Free math worksheets

- Examine the scale as well as the bar heights. - Check whether the starting point and maximum create a fair visual comparison.

1. Compare how the bar heights would look with each starting value. 2. If the vertical axis began at \(18\), differences of only a few degrees would take up most of the graph and appear much larger than they are. 3. Therefore, the vertical axis should begin at \(0\) to avoid exaggerating the differences.

The vertical axis should begin at \(0\). Starting at \(18\) would make small temperature differences appear much larger than they are.

- Ask whether moving from one category label to the next represents a measurable interval. - Consider what a connecting segment normally communicates in a graph. - Choose a display that compares categories without implying continuity.

1. The field-trip choices are categories with no meaningful numerical order or continuous interval between adjacent labels. 2. Connecting the category counts creates slopes that suggest continuous change and a trend that the data do not contain. 3. A bar graph is more appropriate because separate bars compare category frequencies without implying continuity. 4. The category labels, frequency scale, and readable bar heights should remain visible.

The connecting line falsely suggests continuous change across unordered categories. Use a bar graph with separate labeled bars and a clearly labeled frequency axis.

- Separate what the key explicitly defines from what a reader might assume. - Check whether the half symbol has a stated numerical meaning. - Decide what extra legend statement would make the proposed count unambiguous.

1. The key defines the value of one full ticket symbol as \(8\) visitors. 2. It does not define the value of a half-ticket symbol or say that partial symbols are proportional. 3. Therefore Priya is justified: the exact count cannot be determined from the key as written. 4. Devin's count of \(12\) would be valid if the key explicitly stated that a half-ticket represents \(4\) visitors, or more generally that partial symbols represent the corresponding fraction of \(8\).

Priya is justified. The key does not define the half-ticket symbol. Devin's count of \(12\) is valid only if the key states that a half-ticket represents \(4\) visitors (or that partial symbols are proportional).

- Decide whether the horizontal labels are numerical intervals or distinct names. - Recall what touching bars usually communicate about neighboring values. - Match the spacing to the type of variable displayed.

1. The mural themes are separate categories with no continuous numerical intervals between them. 2. Touching bars are normally used in a histogram to show adjacent intervals of a numerical variable. 3. The touching display falsely suggests continuity and an ordered progression among the themes. 4. A bar graph with equal-width bars separated by gaps would represent the categorical counts more appropriately.

The categories are not continuous intervals, so the bars should not touch. Use separated bars with clear category labels.

- Read the category name beneath each bar rather than relying on position. - Check whether left, middle, and right represent the same things in both displays. - Align comparable categories consistently.

1. A left-position bar represents Art in September but Drama in October, and the other positions also change categories. 2. Comparing positions therefore compares different clubs. 3. The graphs should use the same category order in both panels. 4. Distinct, consistently applied labels or patterns can further help readers match each club.

The category order changes, so matching positions do not represent matching clubs. Use the same club order and clear labels in both graphs.

- Check whether each group remains identifiable without color. - Consider how a reader matches a line to the legend. - Add a second visual feature that encodes group identity.

1. The printed lines cannot be matched reliably to the legend because their color difference disappears. 2. Readers may assign values to the wrong group. 3. Use different line patterns, such as solid and dashed, or different point markers, and name those patterns in the legend. 4. Direct labels placed beside the lines can also identify the groups.

The display relies on color that is unavailable in the printed version. Use distinct line patterns or markers with matching labels.

- Identify what feature of each symbol represents one unit of data. - Count symbols independently of the empty space between them. - Standardize layout features that should not encode values.

1. In a pictograph, frequency is encoded by the number of symbols according to the key, not by the row’s total physical length. 2. The first row represents more checkouts because it contains \(5\) symbols rather than \(4\). 3. Unequal spacing makes the row with fewer symbols look longer. 4. Use equal spacing and alignment for all symbols, and keep a clear common key.

The \(5\)-symbol row represents more checkouts. Use equal symbol spacing so row length does not contradict symbol count.

- Identify what adjacency on a time graph is supposed to mean. - Compare alphabetical neighbors with actual neighboring months. - Use an order that preserves elapsed time.

1. A time trend must follow chronological order. 2. Alphabetical order makes adjacent points represent months that may be far apart in time and reverses other time relationships. 3. The connected slopes therefore do not show month-to-month temperature change. 4. Place January through December in chronological order with equal spacing for equal one-month intervals.

Alphabetical month order destroys the time sequence. Reorder the x-axis chronologically from January through December.

- Compare where the bars begin with the numerical value \(0\). - Bar length, not just the top endpoint, is supposed to encode magnitude. - Ask how much of each bar is hidden below the displayed y-axis minimum.

1. The y-axis begins at \(92\) tons instead of \(0\), so the visible bar heights are measured only above \(92\). 2. The visible height for Year 1 is \(100-92=8\) units, while the visible height for Year 3 is \(95-92=3\) units. 3. Because \(3\) is less than half of \(8\), the truncated axis makes the change look much larger than the actual decrease from \(100\) to \(95\) tons. 4. A neutral bar graph should begin the y-axis at \(0\) so bar lengths represent the values from a common zero baseline.

The y-axis is truncated at \(92\) tons, which exaggerates the visual difference. Start the y-axis at \(0\) tons so the bar lengths use a common zero baseline.

- Add the five measurements and divide by \(5\). - Identify the smallest and largest measurements. - A useful axis includes all values while keeping differences easy to see.

1. The total rainfall is \(0.4+0+1.0+0.6+0.5=2.5\) inches. 2. The mean is \(2.5\div5=0.5\) inch per day. 3. Range A is much larger than the data and would make the bars difficult to compare. 4. Range C cannot show the values \(0\) and \(0.4\). 5. Range B includes every value from \(0\) through \(1.0\) and uses the graph space effectively.

a) \(0.5\) inch per day b) Range B, from \(0\) to \(1.2\) inches, because it includes all data values and gives a readable scale.

- Add all percentages in the graph. - A complete circle normally represents \(100\%\). - For part b), compare the Sports value with the total of all displayed values. - Simplify the resulting fraction.

1. The percentages total \(50\%+35\%+45\%+20\%=150\%\). 2. A circle graph represents one whole, or \(100\%\). Because students could choose multiple activities, the percentages exceed \(100\%\), so the sectors cannot represent the stated percentages as parts of one whole. 3. In the drawn graph, Sports has \(50\) parts out of a total of \(150\), so it occupies \(\frac{50}{150}=\frac{1}{3}\) of the circle.

a) \(150\%\). A circle graph is inappropriate because multiple responses make the total exceed \(100\%\), while a circle represents exactly one whole. b) \(\frac{1}{3}\)

- Read the soccer bar from the y-axis. - Add all five percentages and consider the effect of allowing multiple selections. - Find \(35\%\) of \(40\) for the theater club. - Distinguish an expression of interest from a final commitment.

1. The soccer bar shows \(55\%\). 2. The percentages total \(40\%+35\%+55\%+20\%+10\%=160\%\). The total exceeds \(100\%\) because each student could select more than one club. 3. Theater was selected by \(35\%\) of \(40\), or \(0.35\times40=14\) students. 4. The survey measured nonbinding interest, while registration records final choices. Students may change their minds, and schedules, capacity limits, or other conditions may affect enrollment.

a) \(55\%\) b) \(160\%\). The sum is greater than \(100\%\) because multiple responses were allowed. c) \(14\) students d) Interest survey responses and final registrations measure different decisions, so the totals may differ.

- Compare the y-axis maximum in each graph. - The data values are identical in both graphs. - Add all seven daily values before dividing by \(7\).

1. The values are Monday, \(12\,\text{MWh}\); Tuesday, \(16\,\text{MWh}\); Wednesday, \(13\,\text{MWh}\); Thursday, \(15\,\text{MWh}\); Friday, \(11\,\text{MWh}\); Saturday, \(17\,\text{MWh}\); Sunday, \(14\,\text{MWh}\). 2. Production was highest on Saturday and lowest on Friday. 3. The differences appear larger in Graph 1 because its y-axis ends at \(20\,\text{MWh}\). The same differences occupy a much smaller part of Graph 2, whose y-axis ends at \(80\,\text{MWh}\). 4. The weekly total is \(12+16+13+15+11+17+14=98\,\text{MWh}\). The mean is \(98\div7=14\,\text{MWh}\) per day.

a) Monday: \(12\); Tuesday: \(16\); Wednesday: \(13\); Thursday: \(15\); Friday: \(11\); Saturday: \(17\); Sunday: \(14\), all in \(\text{MWh}\). Highest: Saturday; lowest: Friday. b) Graph 1, because its y-axis has a much smaller range. c) \(14\,\text{MWh}\) per day

- Check the unit attached to each set of bars before comparing heights. - Decide whether equal vertical distances represent equal quantities throughout the graph. - Think about how one shared scale could make the comparison meaningful.

1. The two bars use different quantities and different scales: one shows a number of students, while the other shows a percentage of members. 2. A taller bar in one panel does not necessarily represent a larger or more popular club than a shorter bar in the other panel. 3. A fair redesign would show both clubs using the same quantity and one common scale, such as percentages for both clubs.

The caption is misleading because a bar measured in students cannot be compared directly with a bar measured in percent. The display should use one common unit and one common scale for both clubs.

- Distinguish the amount added on one day from the total accumulated by that day. - Use consecutive totals to recover each later day’s contribution. - Compare the recovered daily amounts, not the running totals.

1. Monday’s daily donation is the first running total, \(12\) books. 2. Tuesday’s donation is \(30-12=18\) books. 3. Wednesday’s donation is \(35-30=5\) books. 4. Thursday’s donation is \(55-35=20\) books. 5. Friday’s donation is \(59-55=4\) books. 6. The daily amounts are \(12, 18, 5, 20, 4\), which do not increase throughout the week. The running total rises because donations accumulate, not because each day’s donation rises.

Monday: \(12\) books Tuesday: \(18\) books Wednesday: \(5\) books Thursday: \(20\) books Friday: \(4\) books The caption is misleading. A rising running total does not show that the amount added each day is rising.

- Separate the numerical height of each bar from its apparent size on the page. - Think about which edges are easy or difficult to line up with scale marks. - Remove decorative features that do not represent data.

1. The bar for \(12\) projects is drawn with the widest front face, so it can look largest even though it represents the smallest value. 2. The added depth and slanted edges make the bars' apparent area and volume compete with their heights, so the small numerical differences are hard to compare accurately. 3. A fair redesign is a flat two-dimensional bar graph with equal-width bars, one common baseline, and a clearly labeled scale.

Perspective makes the \(12\)-project bar look widest and adds apparent area and volume that do not represent the data. Use equal-width two-dimensional bars on one common baseline and scale.

- Connect each percentage to its share of a full circle. - Equal percentages should produce equal sector angles. - Check the numerical labels and the drawn areas separately.

1. Category A should have angle \(0.50\times360^\circ=180^\circ\), so A is correct. 2. Category B should have angle \(0.25\times360^\circ=90^\circ\), not \(120^\circ\). 3. Category C should also have angle \(90^\circ\), not \(60^\circ\). 4. The labels total \(100\%\), but the drawn areas do not match those percentages, so the picture distorts the sizes of B and C.

A is correct at \(180^\circ\). B and C should each be \(90^\circ\). The graph is misleading because the sector areas do not match the labeled percentages.

- Ask what quantity each visual category is supposed to represent. - Check whether a reader can determine the size of a difference from the display. - Consider accessibility for readers who cannot distinguish the colors.

1. Without a legend, readers do not know what numerical tree-cover values the shades represent. 2. The phrase “much more” cannot be judged because the size of the differences between shades is unknown. 3. A useful revision should add labeled numerical intervals or percentages for every shade. 4. It should also add patterns or printed values so the categories can be distinguished without relying on color alone.

The map is ambiguous because the shades have no defined numerical meaning, so “much more” cannot be evaluated. Add a legend with labeled tree-cover ranges and use printed values or patterns in addition to color.

- Identify the unit represented by each bar. - Express both changes as counts, then express both as percentages. - Equal visual heights should correspond to equal quantities measured in the same unit.

1. The first increase is \(10\) items, which is \(\frac{10}{40}=25\%\). 2. The second increase is \(20\) items, which is \(\frac{20}{200}=10\%\). 3. The equal-height bars compare different units: one bar represents a count and the other a percent. 4. A fair count comparison would show \(10\) items and \(20\) items. A fair percent comparison would show \(25\%\) and \(10\%\).

The bars compare unlike units. In counts, the changes are \(10\) and \(20\) items. In percents, the changes are \(25\%\) and \(10\%\). Use one of those common units for both bars.

- Compare the two largest percentages before judging the caption. - Ask which visual features change when the numerical values change. - Separate information encoded by sector size from words added for emphasis.

1. Sponsor A has \(31\%\) and Sponsor B has \(29\%\), a difference of only \(2\) percentage points. 2. The sector angles and printed percentages encode the data because their sizes correspond to the category shares. 3. The word “WINNER” is decorative emphasis; it does not represent an additional amount of data. 4. The graph supports only a small lead for Sponsor A, not a claim of clear domination. 5. A fair caption is, “Sponsor A had the largest share at \(31\%\), narrowly ahead of Sponsor B at \(29\%\).”

No. Sponsor A leads Sponsor B by only \(2\) percentage points. The sector angles and percentages encode the data; “WINNER” is decoration. A fair caption would say that Sponsor A had a narrow lead, \(31\%\) to \(29\%\).

- Calculate the two common measures of center separately. - Compare the displayed value with each measure. - Replace a vague statistical term with the exact name of the measure used.

1. The median is the middle ordered value, \(4\) minutes. 2. The mean is \((2+3+4+5+31)\div5=45\div5=9\) minutes. 3. The word “average” is ambiguous because it can be read as the mean, yet the plotted value is the median and the two measures differ greatly. 4. A precise label is “Median wait: \(4\) minutes.”

Mean \(9\) minutes; median \(4\) minutes. Replace the label with “Median wait: \(4\) minutes.”

- Calculate the change from the numerical values rather than judging the line angle. - Compare the physical dimensions of the plotting areas. - Separate data scaling from page layout.

1. The numerical increase is \(25-20=5\) participants. 2. The percent increase is \(\frac{5}{20}\times100\%=25\%\). 3. Changing the chart’s height-to-width ratio changes the visual angle of the same line. 4. A narrow, tall graph can make the change appear more dramatic even though the data and scales are identical. 5. A fair comparison should use consistent chart dimensions or state the numerical change prominently.

The actual increase is \(25\%\). Graph B looks steeper because its aspect ratio stretches vertical change relative to horizontal change, not because the data differ.

- Check whether the groups contain the same number of people. - Convert each total to a per-person quantity. - Match the graphed quantity to the wording “students ... read more.”

1. Group A’s average is \(600\div20=30\) minutes per student. 2. Group B’s average is \(720\div30=24\) minutes per student. 3. Group B has the larger total because it has more students, but a typical Group A student read longer. 4. A fair graph for the caption’s intended comparison should display minutes per student, with bars at \(30\) and \(24\).

The caption is misleading for a per-student comparison. Group A averaged \(30\) minutes per student, and Group B averaged \(24\). Graph the per-student averages instead of the group totals.

- First find the direct difference between the two percentage values. - Then compare that difference with the original rate. - Use precise wording to distinguish the two kinds of change.

1. The difference between the rates is \(10\%-5\%=5\) percentage points. 2. Relative to the original \(10\%\), the decrease is \(\frac{10\%-5\%}{10\%}=0.50=50\%\). 3. Saying only “decreased by \(5\%\)” can be mistaken for a \(5\%\) relative decrease rather than a drop of \(5\) percentage points.

The rate decreased by \(5\) percentage points, which is a \(50\%\) decrease from the original rate.

- Use the labels attached to the first and last plotted points to find the actual change. - Compare that numerical change with how much vertical space the graph uses. - Ask what scale information a reader would need in order to interpret the vertical distance correctly.

1. The labeled values are \(995\) feet and \(1005\) feet at the first and last points. 2. The actual change is \(1005-995=10\) feet, not hundreds of feet. 3. The graph uses a magnified vertical scale around the data values, but the y-axis numerical labels are hidden, so the amount of magnification is not clear to the reader. 4. Show the y-axis numerical scale clearly and make any shortened nonzero scale explicit, for example with a clearly labeled starting value and an axis-break mark or note.

The actual increase is \(10\,\text{ft}\). The graph magnifies a small change while hiding the numerical y-axis scale, which can exaggerate the visual impression. Show the y-axis numbers clearly and explicitly mark any shortened nonzero scale.

- Identify what a sector’s fraction of a circle represents. - Use each graph’s own survey total. - Compare the resulting counts, not just the sector shapes.

1. Equal sectors in separate circle graphs represent equal relative frequencies, not necessarily equal counts. 2. At School A, \(50\%\) of \(60\) is \(0.5\times60=30\) students. 3. At School B, \(50\%\) of \(100\) is \(0.5\times100=50\) students. 4. The identical-looking sectors hide the different group totals.

School A had \(30\) students and School B had \(50\) students choose the musical. Equal sectors show equal percentages, not equal counts.

- Read the units and tick values on both vertical axes. - Ask what it means for two plotted heights to match when each uses a different scale. - Distinguish a similar pattern from equal numerical values.

1. Rainfall and umbrella sales use different units and different numerical scales. 2. The lines overlap because the two axes were scaled separately, not because the quantities are equal. 3. The display may support a statement that the two quantities rose and fell in a similar pattern over time. 4. It cannot support numerical equality between inches of rain and numbers of umbrellas.

Separate y-axes can make different quantities overlap by design. The graph may show similar month-to-month patterns, but it does not show that rainfall amounts and umbrella counts were equal.

- Identify which dimension is supposed to encode the data. - Compare the actual values with the visual area of the bars. - Remove any size difference that does not represent a numerical variable.

1. In a bar graph, the numerical value is encoded by bar height, not by bar area. 2. Making one bar wider gives it much more visual area even though its count is only slightly greater. 3. The actual ratio of the largest count to the smallest is \(\frac{24}{20}=1.2\), not about \(3\). 4. All bars should have equal width and equal spacing so only height represents the category count.

The extra width exaggerates the \(24\) category. Use equal-width, equally spaced bars; the largest count is only \(1.2\) times the smallest.

- Determine what information is lost when values are rounded. - Find two nearby values on opposite sides of the displayed rounded value. - Match the precision of a claim to the precision of the data shown.

1. Rounded labels hide differences smaller than \(0.1\) million. 2. For example, \(1.16\) million and \(1.24\) million both round to \(1.2\) million to the nearest tenth of a million. 3. Equal rounded labels show only that both values fall in the same rounding interval, not that the original values are equal. 4. The display should provide more precise labels before making an exact-equality claim.

The caption is unsupported because rounding can hide a difference. For example, \(1.16\) million and \(1.24\) million both display as \(1.2\) million.

- Distinguish “not observed” from “not offered.” - Consider what information a labeled position with no bar would communicate. - Make the graph’s title agree with all categories in the survey.

1. Design C was an offered category even though no one selected it. 2. Omitting it and changing the title makes readers think only three designs were available. 3. A zero-height bar communicates both the existence of the category and its frequency of \(0\). 4. The graph should include Design C with a labeled zero-height position and state that four designs were offered.

The omission hides an available choice and changes the meaning of the survey. Include Design C as a labeled category with frequency \(0\).

- Find the elapsed time between each pair of dates. - Compare change per day rather than change per plotted segment. - A time axis should represent equal time lengths with equal distances.

1. From January \(1\) to January \(2\), the increase is \(10\) visitors over \(1\) day, or \(10\) visitors per day. 2. From January \(2\) to January \(10\), the increase is \(10\) visitors over \(8\) days, or \(10\div8=1.25\) visitors per day. 3. Equal horizontal spacing falsely represents the unequal time intervals as equal. 4. The dates should be positioned according to the actual elapsed days.

The rates are \(10\) visitors per day and \(1.25\) visitors per day, so they are not constant. The x-axis must space dates according to elapsed time.

- Read the numerical meaning of each shade on both displays. - Decide whether matching visual marks encode matching quantities. - Comparable displays need a common scale.

1. The same shade represents different numerical ranges on the two maps. 2. A darkest area on Map A could have far fewer vehicles than a darkest area on Map B. 3. Matching colors therefore do not represent matching values. 4. Both maps should use one shared numerical legend, or each area should display its count so comparisons use a common scale.

The maps use inconsistent color scales. Use the same numerical intervals and one shared legend for both maps.

- Determine what one unit on the vertical scale represents. - Check whether the printed data label uses that same unit. - Make the tick labels, axis title, and data labels follow one convention.

1. On an axis measured in thousands, a height of \(2.5\) represents \(2.5\) thousand, or \(2500\), visitors. 2. Printing \(2500\) on the bar while retaining “in thousands” can make readers multiply by \(1000\) twice. 3. One correction is to label the bar \(2.5\) and keep the axis unit “in thousands.” 4. Another correction is to label the bar \(2500\) and change the axis to ordinary visitor counts with ticks such as \(0,1000,2000,3000\).

Use either \(2.5\) with an “in thousands” axis or \(2500\) with an axis in individual visitors. Do not mix the two formats.

- Check whether one person can contribute to multiple categories. - Add the displayed percentages. - Use a display that does not require categories to form one whole.

1. Students could appear in more than one category, so the categories are not exclusive parts of one whole. 2. The percentages total \(60\%+50\%+40\%=150\%\), which cannot be represented as sectors totaling one circle. 3. A separate bar for each activity can show each selection rate without implying that the categories partition the students.

The overlapping categories total \(150\%\), so they cannot form one circle. Use a bar graph of the three activity percentages.

- Compare each visitor count with its county population. - Separate “more people” from “greater proportion of people.” - Match the mapped quantity to the wording of the conclusion.

1. County A’s visitor rate is \(\frac{800}{4000}=0.20=20\%\). 2. County B’s visitor rate is \(\frac{1200}{12000}=0.10=10\%\). 3. County B has more visitors because it has a larger population, but its proportion of residents visiting is lower. 4. A map supporting a likelihood claim should shade by visitor rate, not raw visitor count.

County A’s rate is \(20\%\), and County B’s rate is \(10\%\). County A residents were more likely to visit, so the raw-count shading does not support the caption.

- Calculate the numerical difference before judging the colors. - Check where the category boundary falls relative to the two values. - Use a display that preserves small numerical differences as small visual differences.

1. The actual difference is \(50-49=1\) report. 2. The category boundary places the two nearly equal values in visually opposite groups. 3. The strong shade contrast makes a one-report difference appear large. 4. Show the exact values, use more and narrower numerical intervals, or use a bar graph with a common scale.

The towns differ by only \(1\) report. The two-bin color scheme exaggerates a boundary crossing; show exact values or use a finer common scale.

- Identify what the shaded portion represents before using the normalized bars. - Apply the common percentage to each batch's own total. - Distinguish equal composition from equal frequency.

1. A \(100\%\) stacked bar shows composition, so every bar has the same total length regardless of group size. 2. Batch A has \(0.60\times40=24\) light items. 3. Batch B has \(0.60\times100=60\) light items. 4. Equal segment proportions do not imply equal counts when the batch totals differ.

Batch A has \(24\) light items, and Batch B has \(60\). The equal shaded segments show equal percentages, not equal counts.

- Separate plotted observations from values implied between points. - Check whether the caption claims measurement or estimation. - Use a visual convention that makes missing data visible.

1. The line segment connects two observed values but does not create an observed February value. 2. The crossing at \(20\) is an interpolation based on a straight-line assumption. 3. Actual February repairs could have been different, and no data were collected to verify the value. 4. The graph should mark February as missing, leave a gap, or use a dashed segment clearly labeled as an estimate.

The value \(20\) is only an implied estimate, not observed data. Show February as missing or label any dashed interpolation as an estimate.

- Compare the time period named in the title with the period plotted. - Use the omitted beginning and displayed ending values. - Make the graph window and conclusion refer to the same interval.

1. The displayed months show an increase, but the omitted January-through-August data show a larger earlier decrease. 2. The title applies the short displayed trend to the entire year. 3. Attendance ended at \(90\), below the January value of \(100\), so it did not increase throughout the year. 4. Show all twelve months on one consistent time axis or title the graph specifically as the September-to-December increase.

The graph cherry-picks the rising part of the year and hides the earlier decline. Display all months or limit the title to the period actually shown.

- Compare where each bar begins as well as where it ends. - A bar's value should be represented by its length from a common baseline. - Ask what would happen if all three bars started at \(0\).

1. The three bars all end at the same horizontal position, but they do not begin at the same position. 2. Their lengths represent \(30\), \(40\), and \(50\), even though their right endpoints all line up at \(60\). 3. Comparing only the aligned endpoints therefore makes unequal counts look nearly equal. 4. Redraw all three bars from a common zero baseline on one consistent horizontal scale.

The bars use different starting points, so their aligned right endpoints hide different lengths of \(30\), \(40\), and \(50\). Start every bar at \(0\) on the same scale.

- Identify which responses remain in the graph’s total. - Match the denominator to the population named in the caption. - Compare the reduced-group rate with the whole-survey rate.

1. After removing “no opinion,” the graph uses only \(40+30=70\) opinion responses. 2. The displayed \(57\%\) comes from \(\frac{40}{70}\approx57.1\%\). 3. The caption claims a percentage of all \(100\) respondents, which requires denominator \(100\). 4. The correct overall percentage is \(\frac{40}{100}=40\%\). 5. The graph must label its reduced denominator clearly or include all three categories.

The \(57\%\) uses only respondents who expressed an opinion. Of all respondents, \(40\%\) said yes.

- Because each person chose exactly one rating, separate category percentages must add to \(100\%\). - Look at whether each successive bar can be interpreted as including earlier ratings. - Recover separate categories by finding the increase from one running total to the next.

1. If the bars represented four separate, mutually exclusive rating percentages, their percentages would total \(100\%\). 2. The displayed heights are \(20\%,55\%,85\%,100\%\), which total more than \(100\%\), so they cannot be separate-category percentages. 3. The heights form running totals: \(20\%\) gave rating \(1\) or below, \(55\%\) gave rating \(2\) or below, \(85\%\) gave rating \(3\) or below, and \(100\%\) gave rating \(4\) or below. 4. The separate percentages are successive differences: rating \(1\): \(20\%\); rating \(2\): \(55\%-20\%=35\%\); rating \(3\): \(85\%-55\%=30\%\); rating \(4\): \(100\%-85\%=15\%\). 5. A correct title is “Cumulative Percentage at or Below Each Rating.”

The title is misleading. The bars are cumulative percentages, not separate-category percentages. Separate ratings: rating \(1\): \(20\%\); rating \(2\): \(35\%\); rating \(3\): \(30\%\); rating \(4\): \(15\%\). A suitable title is “Cumulative Percentage at or Below Each Rating.”

- Calculate the numerical width of each labeled interval. - Compare those numerical widths with the physical widths of the bars. - Keep equal horizontal distances tied to equal changes in the measured variable.

1. The first two numerical intervals each have width \(5\), while the third has width \(20\). 2. Drawing all three bars with equal physical widths makes equal horizontal distances represent unequal numerical changes. 3. For a Grade 6 frequency histogram, use equal-width numerical intervals, such as consecutive intervals of width \(5\), and recompute the frequencies for those bins.

The third interval covers \(20\) units while each other interval covers only \(5\), yet all bars are drawn equally wide. Use equal-width bins, such as intervals of width \(5\), and recompute the frequencies.

- Identify which numerical values are explicitly readable from the axis. - Compare the caption’s precision with the display’s resolution. - Add labels that make exact values verifiable rather than guessed.

1. The unlabeled gridlines make the intended tick values unclear to a reader. 2. Even if equal spacing suggests \(5\) units per interval, a bar not ending exactly on a gridline cannot be read as exactly \(37\) without a data label or finer scale. 3. The caption states more precision than the display provides. 4. Label the tick values and place an exact value label on the bar, or state only an appropriately rounded estimate.

The graph does not provide enough labeled precision to justify exactly \(37\). Add labeled ticks and an exact data label, or use an estimated value.

- Identify the whole used by each percentage. - Decide whether the two statements divide one common population into exclusive parts. - Choose a display that keeps the two group denominators visible.

1. The \(60\%\) uses the surveyed boys as its whole, while the \(40\%\) uses the surveyed girls as its whole. 2. The percentages therefore have different denominators and are not complementary parts of one group. 3. A circle graph requires categories that divide one common whole. 4. Separate or grouped bars labeled with each percentage and each group size would make the denominators clear.

The graph combines percentages based on different groups, so its sectors are not parts of one whole. Use separate or grouped bars with the percentages and sample sizes labeled.

- Test values that lie exactly on the displayed boundaries. - A complete legend should neither overlap nor leave gaps. - Decide which neighboring interval owns each shared endpoint.

1. The displayed categories overlap at \(10\) and \(20\), so those readings do not have unique classifications. 2. A valid set of nonoverlapping intervals is \(0\leq m<10\), \(10\leq m<20\), and \(20\leq m\leq30\). 3. The revised intervals cover the full stated range and assign every possible reading to exactly one category.

The original legend is ambiguous at \(10\) and \(20\). One valid correction is \(0\leq m<10\), \(10\leq m<20\), and \(20\leq m\leq30\).

- Check whether the two numbers use the same measurement unit. - Use the stated equivalence to decide whether the physical quantities differ. - A valid common scale must represent equal quantities with equal heights.

1. The two numerical values use different temperature scales, so their numbers cannot be compared directly as bar heights. 2. The report states that \(68\,^{\circ}\text{F}\) and \(20\,^{\circ}\text{C}\) represent the same temperature. 3. A fair graph must convert both measurements to the same unit and label the y-axis with that unit. The two bars would then have equal height.

The graph is misleading because it treats Fahrenheit and Celsius numbers as though they were measured on one common scale. Convert both temperatures to the same unit and use a labeled y-axis; the bars should be equal in height.

- Compare the numerical change between every pair of neighboring y-axis labels. - Equal physical spacing on a linear axis should represent equal numerical changes. - Choose one interval size that reaches \(60\) without changing.

1. Equal visual spaces should represent equal numerical changes on a linear axis. 2. The displayed labels increase by \(10,10,30,10\), so one grid space sometimes represents \(10\) units and sometimes \(30\) units. 3. This compresses the change from \(20\) to \(50\) and makes visual comparisons unreliable. 4. A fair scale could label equally spaced grid lines \(0,10,20,30,40,50,60\).

The y-axis uses unequal numerical intervals at equal visual spacing. Use one constant interval, such as \(10\) units, with labels \(0,10,20,30,40,50,60\).

- Compare each data value with the largest value shown on the axis. - Determine whether the visible bar tops are actual endpoints. - Make the display range include every plotted value.

1. The axis maximum is below both data values, so the tops of both bars are clipped. 2. Clipping hides the difference of \(92\%-85\%=7\) percentage points. 3. Equal visible heights are caused by the graph boundary, not equal data. 4. Extend the axis to at least \(100\%\) and draw both complete bars with labeled values.

The graph clips both bars and hides a \(7\)-percentage-point difference. Extend the scale so both \(85\%\) and \(92\%\) are fully visible.

- There are three possible adjacent pairs; evaluate each one rather than assuming which pair should be merged. - Compute the new group counts after each possible merge. - A label must still accurately describe every original rating placed inside that group.

1. Combining Excellent and Good gives \(8+15=23\), with the other group counts \(17\) and \(10\). The positive group is the largest. 2. Combining Good and Fair gives \(15+17=32\), but that group includes Fair responses, so it cannot accurately be described as a positive-rating group without changing category meaning. 3. Combining Fair and Poor gives \(17+10=27\), which creates a large lower-rating group rather than a positive group. 4. Therefore the defensible merge is Excellent with Good, producing counts \(23,17,10\).

Combine Excellent and Good. Positive: \(23\); Fair: \(17\); Poor: \(10\). The other merges either combine Fair with Good or combine Fair with Poor, so their largest merged group cannot accurately be described as students who gave a positive rating.

- Find the difference in votes, then use the graph scale. - Add all votes before finding the Cat percentage. - Think about what happens to a bar with value \(2\) if the axis begins at \(2\).

1. Dog and Hamster differ by \(9-3=6\) votes. At \(2\) votes per centimeter, the difference is \(6\div2=3\,\text{cm}\). 2. The total number of votes is \(9+6+3+2=20\). 3. Cat received \(\frac{6}{20}=30\%\) of the votes. 4. If the y-axis starts at \(2\), the Bird bar would have no visible height even though Bird received votes. The other differences would also appear larger than their actual proportions.

a) \(3\,\text{cm}\) b) \(30\%\) c) The truncated axis would exaggerate differences and make the Bird category appear to have no votes.

- Identify which visual feature people are most likely to compare. - Compare the multiplicative change in the data with the multiplicative change in that visual feature. - Consider a display in which only one dimension represents the values.

1. The student count increases from \(40\) to \(60\), a factor of \(\frac{60}{40}=1.5\), or a \(50\%\) increase. 2. The square areas are \(4^2=16\,\text{cm}^2\) and \(6^2=36\,\text{cm}^2\). 3. The displayed area increases by a factor of \(\frac{36}{16}=2.25\), or \(125\%\), so the visual difference is much larger than the data difference. 4. A fair redesign could use equal-width bars with heights proportional to \(40\) and \(60\), or use equal-size symbols with a stated key.

The display is misleading because the student count is multiplied by \(1.5\), but the square area is multiplied by \(2.25\). A fair redesign is an equal-width bar graph with bar heights proportional to \(40\) and \(60\).

- Use the scale of \(10\) members per part to recover each whole and shaded count. - Compare each shaded amount with its own whole bar. - A fair rate comparison should make the two wholes visually equal in length.

1. Bar a) has \(4\) parts, so it represents \(40\) members; \(2\) shaded parts represent \(20\) instrument players. The relative frequency is \(\frac{20}{40}=50\%\). 2. Bar b) has \(8\) parts, so it represents \(80\) members; \(3\) shaded parts represent \(30\) instrument players. The relative frequency is \(\frac{30}{80}=37.5\%\). 3. Club b) has more instrument players as a count but a smaller share of its membership. 4. A fair display should use equal-length \(100\%\) bars or graph the two percentages directly.

The claim is misleading. The shares are \(50\%\) in club a) and \(37.5\%\) in club b). Use equal-length \(100\%\) bars or graph the percentages directly.

- Look beyond the percentages to the number of observations behind each bar. - Convert each percentage to a count to see the scale of the samples. - Decide whether the caption claims more than the displayed information supports.

1. Event A's \(90\%\) represents \(0.90\times10=9\) satisfied respondents. 2. Event B's \(80\%\) represents \(0.80\times500=400\) satisfied respondents. 3. The percentages describe the observed shares, but the graph hides the very different sample sizes. 4. The bar heights alone do not justify the claim about stronger evidence. 5. The display should state the sample size and satisfied count for each event alongside the percentages.

The caption is not supported by the graph. Event A is based on only \(10\) responses, while Event B is based on \(500\). Add both sample sizes and counts to the display.

- Recover the counts behind each percentage. - Combine the parts and wholes before finding one overall share. - Check whether the two original percentages represent equally sized groups.

1. The two class percentages represent groups of different sizes, so they should not receive equal weight. 2. The combined satisfied count is \(9+18=27\). 3. The combined student count is \(10+30=40\). 4. The correct combined rate is \(\frac{27}{40}=0.675=67.5\%\). 5. Averaging \(90\%\) and \(60\%\) gives too much weight to the smaller class.

The correct combined satisfaction rate is \(67.5\%\), not \(75\%\). The displayed calculation treats unequal class sizes as if they were equal.