Transcript for: Understanding Histograms and Their Construction
WELCOME TO A LESSON
ON HISTOGRAMS. A HISTOGRAM IS A BAR GRAPH THAT REPRESENTS A FREQUENCY
DISTRIBUTION AS WE SEE HERE BELOW. THE WIDTH REPRESENTS
THE INTERVAL, AND THE HEIGHT REPRESENTS
THE CORRESPONDING FREQUENCY. NOTICE THERE ARE NO SPACES
BETWEEN THE BARS. SO LOOKING
AT THE GIVEN HISTOGRAM, NOTICE HOW THE INTERVALS HAVE
THE SAME WIDTH OF THE SAME SIZE, WHICH ONCE AGAIN, MEANS THE BARS
HAVE THE SAME WIDTH. THESE CATEGORIES HERE
OR THE INTERVALS ARE OFTEN CALLED THE BINS, AND THEN NOTICE THE VERTICAL
AXIS INDICATES THE FREQUENCY. SO WE CAN EASILY SEE
THAT THE MODE OF WAIT TIMES IS FOUR TO FIVE MINUTES. LOOKS LIKE IT OCCURRED 13 TIMES. NOW, LET'S CONSTRUCT
OUR OWN HISTOGRAM. HERE WE'RE GIVEN DATA ON AVERAGE
GAS MILEAGE OF 20 CARS. SO BEFORE WE QUIT THE HISTOGRAM, WE'LL COMPLETE THIS FREQUENCY
TABLE WHERE WE ALREADY SET UP
THE INTERVALS. ONCE AGAIN, NOTICE HOW THEY HAVE
THE SAME WIDTH OF THE SAME SIZE, AND THERE ARE SEVEN INTERVALS
HERE. WE CAN SAY WE HAVE SEVEN BINS. SO WE'LL FIRST GO THROUGH
AND TALLY THE DATA IN THE CORRECT INTERVAL AND THEN FIND THE FREQUENCY. SO STARTING WITH 24,
WHICH WOULD BE IN THIS INTERVAL, THEN WE HAVE 17, 14, 22, 25
AND 26 BOTH IN THIS INTERVAL, 38, 42, 24, 12, 28, 19, 32, 21, 35, 28,
21, 31, 18 AND 19. AND THEREFORE, THE FREQUENCIES
WILL BE TWO, FOUR, FIVE, FOUR, TWO, TWO AND ONE. SO WE'LL USE THESE INTERVALS
OR THESE BINS ALONG THE HORIZONTAL AXIS, AND WE'LL USE THE FREQUENCIES
ALONG THE VERTICAL AXIS. LET'S GO AHEAD AND SET THIS UP. STARTING WITH THE INTERVALS
OF THE BINS, WE'LL DIVIDE THE HORIZONTAL AXIS
INTO SEVEN EQUAL INTERVALS. SO ONE, TWO, THREE, FOUR, FIVE,
SIX AND SEVEN, SO LET'S GO AHEAD
AND LABEL THEM. THE HIGHEST FREQUENCY IS FIVE. LET'S GO AHEAD AND HAVE
A MAXIMUM VALUE HERE OF SIX, BREAK THIS INTO
SIX EQUAL PIECES, AND NOW, WE'LL CONSTRUCT A BAR
FOR EACH BIN OR EACH INTERVAL. SO IF WE INTERVAL FROM 10 TO 14,
THE FREQUENCY IS TWO. SO WE'LL MAKE A BAR
WITH A HEIGHT OF TWO. THE INTERVAL FROM 15 TO 19
HAS A FREQUENCY OF FOUR. SO WE'LL MAKE A BAR WITH A HEIGHT OF FOUR
OVER THIS INTERVAL. THE NEXT INTERVAL
HAS A FREQUENCY OF FIVE. SO WE'LL MAKE A BAR
OF HEIGHT FIVE. THE NEXT FREQUENCY IS FOUR. THE NEXT TWO FREQUENCIES
ARE TWO, AND THE LAST FREQUENCY IS ONE. AND HERE'S OUR HISTOGRAM, AND ONCE AGAIN, WE CAN SEE THAT
THE MODE WOULD BE THE INTERVAL FROM 20 TO 24 FOR THIS DATA OF AVERAGE GAS
MILEAGE IN MILES PER GALLON. I WENT AHEAD AND CREATED
THE SAME HISTOGRAM USING SOME SOFTWARE
WITH THE SAME RESULTS. THIS HISTOGRAM WAS MADE
WITH EXCEL. THERE IS ONE MORE THING I'D LIKE
TO MENTION BEFORE WE GO. IF THERE ARE NO BREAKS
IN THE DATA, THEN NORMALLY, THE LAST DATA
VALUE OF A BIN IS THE FIRST VALUE
IN THE NEXT BIN. FOR EXAMPLE, LOOKING
AT THE GIVEN HISTOGRAM ON HEIGHTS
OF BLACK CHERRY TREES, NOTICE HOW THE HORIZONTAL AXIS
IS LABELED BY FIVES. SO THIS FIRST INTERVAL
OR FIRST BIN LOOKS LIKE IT GOES FROM 60
TO 65, AND THE SECOND BIN
OR SECOND INTERVAL GOES FROM 65 TO 70. SO THE QUESTION BECOMES,
"WHERE IS THE 65?" IS IT IN THE FIRST BIN
OR IS IT IN THE SECOND BIN? AND IT'S ACTUALLY
IN THE SECOND BIN. THIS FIRST INTERVAL
IS FROM 60 TO 65 WHERE IF WE USE
INTERVAL NOTATION, IT WOULD BE CLOSED ON 60, MEANING IT INCLUDES 60,
BUT IT WOULD BE OPEN ON 65, MEANING IT DOES NOT INCLUDE 65. THE NEXT INTERVAL WOULD BE
CLOSED ON 65 AND OPEN ON 70. THE VALUE OF 70 WOULD BE
IN THIS THIRD INTERVAL. THE THIRD INTERVAL WOULD BE
CLOSED ON 70, MEANING IT INCLUDES 70
BUT OPEN ON 75, MEANING IT DOES NOT INCLUDE 75
AND SO ON. SO IT IS IMPORTANT
TO KEEP THIS IN MIND IF YOU SEE A HISTOGRAM
LABELED AS WE DO HERE. I HOPE YOU FOUND THIS HELPFUL.