- 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.