Describing Data - Box plot

- WELCOME TO A LESSON ON BOX PLOTS, ALSO CALLED BOX AND WHISKER GRAPHS. FOR VISUALIZING DATA THERE IS A GRAPHICAL REPRESENTATION OF A FIVE-NUMBER SUMMARY CALLED A BOX PLOT OR BOX AND WHISKER PLOT. TO CREATE A BOX PLOT A NUMBER LINE IS DRAWN FIRST, WHICH WE SEE HERE BELOW THE BOX PLOT. A BOX IS DRAWN FROM THE FIRST QUARTILE TO THE THIRD QUARTILE WHICH WOULD BE THIS BOX HERE. QUARTILE 1 = 16, AND QUARTILE 3 = 28. A LINE IS DRAWN THROUGH THE BOX AT THE MEDIAN, WHICH WOULD BE THIS LINE HERE. AND NOTICE HOW OUR MEDIAN IS 20, AND THEN WHISKERS ARE EXTENDED OUT TO THE MINIMUM AND MAXIMUM VALUES. SO ON THE LEFT, NOTICE HOW THIS WHISKER IS DRAWN OUT TO THE MINIMUM OF 4. WE HAVE ANOTHER WHISKER ON THE RIGHT DRAWN OUT TO THE MAXIMUM OF 32. WE SHOULD RECALL FROM THE FIVE-NUMBER SUMMARY THIS DIVIDES THE DATA INTO FOUR QUARTERS WHERE EACH QUARTER REPRESENTS 25% OF THE TOTAL DATA. SO 25% OF THE DATA FALLS IN THIS INTERVAL HERE BETWEEN THE MINIMUM AND Q1. THEN THERE'S 25% OF THE DATA BETWEEN Q1 AND THE MEDIAN. 25% BETWEEN THE MEDIAN AND Q3, AND 25% BETWEEN Q3 AND THE MAXIMUM. THERE IS ALSO SOMETHING CALLED OUTLIERS WHEN CREATING BOX PLOTS, WHICH WE'LL TALK ABOUT AFTER OUR FIRST EXAMPLE. SO THE KEY TO CREATING A BOX PLOT IS TO FIND THE FIVE-NUMBER SUMMARY OR THE MINIMUM Q1, MEDIAN Q3 AND MAXIMUM. IN OUR LESSON ON THE FIVE-NUMBER SUMMARY WE DID DISCUSS HOW TO FIND THESE VALUES, BUT REMEMBER THERE ARE SEVERAL WAYS TO FIND THE QUARTILES OR Q1 AND Q3, BUT IN THIS CLASS WE'LL BE USING THE LOCATOR OR PERCENTILE METHOD TO FIND Q1 AND Q3. IF YOU USE THE TI-83 OR 84 THE METHOD IS DIFFERENT AND CAN YIELD DIFFERENT RESULTS. SO JUST FOR A QUICK REVIEW, USING THE LOCATOR METHOD TO FIND Q1, WE FIND L = 0.25 x N, WHERE N IS THE NUMBER OF DATA VALUES. IF L IS A DECIMAL WE ROUND UP TO THE NEXT WHOLE NUMBER AND USE A DATA VALUE IN THE ROUNDED UP POSITION AS Q1. IF L IS A WHOLE NUMBER THEN WE FIND THE MEAN OR THE AVERAGE OF THE DATA VALUES IN THE L AND L + 1th POSITIONS. WE USE A SIMILAR METHOD TO FIND Q3 EXCEPT WE USE L = 0.75 x N INSTEAD OF 0.25 x N. SO IN OUR LESSON ON THE FIVE-NUMBER SUMMARY WE TOOK THIS DATA HERE AND FOUND THESE VALUES FOR OUR FIVE-NUMBER SUMMARY, WHICH WE CAN NOW USE TO CREATE A BOX PLOT. BUT LET'S QUICKLY REVIEW HOW WE FOUND THESE VALUES. FIRST WE DECIDED TO WRITE THE DATA VALUES HORIZONTALLY IN ORDER FROM LEAST TO GREATEST, AS WE SEE HERE, WHERE THE MINIMUM WOULD BE SIX. THE MAXIMUM WOULD BE 97, AND THERE ARE 15 DATA VALUES OR AN ODD NUMBER OF DATA VALUES, SO THE MEDIAN IS IN THE LIST. IT WOULD BE THE VALUE IN THE EIGHTH POSITION, WHICH IS 49. THERE ARE SEVEN VALUES BELOW THIS, SEVEN VALUES ABOVE THIS. THEREFORE, THIS IS THE MEDIAN. AND THEN FINALLY TO FIND THE QUARTILES WE'LL BEGIN WITH Q1. SO WE FIND L = 0.25 x 15. AGAIN, THIS IS 15 BECAUSE WE HAVE 15 DATA VALUES. THIS GIVES US A DECIMAL OF 3.75, WHICH WE ROUND UP TO THE NEXT WHOLE NUMBER OF 4. THE DATA VALUE IN THE 4th POSITION WILL BE Q1, SO 18 IS QUARTILE 1. AND THEN FOR Q3 WE FIND L, WHICH IS 0.75 x 15, WHICH IS 11.25, WHICH ROUNDS TO 12. THE VALUE IN THE 12th POSITION WILL BE Q3. SO 8, 9, 10, 11, 12, WHICH IS 82. THESE ARE THE FIVE VALUES WE'LL USE TO CREATE OUR BOX PLOT. SO THE FIRST STEP IS TO CREATE A BOX USING Q1 AND Q3, WHICH IS HERE. NEXT, WE DRAW A LINE AT THE MEDIAN, WHICH IS HERE AT 49. AND THEN WE DRAW A WHISKER DOWN TO THE MINIMUM, AND ANOTHER WHISKER UP TO THE MAXIMUM, WHICH WOULD BE HERE TO 6 AND HERE TO 97. THIS IS OUR BOX AND WHISKER PLOT FOR THE GIVEN DATA. BUT WE DO HAVE TO MODIFY OUR BOX PLOT IF WE DO HAVE OUTLIERS IN THE DATA. SO NOW LET'S TALK ABOUT OUTLIERS. SOMETIMES THE MOST INTERESTING POINTS OF A DATA SET ARE THE POINTS THAT DO NOT SEEM TO BELONG, MEANING THEY SEEM TO DIFFER BY A SUBSTANTIAL AMOUNT FROM THE REST OF THE DATA. WE CALL THESE POINTS OUTLIERS. OUTLIERS ARE VALUES THAT ARE BEYOND 1.5 x THE INNER QUARTILE RANGE, WHICH WOULD BE Q3 - Q1. WE INDICATE OUTLIERS BY PLOTTING THE POINTS RATHER THAN EXTENDING THE WHISKER OUT TO THE OUTLIERS. NOTICE HERE Q3 - Q1 WOULD BE 82 - 70 OR 12, AND 1.5 x 12 = 18. SO THE VALUES THAT ARE 18 UNITS ABOVE 82 OR BELOW 70 WOULD BE CONSIDERED OUTLIERS. NOTICE HOW THIS VALUE HERE LOOKS LIKE IT'S 35, WHICH IS MORE THAN 18 UNITS BELOW Q1, THEREFORE, IT'S AN OUTLIER. SO INSTEAD OF EXTENDING THE WHISKER ALL THE WAY OUT TO 35, WE STOP THE WHISKER AT THE NEXT LOWEST VALUE AND ONLY PUT A SINGLE POINT AT THE VALUE OF 35. THIS INDICATES THAT WHILE MOST OF THE DATA IS BETWEEN 62 AND 99, THERE IS ONE OUTLIER WAY DOWN HERE AT 35. AND IT IS POSSIBLE TO HAVE MORE THAN ONE OUTLIER. WE COULD HAVE ANOTHER POINT HERE AT 40, SINCE THIS VALUE WOULD STILL BE MORE THAN 18 UNITS BEYOND 70. SO LET'S TAKE A LOOK AT ONE MORE EXAMPLE. WE WANT TO FIND THE FIVE-NUMBER SUMMARY OF THE DATA AND THEN CREATE A BOX PLOT. FIRST NOTICE HOW I ORDERED THE DATA FROM LEAST TO GREATEST HORIZONTALLY. THE MINIMUM IS 34, AND THE MAXIMUM IS 93. WE HAVE A TOTAL OF 12 DATA VALUES OR AN EVEN NUMBER OF DATA VALUES, SO THE MEDIAN WILL BE THE AVERAGE OR MEAN OF THE 6th AND 7th DATA VALUES, WHICH WOULD BE THESE TWO VALUES HERE. WELL, 80 + 83 DIVIDED BY 2 = 81.5, SO THE MEDIAN IS 81.5. NEXT, USING THE LOCATOR METHOD, WE'LL FIND Q1 AND Q3 AND BEGIN BY DETERMINING L = 0.25 x N FOR Q1. N IS 12. NOTICE HERE THIS GIVES US 3. BECAUSE L IS A WHOLE NUMBER WE USE THE AVERAGE OF THE VALUES IN THE 3rd AND 4th POSITIONS FOR Q1. SO Q1 IS GOING TO BE EQUAL TO, LET'S SEE, THE VALUES IN THE 3rd AND 4th POSITIONS WOULD BE 68 AND 75. SO 68 + 75 DIVIDED BY 2 GIVES US 71.5 FOR Q1. NOW, FOR Q3 WE'LL HAVE L = 0.75 x 12 = 9. SO TO FIND Q3, BECAUSE L IS A WHOLE NUMBER, WE'LL TAKE THE AVERAGE OF THE VALUES IN THE 9th AND 10th POSITIONS, WHICH WOULD BE 88 AND 90. SO Q3 = 88 + 90 DIVIDED BY 2 = 89. NOW, WE'RE ALMOST READY TO MAKE THE BOX PLOT, BUT LET'S CHECK FOR OUTLIERS FIRST. WE WANT TO DETERMINE 1.5 x THE INNER QUARTILE RANGE, SO 1.5 x Q3 - Q1. SO THIS COMES OUT TO 1.5 x 17.5 = 26.25. SO WE'RE LOOKING FOR VALUES THAT ARE MORE THAN 26.25 UNITS BELOW Q1 OR ABOVE Q3. AND NOTICE HOW 34 IS MORE THAN 26.25 UNITS BELOW 71.5, THEREFORE, 34 IS AN OUTLIER. SO I'LL GO AHEAD AND HIGHLIGHT THIS. WE'LL ACTUALLY REPRESENT 34 AS A POINT. NOTICE HOW THE NEXT VALUE WOULD BE 65, SO WE'LL ACTUALLY EXTEND OUR WHISKER OUT TO 65. NOTICE HOW 65 IS NOT AN OUTLIER, SO WE'LL EXTEND THE WHISKER OUT TO 65 AND PUT A POINT AT 34. AND NOW WE CAN GO AHEAD AND MAKE OUR BOX PLOT. WE CREATE THE BOX USING Q1 AND Q3, WHICH IS HERE, 71.5 TO 89. DRAW A LINE AT THE MEDIAN, WHICH IS 81.5 HERE. TO CREATE THE LOWER WHISKER WE ONLY EXTEND THE WHISKER DOWN TO 65, AS WE SEE HERE. AND THEN PUT THIS POINT HERE AT 34, BECAUSE THIS IS AN OUTLIER. AND THE UPPER WHISKER WILL GO OUT TO 93, WHICH WE SEE HERE. AND THAT'S HOW WE CREATE BOX PLOTS WITH AND WITHOUT OUTLIERS. I HOPE YOU HAVE FOUND THIS HELPFUL.