- WELCOME TO A LESSON ON HOW TO FIND THE QUARTILES
AND FIVE-NUMBER SUMMARY OF A SET OF DATA.
LET'S BEGIN BY TALKING ABOUT THE MEDIAN.
THE MEDIAN OF A SET OF DATA IS THE VALUE IN THE MIDDLE
WHEN THE DATA IS IN ORDER FROM LEAST TO GREATEST.
AND SINCE THE MEDIAN IS IN THE MIDDLE,
HALF THE DATA VALUES WOULD BE BELOW THE MEDIAN,
AND HALF WOULD BE ABOVE THE MEDIAN.
THE QUARTILES ARE THE VALUES
THAT DIVIDE THE DATA INTO QUARTERS.
SO THE FIRST QUARTILE, OR Q1,
IS THE VALUE SO THAT 25% OF THE DATA VALUES ARE BELOW IT,
AND THE 3rd QUARTILE, OR Q3,
IS THE VALUE SO THAT 75% OF THE DATA VALUES ARE BELOW IT.
YOU MAY HAVE GUESSED THAT THE SECOND QUARTILE, OR Q2,
IS THE SAME AS THE MEDIAN
SINCE THE MEDIAN IS THE VALUE
SO THAT 50% OF THE DATA VALUES ARE BELOW IT
AND 50% ARE ABOVE IT.
SO THE QUARTILES DIVIDE THE DATA INTO QUARTERS
WHERE 25% OF THE DATA IS BETWEEN THE MINIMUM AND Q1.
25% IS BETWEEN Q1 AND THE MEDIAN, OR Q2.
25% IS BETWEEN THE MEDIAN OR Q2 AND Q3,
AND FINALLY, 25% IS BETWEEN Q3 AND THE MAXIMUM VALUE.
SO TO ILLUSTRATE THIS
IF WE THINK OF THIS BAR AS BEING THIS DATA,
WHERE THIS WOULD BE THE MINIMUM VALUE.
THIS WOULD BE THE MAXIMUM VALUE. THIS WOULD BE THE MEDIAN OR Q2.
THIS VALUE WOULD BE Q1, OR THE 1st QUARTILE,
AND THIS VALUE WOULD BE Q3 OR THE 3rd QUARTILE.
SO 25% OF THE DATA WOULD BE IN THIS INTERVAL.
25% WOULD BE IN THIS INTERVAL AND SO ON.
SO WHILE THE QUARTILES ARE NOT A ONE-NUMBER SUMMARY OF VARIATION,
LIKE STANDARD DEVIATION,
THE QUARTILES ARE USED WITH THE MEDIAN,
MINIMUM, AND MAXIMUM VALUES
TO FORM WHAT'S CALLED A FIVE- NUMBER SUMMARY OF THE DATA.
SO THE FIVE-NUMBER SUMMARY TAKES THE FORM OF THE MINIMUM,
Q1 OR QUARTILE 1, THE MEDIAN, Q3 OR QUARTILE 3,
AND THEN FINALLY THE MAXIMUM.
NOW, THERE ARE SEVERAL WAYS TO FIND THE QUARTILES.
IN THIS LESSON WE'LL BE FINDING THE LOCATOR OR PERCENTILE METHOD
TO FIND THE QUARTILES.
NOW, I DO WANT TO MENTION
IF YOU USE THE TI-83 OR 84 GRAPHING CALCULATOR
TO FIND THE QUARTILES,
THE CALCULATOR DOES USE A DIFFERENT METHOD.
FOR THE LOCATOR METHOD,
WE'LL BEGIN BY ORDERING THE DATA FROM THE SMALLEST TO LARGEST,
OR LEAST TO GREATEST.
AND THEN TO FIND Q1 OR QUARTILE 1,
WE COMPUTE THE LOCATOR, WHICH IS L, WHERE L = 0.25 x N,
WHERE N IS THE NUMBER OF DATA VALUES.
AND HERE'S WHERE WE HAVE TO BE CAREFUL.
IF L IS A DECIMAL VALUE WE ROUND L UP TO THE NEXT WHOLE NUMBER,
WHICH WE'LL INDICATE USING THIS NOTATION HERE.
AND THEN WE USE A DATA VALUE
IN THE ROUNDED UP WHOLE NUMBER POSITION AS QUARTILE 1.
HOWEVER, IF L IS A WHOLE NUMBER,
WE FIND THE MEAN OF THE DATA VALUES IN THE L
AND L + 1th POSITIONS.
SO AS AN EXAMPLE, LET'S SAY L IS 5.2.
SINCE THAT WOULD BE A DECIMAL,
WE'D ROUND THAT UP TO 6
AND USE THE DATA VALUE IN THE 6th POSITION FOR Q1.
HOWEVER, IF L WAS EXACTLY 5,
THEN WE'D FIND THE MEAN OF THE DATA VALUES
IN THE 5th AND 6th POSITIONS FOR QUARTILE 1.
TO FIND QUARTILE 3 WE USE THE SAME PROCEDURE,
BUT NOW THE LOCATOR WOULD BE 0.75 x N.
LETS TAKE A LOOK AT A COUPLE OF EXAMPLES.
WE WANT TO FIND THE FIVE-NUMBER SUMMARY FOR THE GIVEN DATA.
NOTICE HOW IT'S GIVEN IN THIS TABLE.
MY SUGGESTION WOULD BE TO WRITE IT OUT HORIZONTALLY
IN ORDER FROM LEAST TO GREATEST, AS I'VE DONE HERE.
THE FIRST THING YOU PROBABLY RECOGNIZE
IS THAT THE MINIMUM IS 6 AND THAT THE MAXIMUM IS 97.
NOW, LET'S FIND THE MEDIAN.
BECAUSE WE HAVE AN ODD NUMBER OF DATA VALUES,
THE MEDIAN WILL BE ONE OF THE DATA VALUES.
IF WE HAVE AN EVEN NUMBER OF A DATA VALUES,
WE ACTUALLY HAVE TO FIND THE MEAN OF THE TWO MIDDLE VALUES.
BUT HERE BECAUSE THERE ARE 15 VALUES,
THE 8th VALUE WOULD BE THE VALUE IN THE MIDDLE OR THE MEDIAN.
SO 1, 2, 3, 4, 5, 6, 7, 8.
49 IS THE MEDIAN, WHICH WE CAN ALSO CALL QUARTILE 2 OR Q2.
AND, AGAIN, THIS WAS THE MINIMUM AND THIS WAS THE MAXIMUM.
NOW LET'S FIND Q1 AND Q3 USING THE LOCATOR METHOD.
SO TO FIND Q1 WE FIRST NEED TO FIND L,
THE LOCATOR = 0.25 x N, THE NUMBER OF DATA VALUES,
WHICH WE KNOW IS 15.
AND SINCE THIS PRODUCT IS EQUAL TO 3.75, WE HAVE A DECIMAL,
WE'RE GOING TO MOVE UP TO THE NEXT WHOLE NUMBER
AND ROUND THIS UP TO 4,
WHICH MEANS THE NUMBER IN THE 4th POSITION WILL BE QUARTILE 1,
OR Q1.
SO 1, 2, 3, 4. 18 IS Q1.
AND NOW TO FIND QUARTILE 3 OR Q3,
WE'LL FIRST FIND L, WHICH IS EQUAL TO FOR Q3 AT 0.75 x N.
THIS COMES OUT TO 11.25, WHICH MEANS YOU ROUND UP TO 12.
SO THE VALUE IN THE 12th POSITION WILL BE QUARTILE 3.
SO THIS IS 8, 9, 10, 11, 12.
82 IS Q3.
SO OUR FIVE-NUMBER SUMMARY WOULD BE 6, 18, 49, 82, 97.
NOW, IN OUR NEXT LESSON WE'LL ACTUALLY TALK ABOUT
HOW WE CAN TAKE THESE VALUES AND FORM A GRAPH CALLED A BOX PLOT.
BEFORE WE GO, LETS TAKE A LOOK AT ONE MORE EXAMPLE
WHERE THE DATA IS GIVEN IN A FREQUENCY TABLE.
AGAIN, OUR GOAL HERE IS TO FIND THE FIVE-NUMBER SUMMARY,
BUT HERE NOTICE THAT 30 OCCURS 3 TIMES.
40 OCCURS 6 TIMES, 50 OCCURS 8 TIMES, AND SO ON.
IF WE FIND THE SUM OF THE FREQUENCY
WE CAN DETERMINE HOW MANY TOTAL VALUES WE HAVE.
3 + 6 + 8 + 4 + 5 + 4 = 30.
SO WE HAVE 30 DATA VALUES HERE, SO N IS 30.
INSTEAD OF WRITING ALL OF THESE VALUES OUT, WHICH WE COULD DO,
LETS SEE IF WE CAN USE THE FREQUENCY TABLE
TO DETERMINE THESE FIVE VALUES.
WELL, WE CAN TELL THE MINIMUM IS GOING TO BE 30.
THAT'S THE SMALLEST VALUE IN OUR DATA SET.
THE LARGEST VALUE IS 80.
AND NOW LET'S FIND THE MEDIAN.
AGAIN, THE MEDIAN IS GOING TO BE THE VALUE IN THE MIDDLE,
AND SINCE WE HAVE 30 VALUES, OR AN EVEN NUMBER OF VALUES,
WE WANT TO FIND THE AVERAGE OF THE 15th AND 16th DATA VALUE.
SO ON TOP WE START AT 30 AND START COUNTING DOWN.
THE 30s AND 40s MAKE UP THE FIRST NINE VALUES,
BUT THEN THERE ARE EIGHT 50s,
WHICH TAKES US TO THE 17th DATA VALUE.
SO NOTICE HOW BOTH THE 15th AND 16th DATA WOULD BOTH BE IN HERE,
AND THEREFORE, BOTH THE 15th AND 16th DATA VALUE ARE 50.
AND, OF COURSE, THE MEAN OF 50 AND 50 WOULD STILL BE 50,
SO THE MEDIAN IS 50.
NOW LET'S FIND Q1 WHERE L = 0.25 x N AND N IS 30.
THIS PRODUCT COMES OUT TO 7.5,
WHICH MEANS WE ARE GOING TO ROUND UP TO 8,
THE NEXT WHOLE NUMBER.
NOW WE WANT TO FIND THE DATA VALUE
THAT WOULD BE IN THE 8th POSITION.
NOTICE HOW THE SMALLEST THREE VALUES ARE 30,
AND THEN THE 4th THROUGH THE 9th VALUE WOULD BE 40.
SO WE'RE LOOKING FOR THE 8th VALUE.
Q1 WOULD BE 40.
AND THEN, FINALLY, WE WANT TO FIND Q3.
SO L= 0.75 x 30 = 22.5.
SO THEN WE'LL GO UP TO THE NEXT WHOLE NUMBER, WHICH WOULD BE 23.
WE WANT TO FIND THE DATA VALUE IN THE 23rd POSITION
ORDER FROM LEAST TO GREATEST.
SO NOTICE THAT 3 + 9 + 8, THAT'S 17 + 4 IS 21,
WHICH MEANS THE 22nd THROUGH THE 26th DATA VALUE
WOULD BE HERE AT 70,
AND SINCE WE'RE LOOKING FOR THE 23rd DATA VALUE,
WE KNOW IT MUST BE 70.
AGAIN, IF THIS IS CONFUSING
WE COULD WRITE ALL OF THESE VALUES OUT,
BUT AS THE FREQUENCY INCREASES,
WRITING THEM OUT WOULD BE MUCH MORE TIME-CONSUMING.
AND NOW WE HAVE OUR FIVE-NUMBER SUMMARY.
THE FIVE-NUMBER SUMMARY IS 30, 40, 50, 70, 80.
AND, AGAIN, IN OUR NEXT VIDEO
WE'LL GO OVER HOW WE CAN MAKE A BOX AND WHISKER PLOT
OR BOX PLOT USING THE FIVE-NUMBER SUMMARY.
I HOPE YOU HAVE FOUND THIS HELPFUL.