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to create a stem and leaf plot.

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The stem and leaf plot or stemplot is a way to graph data

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and gives a visualization of the distribution.

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It will show an overall pattern of the data

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and any outliers.

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An outliers is a data value

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that does not fit the rest of the data.

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An outlier is sometimes called an extreme value.

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To begin, we create a table as shown here on the right

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where we have the stems on the left

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and the leaves on the right.

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Sometimes instead of a table, you will just see

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a vertical bar separating the stems and the leaves.

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Step two, we determine these stems and leaves from the data.

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The leaves are the right most digit of each number.

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So going down

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to the data, four, three, four, four, and eight,

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three, four, four, and eight are the leaves

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because we have a single digit,

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these digits are considered on the right.

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For 19, nine is a leaf.

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For the numbers in the 20s,

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we have a leaves of two, five, and seven.

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For the numbers in the 30s,

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we have leaves of two and eight.

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The numbers in the 40s, we have leaves of two and five.

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For the 60s,

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we have leaves of zero one, five, and five.

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For the 70s,

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we have leaves of zero, eight and nine.

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For the 80s,

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we have leaves of zero, one and seven.

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For the 90s, we have leaves of three and nine.

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And for 100, we have a leaf of zero.

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Again, the right most digit of each number is the leaf.

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The stems are all the digits to the left of each leaf.

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So for three, four, four and eight,

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because there is no number to the left,

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the stem is zero.

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For 19, the stem is one.

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For the 20s, the stem is two.

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For the 30s, the stem is three.

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For the 40s, the stem is four.

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Notice there are no 50s,

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but we are still going to include a stem of five

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in the stem and leaf plot.

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For the numbers in the 60s, the stem is six.

For the numbers in the 70s, the stem is seven.For the numbers in the 80s, the stem is eight.And for the numbers of the 90s, the stem is nine. And for 100, the stem is 10. Step three, we now record the stems on the left in the two column table and we include missing stems. So again, the first stem is zero. Then we have a stem of one all the way through a stem of 10.So the stems are zero one, two, three, four, fivsix, seven, eight, nine, and 10 Step four, we record the leaves to the right in the table.So for the stem of zero,the leaves are three, four, four, and eight, which we record in the leaf column. For the stem of one, the leaf is nine for the number 19, and we do want to line up the leaves vertically. For the stem of two, we have leaves of two, five, and seven.For a stem of three, we have leaves of two and eight.For the stem of four, we have leaves of two and five. We don't have any numbers in the 50s and therefore we do not record a leaf for the stem of five.For the stem of six,we have leaves of zero, one, five and five, and we do record five twice.You stated value must have a leaf. For the stem of seven, we have leaves of zero, eight and nine. And for the stem of eight,we have leaves of zero, one and seven.For the stem of nine, we have leaves of three and nine. And for the stem of 10, we have a leaf of zero. So here we have our completed stem and leaf plot, but there is one more thing we should include. We should include a key because if the data isn't given, we do have to know what data values the stem and leaf plot represent. So for example, we should say something like if we have the number of 57,this is equal to a stem of five and a leaf of seven. With this key, someone can now interpret all the data from the given stem and leaf plot.

WELCOME TO A LESSON ON HOW TO DISPLAY CATEGORICAL

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OR QUALITATIVE DATA

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GIVEN IN A FREQUENCY TABLE AS A BAR GRAPH, PARETO CHART,

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PIE GRAPH, AND A PICTOGRAM.

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CATEGORICAL OR A QUALITATIVE DATA ARE PIECES OF INFORMATION

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THAT ALLOW US TO CLASSIFY THE OBJECTS UNDER INVESTIGATION

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INTO VARIOUS CATEGORIES.

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WE USUALLY BEGIN WORKING WITH CATEGORICAL DATA

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BY SUMMARIZING THE DATA IN A FREQUENCY TABLE.

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WHERE A FREQUENCY TABLE IS A TABLE WITH TWO COLUMNS

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AS WE SEE HERE, ONE COLUMN LISTS THE CATEGORIES

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AND THE OTHER FOR THE FREQUENCIES WITH WHICH THE ITEMS

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IN THE CATEGORIES OCCUR,

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MEANING HOW MANY ITEMS FIT INTO EACH CATEGORY.

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LOOKING AT THE FREQUENCY TABLE,

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WE CAN TELL THAT 10 STUDENTS RECEIVED AN "A,"

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12 STUDENTS RECEIVED A "B," 15 STUDENTS RECEIVED A "C,"

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AND SO ON.

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IF WE SUM THE FREQUENCIES, WE CAN DETERMINE THE POPULATION,

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WHICH WE CAN SEE HERE WOULD BE 40.

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SO THERE WERE 40 STUDENTS IN THE CLASS.

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LET'S BEGIN BY DISPLAYING THIS INFORMATION AS A BAR GRAPH.

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TO CONSTRUCT A BAR GRAPH,

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WE NEED TO DRAW A VERTICAL AXIS AND A HORIZONTAL AXIS.

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THE VERTICAL AXIS WILL HAVE A SCALE

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AND MEASURE THE FREQUENCY OF EACH CATEGORY.

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HERE IS OUR VERTICAL AXIS.

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NOTICE HOW THE LARGEST FREQUENCY IS 15,

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AND THEREFORE, THE VERTICAL AXIS IS SCALED TO 16 BY TWOS.

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THE HORIZONTAL AXIS SHOWS THE CATEGORIES.

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AGAIN, HERE WE SEE THE LETTER GRADES,

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AND THE BAR HEIGHT SHOWS THE FREQUENCY.

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SO FOR "A" THE FREQUENCY IS 10.

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NOTICE HOW THE BAR HAS A HEIGHT OF 10.

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FOR B THE FREQUENCY IS 12.

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SO FOR B THE BAR HAS A HEIGHT OF 12 AND SO ON.

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SOMETIMES YOU ALSO SEE THE FREQUENCY LISTED

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AT THE TOP OF EACH BAR LIKE THIS.

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NOW, LET'S TALK ABOUT A PARETO CHART.

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SOMETIMES OUR CHART MIGHT BENEFIT FROM BEING REORDERED

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FROM LARGEST TO SMALLEST FREQUENCY.

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THIS ARRANGEMENT CAN MAKE IT EASIER

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TO COMPARE SIMILAR VALUES IN THE CHART,

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EVEN WITHOUT GRIDLINES.

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WHEN WE REARRANGE THE CATEGORIES IN DECREASING FREQUENCY ORDER

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LIKE THIS,

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IT IS CALLED A PARETO CHART.

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SO HERE'S THE ORIGINAL FREQUENCY TABLE.

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IF WE WANTED TO REORDER THIS FROM LARGEST FREQUENCY

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TO SMALLEST FREQUENCY,

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WE'D HAVE TO SWITCH THE As AND THE Cs

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SO THAT THE HIGHEST FREQUENCY OF 15 IS FIRST,

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FOLLOWED BY 12, 10, 2 AND THEN 1.

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NOW, IF WE USE THIS FREQUENCY TABLE TO MAKE A BAR GRAPH,

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IT'LL BE A PARETO CHART.

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SO THE ORANGE GRAPH IS A BAR GRAPH.

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THIS PURPLE GRAPH IS A PARETO CHART.

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AGAIN, LOOKING AT THE FREQUENCIES,

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NOTICE HOW THEY GO FROM LARGEST TO SMALLEST.

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SO NOW, THE CATEGORIES ARE IN THE ORDER OF C, B, A, D, F,

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AND AGAIN, SOMETIMES YOU WILL SEE THE FREQUENCY

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LISTED AT THE TOP OF EACH BAR.

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AND NOW, LET'S DISPLAY THE SAME DATA AS A PIE CHART.

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TO SHOW RELATIVE SIZES, IT IS COMMON TO USE A PIE CHART.

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A PIE CHART IS A CIRCLE WITH WEDGES CUT OUT OF VARYING SIZES

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MARKED OUT LIKE SLICES OF PIE OR PIZZA.

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THE RELATIVE SIZES OF THE WEDGES

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CORRESPOND TO THE RELATIVE FREQUENCIES OF THE CATEGORIES.

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SO HERE'S THE PIE CHART.

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THIS WEDGE REPRESENTS THE As.

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THIS WEDGE REPRESENTS THE Bs, Cs, Ds AND Fs.

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NOTICE HOW THIS IS CREATED USING SOFTWARE,

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WHICH IS VERY COMMON THESE DAYS,

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BUT IF WE HAD TO DO THIS BY HAND,

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ONE CIRCLE REPRESENTS 360 DEGREES.

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SO ONE WAY TO DETERMINE THE SIZE OF EACH WEDGE

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WOULD BE TO USE A PROTRACTOR, WHICH MEASURES ANGLES.

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AND BECAUSE THE TOTAL NUMBER OF STUDENTS IS 40,

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THE SUM OF THE FREQUENCIES, AND 10 OF THEM HAVE As,

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THE LETTER GRADE OF "A" REPRESENTS

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25% OF THE TOTAL POPULATION

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OR IN THIS CASE, 25% OF THE CIRCLE.

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AND SO IF WE FIND 25% OF 360 DEGREES, WHICH IS 90 DEGREES,

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WE CAN MEASURE OUT 90 DEGREES TO CREATE THIS WEDGE.

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AND WE CAN DO THE SAME FOR THE OTHER LETTER GRADES,

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BUT AGAIN, NORMALLY, WE JUST USE SOFTWARE TO CREATE PIE CHARTS.

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THE LAST GRAPH WE'LL TAKE A LOOK AT IS CALLED A PICTOGRAM.

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A PICTOGRAM IS A STATISTICAL GRAPH

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IN WHICH THE SIZE OF THE PICTURE

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IS INTENDED TO REPRESENT THE FREQUENCIES

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OR SIZE OF THE VALUES BEING REPRESENTED,

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AND THERE ARE SOME VARIATIONS OF PICTOGRAMS.

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NOTICE HERE INSTEAD OF BARS,

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WE'RE USING PICTURES OF As, Bs, Cs, Ds AND Fs.

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SO IT IS VERY SIMILAR TO A BAR GRAPH.

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OFTEN YOU WILL SEE THE FREQUENCY

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LISTED AT THE TOP OF EACH PICTURE.

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NOTICE HOW HERE THE IMAGE OF EACH LETTER IS THE SAME SIZE.

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ANOTHER OPTION IS TO STRETCH EACH IMAGE

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TO THE CORRECT HEIGHT, AS WE SEE HERE.

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A LABOR UNION MIGHT PRODUCE THE GRAPH TO THE RIGHT OR BELOW HERE

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TO SHOW THE DIFFERENCE BETWEEN THE AVERAGE MANAGER SALARY

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AND THE AVERAGE WORKER SALARY.

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LOOKING AT THE PICTURES,

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IT WOULD BE REASONABLE TO GUESS THAT THE MANAGERS' SALARIES

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ARE FOUR TIMES AS LARGE AS THE WORKERS' SALARIES,

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BECAUSE IT DOES APPEAR THE AREA OF THIS LARGER BAG

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IS FOUR TIMES AS LARGE AS THE AREA OF THE SMALLER BAG.

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HOWEVER, THE MANAGERS' SALARIES ARE, IN FACT,

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ONLY TWICE AS LARGE AS THE WORKERS' SALARIES,

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WHICH ARE REFLECTED IN THE PICTURE

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BY MAKING THE MANAGER BAG TWICE AS TALL.

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SO HERE THIS PICTOGRAM CAN BE DECEPTIVE

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UNLESS WE PAY CLOSE ATTENTION TO THE HEIGHT OF EACH OF THESE

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RATHER THAN THEIR SIZE.

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LOOKING AT THE SCALING ON THE AXES,

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NOTICE HOW THIS LARGER BAG IS TWICE AS TALL

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AS THIS SMALLER BAG,

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REPRESENTING THE SALARIES ARE ONLY TWICE AS LARGE,

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NOT FOUR TIMES AS LARGE.

n this table we have two rows.

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In one we will have the class interval,

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which will be the price range of pens and in the second,

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we will have the frequency, that is the number of pens sold of each range.

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So these are the different class intervals that are given.

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The first range is pens that cost between 10 and 20 rupees.

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We can see that the upper limit of one interval

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is equal to the lower limit of the next interval

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making it continuous.

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As It's continuous the first group would mean

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10 rupees is less than,

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or equal to the price of the pen which will be less than 20 rupees.

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The lower limit is included, but the upper limit is not.

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So in the second group 20 is included,

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along with all numbers less than 30 but not equal to 30.

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In the second row, we have the number of pens sold for each category.

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15 pens of this category was sold, 20 of this category and so on...

Histogram

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Until now, we have seen examples of bar graphs,

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but never seen bar graphs used to represent

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continuous form of grouping.

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Let's see how we can do that.

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The first thing that we do is draw the 'x' and the 'y' axes.

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We draw the axes and make equal intervals on each of them.

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Then we label the axes.

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Since we want to represent the quantity of each class interval,

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we have the pens sold in the 'y' axis, and the price range of pens on the 'x' axis.

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The steps are similar to the ones we use to draw a Bar graph.

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Now we have to choose a suitable scale for the 'y' axis.

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The number of pens sold range between 5 and 30.

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A scale of 1 centimetre equals to 5 pens on the 'y' axis

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would be perfect as we would need just 6 divisions.

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We start writing the values on the 'y' axis.

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5, 10, 15 and so on up to 30.

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And on the 'x' axis, we start writing the price range of pens.

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The first range is 10 to 20.

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Then 20 to 30 and so on up to 50 to 60.

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That's it! Now we just have to draw

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appropriate bars for each class interval.

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For the range 10 to 20, the frequency is 15.

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So based on the scale we draw a Bar of length 3 centimetres.

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For a frequency of 20, we draw a bar of length 4 centimeters.

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And we continue the process till we reach the last range.

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This is called a Histogram.

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Bars used to represent continuous class intervals.

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Remember, for Bar graphs we did not have intervals here.

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We just had plain simple categories.

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But here, we have intervals on one axis and quantity on the other.

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What we did here was simple.

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We choose a scale and draw bars for each continuous interval.

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But note that the intervals were continuous.

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20 which is the upper limit of the first interval,

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is used as the lower limit for the next interval

Give a 2-bullet point summary/reflection of what you learned or found to be interesting and/or important for each of the 3 lesson pages on displaying data with graphs that are just before this assignment in the STAT 32/532 Week 2 module. That is, the following STAT 32/532 pages: stem-and leaf plots, categorical displays-bar graphs, etc., and histograms. So, 6 bullet points in total.This lesson we'll show how