also any one of the four classes The information within brackets that precedes the definition is known as the etymology of the word An etymology traces the history of a word in English and tells what language it came from and in what form it is used in English Many words are traced to either Middle English (ME) or Old English (OE), or both In the case of mathematical terms, many have a label of ISV which stands for International Scientific Vocabulary Such a reference means the etymology recognizes the use of this word as international and the possibility exists that it originated somewhere other than the English language The fr L quartus states that the word comes from the Latin word quartus meaning fourth When a word is changed slightly for the English language, the original word in the original language is called a transliteration 1 Check the etymology of quartile in a different dictionary to see if more detail of the history of the word is given 2 Choose another mathematical term and investigate its etymology 75 Quartiles and Box-and-Whisker Plots 419
Lesson Assessment Think and Discuss 1 Can you find the mean of the data from a box-and-whisker plot? Explain 2 Excluding any outliers, how can you find the range of the data from just looking at a box-and-whisker plot? 3 Describe how to find each of the quartiles Practice and Problem Solving 11,300 10,500 11,400 11,000 Car Prices 12,000 11,700 11,800 11,400 10,500 11,800 12,100 4 Ella plans to sell her sports car On the Internet she finds many prices from other people trying to sell cars of the same kind and age Ella made a list of the prices, as shown, rounded to the nearest $100 a What is the Q 2 value? $ b What is the Q 1 value? $11,300 c What is the Q 3 value? $11,800 d Name the whiskers $10,500 and $12,100 e Construct a box-and-whisker plot of these values see margin f After seeing your plot, do you feel that you might have an outlier in your data? If so, remove the outlier and use your graphing calculator to redo your plot $10,500 may be an outlier; see margin for graph g Which plot, with or without the outlier, is a better guide for Ella to use to determine the price at which she should sell her car? without the outlier h Find the quartile values and the mean of the data excluding the outlier What affect does excluding the outlier have on each value? Q 2 $, Q 1 $11,400, Q 3 $11,800; Removing the lowest value did not affect the quartile values Mixed Review Name the property illustrated in each statement Determine the value of x 5 2x 0 Property of Zero; 0 6 6 x x 6 1 Reciprocal Property; any number except 0 7 4(x 3) 20 12 32 Distributive 12 Property; 5 8 5x 6 7x 12x 6 30 Associative Property, Commutative Property; 2 420 Chapter 7 Statistics and Probability