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Measuring Variation: Quartiles and Five Number Summary - YouTube
Channel: Mathispower4u
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- WELCOME TO A LESSON
ON HOW TO FIND THE QUARTILES
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AND FIVE-NUMBER SUMMARY
OF A SET OF DATA.
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LET'S BEGIN BY TALKING
ABOUT THE MEDIAN.
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THE MEDIAN OF A SET OF DATA
IS THE VALUE IN THE MIDDLE
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WHEN THE DATA IS IN ORDER
FROM LEAST TO GREATEST.
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AND SINCE THE MEDIAN
IS IN THE MIDDLE,
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HALF THE DATA VALUES
WOULD BE BELOW THE MEDIAN,
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AND HALF WOULD BE
ABOVE THE MEDIAN.
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THE QUARTILES ARE THE VALUES
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THAT DIVIDE THE DATA
INTO QUARTERS.
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SO THE FIRST QUARTILE, OR Q1,
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IS THE VALUE SO THAT 25%
OF THE DATA VALUES ARE BELOW IT,
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AND THE 3rd QUARTILE, OR Q3,
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IS THE VALUE SO THAT 75%
OF THE DATA VALUES ARE BELOW IT.
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YOU MAY HAVE GUESSED
THAT THE SECOND QUARTILE, OR Q2,
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IS THE SAME AS THE MEDIAN
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SINCE THE MEDIAN IS THE VALUE
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SO THAT 50% OF THE DATA VALUES
ARE BELOW IT
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AND 50% ARE ABOVE IT.
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SO THE QUARTILES
DIVIDE THE DATA INTO QUARTERS
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WHERE 25% OF THE DATA IS BETWEEN
THE MINIMUM AND Q1.
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25% IS BETWEEN Q1
AND THE MEDIAN, OR Q2.
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25% IS BETWEEN THE MEDIAN
OR Q2 AND Q3,
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AND FINALLY, 25% IS BETWEEN Q3
AND THE MAXIMUM VALUE.
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SO TO ILLUSTRATE THIS
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IF WE THINK OF THIS BAR
AS BEING THIS DATA,
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WHERE THIS WOULD BE
THE MINIMUM VALUE.
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THIS WOULD BE THE MAXIMUM VALUE.
THIS WOULD BE THE MEDIAN OR Q2.
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THIS VALUE WOULD BE Q1,
OR THE 1st QUARTILE,
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AND THIS VALUE WOULD BE Q3
OR THE 3rd QUARTILE.
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SO 25% OF THE DATA WOULD BE
IN THIS INTERVAL.
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25% WOULD BE IN THIS INTERVAL
AND SO ON.
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SO WHILE THE QUARTILES ARE NOT A
ONE-NUMBER SUMMARY OF VARIATION,
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LIKE STANDARD DEVIATION,
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THE QUARTILES ARE USED
WITH THE MEDIAN,
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MINIMUM, AND MAXIMUM VALUES
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TO FORM WHAT'S CALLED A FIVE-
NUMBER SUMMARY OF THE DATA.
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SO THE FIVE-NUMBER SUMMARY TAKES
THE FORM OF THE MINIMUM,
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Q1 OR QUARTILE 1, THE MEDIAN,
Q3 OR QUARTILE 3,
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AND THEN FINALLY THE MAXIMUM.
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NOW, THERE ARE SEVERAL WAYS
TO FIND THE QUARTILES.
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IN THIS LESSON WE'LL BE FINDING
THE LOCATOR OR PERCENTILE METHOD
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TO FIND THE QUARTILES.
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NOW, I DO WANT TO MENTION
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IF YOU USE THE TI-83 OR 84
GRAPHING CALCULATOR
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TO FIND THE QUARTILES,
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THE CALCULATOR DOES USE
A DIFFERENT METHOD.
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FOR THE LOCATOR METHOD,
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WE'LL BEGIN BY ORDERING THE DATA
FROM THE SMALLEST TO LARGEST,
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OR LEAST TO GREATEST.
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AND THEN TO FIND Q1
OR QUARTILE 1,
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WE COMPUTE THE LOCATOR,
WHICH IS L, WHERE L = 0.25 x N,
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WHERE N IS THE NUMBER
OF DATA VALUES.
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AND HERE'S WHERE
WE HAVE TO BE CAREFUL.
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IF L IS A DECIMAL VALUE WE ROUND
L UP TO THE NEXT WHOLE NUMBER,
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WHICH WE'LL INDICATE
USING THIS NOTATION HERE.
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AND THEN WE USE A DATA VALUE
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IN THE ROUNDED UP WHOLE NUMBER
POSITION AS QUARTILE 1.
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HOWEVER, IF L IS A WHOLE NUMBER,
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WE FIND THE MEAN OF THE DATA
VALUES IN THE L
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AND L + 1th POSITIONS.
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SO AS AN EXAMPLE,
LET'S SAY L IS 5.2.
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SINCE THAT WOULD BE A DECIMAL,
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WE'D ROUND THAT UP TO 6
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AND USE THE DATA VALUE
IN THE 6th POSITION FOR Q1.
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HOWEVER, IF L WAS EXACTLY 5,
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THEN WE'D FIND THE MEAN
OF THE DATA VALUES
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IN THE 5th AND 6th POSITIONS
FOR QUARTILE 1.
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TO FIND QUARTILE 3
WE USE THE SAME PROCEDURE,
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BUT NOW THE LOCATOR WOULD BE
0.75 x N.
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LETS TAKE A LOOK
AT A COUPLE OF EXAMPLES.
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WE WANT TO FIND THE FIVE-NUMBER
SUMMARY FOR THE GIVEN DATA.
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NOTICE HOW IT'S GIVEN
IN THIS TABLE.
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MY SUGGESTION WOULD BE TO WRITE
IT OUT HORIZONTALLY
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IN ORDER FROM LEAST TO GREATEST,
AS I'VE DONE HERE.
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THE FIRST THING YOU PROBABLY
RECOGNIZE
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IS THAT THE MINIMUM IS 6
AND THAT THE MAXIMUM IS 97.
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NOW, LET'S FIND THE MEDIAN.
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BECAUSE WE HAVE AN ODD NUMBER
OF DATA VALUES,
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THE MEDIAN WILL BE
ONE OF THE DATA VALUES.
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IF WE HAVE AN EVEN NUMBER
OF A DATA VALUES,
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WE ACTUALLY HAVE TO FIND THE
MEAN OF THE TWO MIDDLE VALUES.
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BUT HERE BECAUSE THERE ARE
15 VALUES,
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THE 8th VALUE WOULD BE THE VALUE
IN THE MIDDLE OR THE MEDIAN.
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SO 1, 2, 3, 4, 5, 6, 7, 8.
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49 IS THE MEDIAN, WHICH WE CAN
ALSO CALL QUARTILE 2 OR Q2.
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AND, AGAIN, THIS WAS THE MINIMUM
AND THIS WAS THE MAXIMUM.
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NOW LET'S FIND Q1 AND Q3
USING THE LOCATOR METHOD.
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SO TO FIND Q1
WE FIRST NEED TO FIND L,
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THE LOCATOR = 0.25 x N,
THE NUMBER OF DATA VALUES,
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WHICH WE KNOW IS 15.
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AND SINCE THIS PRODUCT IS EQUAL
TO 3.75, WE HAVE A DECIMAL,
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WE'RE GOING TO MOVE UP
TO THE NEXT WHOLE NUMBER
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AND ROUND THIS UP TO 4,
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WHICH MEANS THE NUMBER IN THE
4th POSITION WILL BE QUARTILE 1,
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OR Q1.
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SO 1, 2, 3, 4. 18 IS Q1.
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AND NOW TO FIND QUARTILE 3
OR Q3,
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WE'LL FIRST FIND L, WHICH IS
EQUAL TO FOR Q3 AT 0.75 x N.
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THIS COMES OUT TO 11.25,
WHICH MEANS YOU ROUND UP TO 12.
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SO THE VALUE IN THE 12th
POSITION WILL BE QUARTILE 3.
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SO THIS IS 8, 9, 10, 11, 12.
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82 IS Q3.
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SO OUR FIVE-NUMBER SUMMARY
WOULD BE 6, 18, 49, 82, 97.
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NOW, IN OUR NEXT LESSON
WE'LL ACTUALLY TALK ABOUT
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HOW WE CAN TAKE THESE VALUES AND
FORM A GRAPH CALLED A BOX PLOT.
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BEFORE WE GO, LETS TAKE A LOOK
AT ONE MORE EXAMPLE
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WHERE THE DATA IS GIVEN
IN A FREQUENCY TABLE.
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AGAIN, OUR GOAL HERE IS TO FIND
THE FIVE-NUMBER SUMMARY,
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BUT HERE NOTICE THAT 30 OCCURS
3 TIMES.
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40 OCCURS 6 TIMES,
50 OCCURS 8 TIMES, AND SO ON.
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IF WE FIND THE SUM
OF THE FREQUENCY
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WE CAN DETERMINE HOW MANY TOTAL
VALUES WE HAVE.
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3 + 6 + 8 + 4 + 5 + 4 = 30.
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SO WE HAVE 30 DATA VALUES HERE,
SO N IS 30.
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INSTEAD OF WRITING ALL OF THESE
VALUES OUT, WHICH WE COULD DO,
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LETS SEE IF WE CAN USE
THE FREQUENCY TABLE
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TO DETERMINE THESE FIVE VALUES.
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WELL, WE CAN TELL THE MINIMUM
IS GOING TO BE 30.
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THAT'S THE SMALLEST VALUE
IN OUR DATA SET.
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THE LARGEST VALUE IS 80.
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AND NOW LET'S FIND THE MEDIAN.
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AGAIN, THE MEDIAN IS GOING TO BE
THE VALUE IN THE MIDDLE,
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AND SINCE WE HAVE 30 VALUES,
OR AN EVEN NUMBER OF VALUES,
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WE WANT TO FIND THE AVERAGE
OF THE 15th AND 16th DATA VALUE.
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SO ON TOP WE START AT 30
AND START COUNTING DOWN.
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THE 30s AND 40s MAKE UP
THE FIRST NINE VALUES,
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BUT THEN THERE ARE EIGHT 50s,
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WHICH TAKES US
TO THE 17th DATA VALUE.
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SO NOTICE HOW BOTH THE 15th AND
16th DATA WOULD BOTH BE IN HERE,
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AND THEREFORE, BOTH THE 15th
AND 16th DATA VALUE ARE 50.
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AND, OF COURSE, THE MEAN OF 50
AND 50 WOULD STILL BE 50,
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SO THE MEDIAN IS 50.
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NOW LET'S FIND Q1
WHERE L = 0.25 x N AND N IS 30.
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THIS PRODUCT COMES OUT TO 7.5,
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WHICH MEANS WE ARE GOING
TO ROUND UP TO 8,
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THE NEXT WHOLE NUMBER.
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NOW WE WANT TO FIND THE DATA
VALUE
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THAT WOULD BE
IN THE 8th POSITION.
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NOTICE HOW THE SMALLEST
THREE VALUES ARE 30,
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AND THEN THE 4th THROUGH
THE 9th VALUE WOULD BE 40.
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SO WE'RE LOOKING
FOR THE 8th VALUE.
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Q1 WOULD BE 40.
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AND THEN, FINALLY,
WE WANT TO FIND Q3.
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SO L= 0.75 x 30 = 22.5.
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SO THEN WE'LL GO UP TO THE NEXT
WHOLE NUMBER, WHICH WOULD BE 23.
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WE WANT TO FIND THE DATA VALUE
IN THE 23rd POSITION
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ORDER FROM LEAST TO GREATEST.
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SO NOTICE THAT 3 + 9 + 8,
THAT'S 17 + 4 IS 21,
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WHICH MEANS THE 22nd
THROUGH THE 26th DATA VALUE
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WOULD BE HERE AT 70,
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AND SINCE WE'RE LOOKING
FOR THE 23rd DATA VALUE,
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WE KNOW IT MUST BE 70.
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AGAIN, IF THIS IS CONFUSING
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WE COULD WRITE
ALL OF THESE VALUES OUT,
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BUT AS THE FREQUENCY INCREASES,
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WRITING THEM OUT WOULD BE
MUCH MORE TIME-CONSUMING.
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AND NOW WE HAVE
OUR FIVE-NUMBER SUMMARY.
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THE FIVE-NUMBER SUMMARY IS 30,
40, 50, 70, 80.
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AND, AGAIN, IN OUR NEXT VIDEO
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WE'LL GO OVER HOW WE CAN MAKE
A BOX AND WHISKER PLOT
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OR BOX PLOT USING
THE FIVE-NUMBER SUMMARY.
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I HOPE YOU HAVE FOUND THIS
HELPFUL.
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