Reading Data and Graphs: Free Response

A row of circles is a count of symbols, and the key under the rows is what turns that count into a weight. Take glass first, since it is the row with a part symbol: two full circles and one half circle make $2.5$ symbols, and the half circle is worth half of the key rather than one kilogram.

$2.5 \times 12 = 30$

So glass brought in $30$ kilograms. Paper's row is four full circles, a whole number of symbols, and the same multiplication applies.

$4 \times 12 = 48$

So paper brought in $48$ kilograms. Naming the material and the unit is part of the answer here: a bare $30$ could be read as a count of circles, of bags, or of anything else on the page.

The claim sets one material against two others combined, so every row involved has to be a weight before anything is added. Cans is three full circles and glass is two and a half:

$3 \times 12 = 36, \qquad 2.5 \times 12 = 30$

Together they are $36 + 30 = 66$ kilograms, while paper is $48$ kilograms. Since $48 < 66$, the pictograph refutes the poster rather than supporting it.

It is worth seeing why the claim was tempting. Paper's row is the longest row on the display, so paper really is the largest single material, and the poster has quietly slid from that true statement to a stronger one. Being ahead of every other material one at a time says nothing about being ahead of two of them added together, and only the first of those is something the display shows.

Count every circle on the display: four for paper, two and a half for glass, three for cans, and one and a half for plastic.

$4 + 2.5 + 3 + 1.5 = 11$

That $11$ is exactly the figure in the report, so the helper counted symbols and then wrote kilograms beside the count. A symbol is not a kilogram. The key fixes each circle at $12$ kilograms, so the symbol count still has to be multiplied by it.

$11 \times 12 = 132$

The report should say $132$ kilograms. Adding the four row weights gives the same figure, $48 + 30 + 36 + 18 = 132$, which is a useful check that no row was miscounted along the way. Notice that the helper's arithmetic was never wrong; only the unit attached to it was, and that is what makes this slip so easy to miss.

Read a month by finding its point and following it across to the scale. The highest point on the display is the one above July, which sits on the line marked $42$, and the lowest is the one above June, on the line marked $18$.

The range measures spread: the largest reading minus the smallest.

$42 - 18 = 24$

So the range is $24$ millimetres. Notice that the range is a size, not a month, so it carries the unit of the readings and no month name of its own. Reading the highest and lowest points off the scale, rather than judging which part of the line looks tallest, is what makes both answers exact.

A mean needs two things: the sum of the readings and how many readings there are. The display supplies both, once each point has been read off the scale. In order they are $24$, $36$, $30$, $18$ and $42$ millimetres.

$24 + 36 + 30 + 18 + 42 = 150$

There are five months, so divide the total by five.

$150 \div 5 = 30$

The mean is $30$ millimetres per month. A quick check: measure each reading against $30$ and the gaps are $-6$, $+6$, $0$, $-12$ and $+12$, which cancel to zero, exactly as the gaps from a correct mean must. Note also what the mean is not. May happens to land exactly on it, which is a coincidence rather than the rule: the mean is the level at which five equal months would give the same five-month total, and no month need ever record it.

Four segments join the five points, so all four have to be checked rather than the first downward one you notice. March to April rises by $12$, April to May falls by $6$, May to June falls by $12$, and June to July rises by $24$. Two segments fall, and the larger of the two drops is May to June.

$30 - 18 = 12$

So the largest fall is from May to June, a drop of $12$ millimetres.

August is a different matter. The last plotted point is July, and a line graph reports the readings it plots together with the change between consecutive ones. There is no segment past July, so nothing on the display is a statement about August. It would be easy to look at the sharp rise into July and carry the line onwards in your head, but that line would be your guess, not the reserve's data.

Every lunch served was exactly one of the four dishes, so the slices account for the whole day between them and their percents fill $100\%$. That is what recovers a missing share: subtract the labelled ones.

$100 - 48 - 28 - 16 = 8$

So the unlabelled slice is $8\%$.

A percent is a share, not a count, and it becomes a count only against a stated total. Here the total is given as $175$ lunches, so take $8\%$ of it.

$\frac{8}{100} \times 175 = 0.08 \times 175 = 14$

So the wrap slice stands for $14$ lunches. The two answers are different kinds of thing and both are wanted: $8$ is a share of the day's service, and $14$ is a number of meals actually handed over the counter.

Comparing two slices means comparing two counts, so convert each share against the day's total before subtracting anything.

$0.48 \times 175 = 84, \qquad 0.28 \times 175 = 49$

The difference is then a straight subtraction.

$84 - 49 = 35$

So $35$ more pasta lunches were served than curry ones. Subtracting the percents on their own would have given $20$, which is a share of the day and not a number of meals; that gap becomes meals only after it meets the total, which is what the alternate method below does deliberately.

A slice measures a share of its own circle, and each circle here is a different canteen. So two slices of equal size say that curry took the same fraction of each day's service, which is not yet a statement about meal counts: a fraction becomes a count only against its own total.

$0.28 \times 175 = 49, \qquad 0.28 \times 250 = 70$

The first school served $49$ curry lunches and the second $70$, so the conclusion does not follow.

The cook's reasoning would be sound if the two canteens had served the same number of lunches, and that is the assumption doing the hidden work. It is the same trap as reading a slice's percent straight off as a count, one step further along: a percent settles a share, and only a share, until a total is supplied. What the two circle graphs really do agree on is that curry took the same portion of each school's day, which is a genuine and useful fact, just not the one the cook stated.

Follow the top of each bar across to the vertical scale and read the number printed there. Tray A's bar reaches $96$ and Tray B's reaches $84$. The heights on the page are not the values; the scale is.

$96 - 84 = 12$

So Tray A produced $12$ more sprouted seeds than Tray B. Twelve seeds out of numbers in the nineties is a small margin, and it is worth holding that thought against the picture, which does not look like a small margin at all.

A bar is drawn upwards from the axis line, so its length stands for however far its value sits above the number at the axis, and here that number is $80$. Measure both bars in scale units.

$96 - 80 = 16, \qquad 84 - 80 = 4$

Dividing those lengths compares the bars as drawn.

$16 \div 4 = 4$

So Tray A's bar is drawn four times as tall as Tray B's, which is what a reader's eye picks up first.

The values compare quite differently, because a value is measured from zero whatever the graph chooses to show.

$\frac{96}{84} = \frac{8}{7} \approx 1.14$

So Tray A produced about $1.14$ times what Tray B produced, a seventh more. One picture, two comparisons that disagree by a wide margin: four against roughly one and a seventh.

Start from where a bar begins. Every bar rises from the axis line, and the axis line here is not zero seeds, it is $80$ seeds. So a bar of value $v$ is drawn with a length proportional to $v - 80$, and the first $80$ seeds of each tray are simply not on the page.

That is the whole distortion. Tray B is only $4$ above the cut, Tray A is $16$ above it, and cutting the same $80$ off both leaves a lopsided pair.

$\frac{96 - 80}{84 - 80} = \frac{16}{4} = 4$

Subtracting a fixed amount from two close numbers does not preserve their ratio; it enlarges it, and the closer the two numbers are to the cut, the worse the enlargement gets. The reader is invited to compare lengths, as bar graphs normally allow, and lengths here answer a different question from the one being asked.

The change that repairs it is to start the value axis at $0$. Then each bar's length is proportional to the value itself, and the two bars stand in the ratio

$96 : 84 = 8 : 7,$

so Tray A's bar would be just a seventh longer than Tray B's, which is the honest picture of a $12$ seed lead. None of this makes a truncated scale automatically dishonest, since a chart of temperatures near freezing has good reason to cut, but it does mean the reader must look at where the scale begins before trusting any comparison of lengths.

A frequency table is read straight off, one row at a time: flutes $15$, clarinets $21$, trumpets $9$, drums $6$. Because every player sits in exactly one section, the four frequencies account for the whole band with nobody counted twice and nobody left out, so adding them gives the band's size.

$15 + 21 + 9 + 6 = 51$

So there are $51$ players.

The two columns hold different kinds of thing, and keeping them apart is the whole skill. The left column names what was counted, the section, and those names are the categories; they are not numbers at all. The right column says how many players fall into each one, and those are the frequencies. "Clarinets" is a category and $21$ is its frequency, and it makes no sense to add category names or to ask which frequency is a woodwind.

A segment drawn between two points asserts a progression: that one quantity moved from the first reading to the second across some stretch the horizontal axis measures. Test that assertion here. The first segment climbs from flutes to clarinets:

$21 - 15 = 6$

and a rise of $6$ ought to be six extra players arriving across some stretch. Across what? Put a finger halfway along that segment and ask which section it stands over. There is no section between flutes and clarinets, and no sense in which flutes come before clarinets in the first place, so the point under your finger measures nothing and the height it reaches counts nobody.

The "climbing" and the "falling away" are therefore artefacts of the drawing rather than facts about the band. The clearest sign of it is that the shape depends on an arbitrary decision: the sections could have been listed in any order, and a different order would produce a different-looking trend from identical data. A display whose story changes when you rearrange the rows is not reporting the data.

For joining points to mean anything, the horizontal axis has to carry an order, so that consecutive readings really are consecutive and the stretch between them is something. In practice that means time or another measured quantity. Sections are separate categories with no order and no in between, so this record wants bars standing apart, one per section, precisely because separated bars make no claim about the space between them.

Match each question to what its data looks like rather than to a favourite display.

The first question tracks one quantity, the band's size, across five years in order. Years come in sequence, so consecutive readings really are consecutive and a segment between them reports the change from that year to the next. Membership never takes a fractional value in between, and a line graph does not claim it does; what the segment carries is the direction and size of the change, which is exactly what the question asks. That is the situation a line graph is built for, and the same reasoning that ruled it out in part B rules it in here.

The second question asks what portion of one whole each section takes, and the sections do split the band with nobody counted twice. A circle graph shows one whole divided into parts, so each section becomes a slice, and the clarinets' slice for instance would be

$\frac{21}{51} = \frac{7}{17},$

a little over two fifths of the circle. Had the question instead asked for exact player counts, the table already answers it better than any picture: a circle graph is the right choice precisely because portions, not counts, are what was asked for.