Free math worksheets

- Check the category labels and the y-axis increments before reading any value. - Read the two bar heights from the scale, then subtract in the requested order.

1. Fantasy has the tallest bar at \(45\). 2. History has \(20\) selections. 3. The difference is \(45-20=25\).

Fantasy, by \(25\) selections.

- Use the fact that a full pie represents the entire distribution. - Subtract the combined labeled percentage from \(100\%\), not from the number of slices.

1. The labeled slices total \(37\%+29\%+18\%=84\%\). 2. The remaining slice is \(100\%-84\%=16\%\).

\(16\%\).

- Consider how much space each category label needs. - Separate a formatting decision from a data transformation. - Ask which axis carries the category names after rotation.

1. The category labels are long, so placing them beside horizontal bars allows the text to remain readable without rotation or crowding. 2. Orientation changes only the layout of the graph. 3. The category frequencies, relative frequencies, ordering, and comparisons among bar lengths remain unchanged.

A horizontal bar chart gives the long labels more space. Changing orientation does not change any frequency, relative frequency, order, or comparison.

- Translate the bar height into a percentage and compare it with the remainder. - The complement of the recycled share is the combined share of the other categories.

1. The Recycle bar has relative frequency \(0.45=45\%\). 2. The other categories total \(0.30+0.25=0.55=55\%\). 3. Recycling is not more than the other two combined.

\(45\%\) were recycled; no, the other two categories total \(55\%\).

- Use the slice’s share of the full sample. - Convert the Walk slice from a percent to a decimal before applying it to the total.

1. The Walk slice represents \(25\%\). 2. The count is \(0.25\cdot 360=90\).

\(90\) students.

- Combine every slice named in the claim before comparing with the benchmark. - A majority means a combined share strictly greater than \(50\%\).

1. Drama and Comedy account for \(32\%+28\%=60\%\). 2. Since \(60\%>50\%\), the claim is supported.

Yes. Together, Drama and Comedy account for \(60\%\).

- Combine the relevant bar lengths and compare with the total sample size. - Add the South and West frequencies, place that sum over the chart total, and simplify.

1. South and West total \(30+20=50\). 2. The fraction is \(\frac{50}{80}=\frac{5}{8}\).

\(\frac{5}{8}\).

- Find the difference between the two slices before applying it to the total. - A percentage-point gap can be converted to a count gap because both slices refer to the same total.

1. The percentage-point difference is \(45\%-35\%=10\%\). 2. The count difference is \(0.10\cdot 500=50\).

\(50\) more people answered Yes.

- Check each statement against the exact bar heights. - Test each choice separately rather than stopping after the first false statement.

1. Often or Always totals \(28+12=40\), not a majority of \(100\). 2. Sometimes is \(36\), not exactly one-third of \(100\). 3. The Sometimes bar is the tallest at \(36\).

c) More selected Sometimes than any other single category.

- Read the category count and compare it with the stated total. - Use \(\frac{\text{category frequency}}{\text{sample size}}\) for the requested bar.

1. Category \(C\) has frequency \(24\). 2. Its relative frequency is \(\frac{24}{80}=0.30\).

\(0.30\), or \(30\%\).

- Identify every slice included in the definition of a favorable response. - Compare their combined share with the strict \(50\%\) majority benchmark.

1. Favorable responses total \(26\%+31\%=57\%\). 2. Since \(57\%>50\%\), a majority gave a favorable response.

Yes. Favorable responses account for \(57\%\).

- Apply the relative frequency to the stated total. - Use \(\text{count}=\text{percentage as a decimal}\cdot\text{total}\).

1. The Late bar represents \(20\%=0.20\). 2. The frequency is \(0.20\cdot 250=50\).

The Late frequency is \(50\).

- Read both bar heights and simplify their comparison. - Preserve the order \(B:C\) and divide both counts by a common factor.

1. The ratio is \(40:16\). 2. Divide both terms by \(8\) to obtain \(5:2\).

\(5:2\).

- Read each bar height before deciding the order. - A rearrangement does not add, remove, or recode observations. - Separate visual position from the numerical information represented by a bar.

1. The bar frequencies are Transit \(35\), Parks \(30\), Housing \(20\), and Safety \(15\). 2. In descending order, the categories are Transit, Parks, Housing, and Safety. 3. Reordering changes only the horizontal placement of the bars. Each category's frequency, relative frequency, modal category, and the total sample size remain unchanged.

The order is Transit, Parks, Housing, Safety. Only the positions of the bars change; the frequencies, relative frequencies, mode, and total remain the same.

- Check whether one respondent can belong to more than one category. - Consider what the full circle of a pie chart is supposed to represent. - Choose a display that compares category amounts without requiring them to total \(100\%\).

1. The category counts sum to \(68+54+47+39=208\), which is greater than the \(120\) respondents because students could select more than one category. 2. Pie-chart sectors represent mutually exclusive parts of one whole and must total \(100\%\). These overlapping selections do not meet that condition. 3. A bar chart can display each activity's count or selection rate without implying that the categories partition the students.

A pie chart is not appropriate because the categories overlap and do not form parts of one whole. A bar chart is appropriate.

- Decide whether the horizontal axis represents measurements or category labels. - Ask whether gaps between category positions encode any recorded variable. - Separate a layout choice from the bar heights that represent frequencies.

1. The x-axis positions represent category placement, not measured numerical distances. 2. The extra space before C therefore has no statistical meaning and can create a false impression of separation. 3. The bars should be evenly spaced while preserving their heights of \(20\), \(35\), and \(25\).

The statement is incorrect. Nominal categories have no meaningful numerical distance, so the bars should be evenly spaced. Their frequencies remain A: \(20\), B: \(35\), and C: \(25\).

- Add the heights of all bars that are already shown. - Compare that sum with the stated total number of responses. - Convert the missing count to a share of the full sample.

1. The displayed frequencies total \(45+35+40=120\). 2. The missing West frequency is \(150-120=30\). 3. Its relative frequency is \(\frac{30}{150}=0.20\), or \(20\%\).

The West bar should have height \(30\), representing \(20\%\) of the responses.

- Compare the sum of the bar heights with the stated sample size. - Decide whether each height is a count or a share. - Convert every count using the same denominator.

1. The bar heights are counts \(54\), \(36\), and \(30\), not percentages, because they total \(120\). 2. The correct percentages are \(\frac{54}{120}=45\%\), \(\frac{36}{120}=30\%\), and \(\frac{30}{120}=25\%\). 3. The y-axis should either be relabeled “Frequency” or the bar heights should be changed to \(45\), \(30\), and \(25\) on a percent scale.

The y-axis incorrectly labels counts as percentages. The correct percentages are A: \(45\%\), B: \(30\%\), and C: \(25\%\).

- Add all displayed sector angles. - Compare the result with one full rotation. - Hold the stated correct angles fixed and find the remainder.

1. The labeled angles total \(110^\circ+95^\circ+80^\circ+85^\circ=370^\circ\), but a full circle contains only \(360^\circ\). 2. If the first three angles are correct, they total \(285^\circ\). 3. The fourth angle must be \(360^\circ-285^\circ=75^\circ\).

The labels are impossible because they sum to \(370^\circ\). The fourth angle should be \(75^\circ\).

- Distinguish whether a graph is mathematically valid from whether it serves the stated communication goal well. - Consider what a zero frequency looks like in each display. - Preserve the category label even though its represented amount is zero.

1. The “Abstain” category has frequency \(0\), but it is still part of the ballot design. 2. A bar chart can display the “Abstain” label at a position with bar height \(0\), so the option remains immediately visible. 3. A pie chart can list “Abstain” in a complete legend, but a zero-frequency category has no visible sector. Therefore, a bar chart communicates the office’s documentation goal more directly.

A bar chart is preferable because it can show “Abstain” as a labeled category with bar height \(0\). A pie chart is not inherently invalid if its legend includes the category, but the category would have no visible sector.

- Compare the sector angle with a complete \(360^\circ\) circle. - Convert the displayed angle to a fraction of the whole. - Independently convert the stated percentage to an angle.

1. The displayed sector represents \(\frac{72}{360}=0.20=20\%\). 2. A \(25\%\) sector should have angle \(0.25(360^\circ)=90^\circ\). 3. Therefore, either the legend should read \(20\%\) or the sector should be redrawn as \(90^\circ\).

A \(72^\circ\) sector represents \(20\%\), not \(25\%\). A true \(25\%\) sector would have central angle \(90^\circ\).

- Identify the visual feature that is intended to encode category share. - Ask whether viewpoint changes that feature independently of the data. - Prefer a display in which equal data values always have equal visual size.

1. In a pie chart, sector angle and area should represent relative frequency. 2. Tilting the disk introduces perspective, so apparent area depends on position as well as on the data. 3. A flat two-dimensional pie chart avoids the perspective distortion; a bar chart would make category comparisons even easier.

Perspective changes the apparent sector areas, so visual size no longer reflects only relative frequency. Use a flat pie chart or, preferably for comparison, a bar chart.

- Look at what the horizontal axis represents rather than only at the spacing between bars. - Decide whether the possible values are categories or numerical intervals. - Classification depends on the variable, not on a decorative choice.

1. Red, Blue, and Green are categories rather than numerical intervals. 2. A graph of frequencies for categories is a bar chart, regardless of whether the bars happen to touch. 3. Histograms require a quantitative variable divided into ordered numerical intervals.

The student is not correct. The graph is a bar chart because its x-axis represents categories, not quantitative intervals.

- Decide whether the category names have an inherent ranking. - Think about what a reader should be able to see across adjacent bars. - Frequency order and meaning order serve different purposes.

1. The response categories have a natural order from least to most satisfied. 2. A suitable order is Poor, Fair, Good, Excellent. 3. This order preserves the meaning of the ordinal scale and makes shifts toward lower or higher satisfaction visible. Sorting by frequency would hide that progression.

Use Poor, Fair, Good, Excellent. The natural ordinal order communicates the scale's progression better than a frequency ranking.

- Focus on the report's specific comparison goal. - Consider which visual encoding supports precise judgments between close values. - A graph can be valid yet still be less effective for a particular purpose.

1. The two largest categories differ by only \(31-29=2\) observations. 2. A bar chart encodes frequency by position along a common numerical scale, so the two heights can be compared directly. 3. A pie chart encodes frequency by sector angle and area, which makes a small difference harder to judge accurately.

Use a bar chart. Its common baseline and numerical scale make the difference of \(2\) easier to compare than pie-sector angles or areas.

- Separate a relative frequency from an absolute frequency. - Ask what additional quantity is needed to turn a percentage into a count. - Test more than one total that is compatible with the stated percentage.

1. The chart gives only the proportion \(0.40\), not the total number of responses. 2. For a sample of \(5\), the category count could be \(0.40\cdot5=2\). 3. For a sample of \(100\), the category count could be \(0.40\cdot100=40\). 4. Because different totals give different counts with the same proportion, the exact count cannot be determined.

No. For example, the sector could represent \(2\) of \(5\) responses or \(40\) of \(100\) responses.

- Compare the intended category list with the legend entries. - Distinguish information carried by sector size from information carried by labels. - Do not guess which sector has the incorrect name.

1. The four sector sizes still show that the data were divided into four category shares. 2. Because East appears twice and West is absent, at least one sector is mislabeled. 3. Without knowing which repeated East label should be West, the sector shares cannot be assigned reliably to all four named regions.

The chart shows four shares, but the regional assignments are unreliable. One “East” label must be corrected to “West” before all sectors can be interpreted.

- Recall the possible values of a count. - Ask whether the vertical axis might actually represent a different quantity. - Identify the inconsistency before proposing a replacement value.

1. A frequency counts observations, so it cannot be negative. 2. The bar at \(-5\) shows that either the axis label is wrong or the plotted value is not a raw frequency. 3. One plausible issue is that a change from a baseline, such as a net gain or loss, was plotted but labeled as frequency; another is an erroneous subtraction during data processing.

The chart is invalid as a frequency graph because counts cannot be negative. The analyst should check for a mislabeled change score or a data-processing error.

- A mode and a majority describe different features. - Compare the largest single bar with half of the total. - Then combine the two tallest bars and repeat the comparison.

1. The modal category has relative frequency \(\frac{42}{120}=0.35\), so it does not represent more than half of the respondents. 2. The two largest categories have combined frequency \(42+33=75\). 3. Their combined relative frequency is \(\frac{75}{120}=0.625\), or \(62.5\%\), so together they do represent a majority.

The modal category is not a majority because it represents \(35\%\). The two largest categories together are a majority because they represent \(62.5\%\).

- Combining categories combines the portions of the circle assigned to them. - Relate the merged angle to a full circle. - Identify which observations are unaffected by the recoding.

1. The merged angle is \(72^\circ+54^\circ=126^\circ\). 2. Its relative frequency is \(\frac{126}{360}=0.35\), or \(35\%\). 3. The angles and relative frequencies for A and B remain unchanged because only C and D are combined.

The merged sector has angle \(126^\circ\) and relative frequency \(35\%\). Sectors A and B remain unchanged.

- Apply the legend consistently to each complete symbol. - Decide what proportional meaning a partial symbol would need to have. - Separate a reasonable interpretation from information actually supplied by the graph.

1. The first category represents \(4\cdot8=32\) responses. 2. The second represents \(3\cdot8+\frac{1}{2}\cdot8=28\) responses. 3. The third represents \(2\cdot8=16\) responses. 4. The graph is inadequately labeled because the meaning of a partial icon is not stated; a reader should not have to assume that its area is interpreted proportionally.

Under the stated assumption, the frequencies are \(32\), \(28\), and \(16\). The legend should explicitly define the half icon.

- Find the total number of observations in each panel. - Convert category \(A\) to a relative frequency within each group. - Compare the two shares rather than the raw counts.

1. In a), the total is \(48+32+20=100\), so the relative frequency is \(0.48\). 2. In b), the total is \(30+24+6=60\), so the relative frequency is \(0.50\). 3. Category \(A\) has a slightly larger relative frequency in group b).

Group a): \(0.48\). Group b): \(0.50\). Category \(A\) is relatively more common in b).

- Compare the two bar heights multiplicatively, not just by subtraction. - Use a ratio of relative frequencies to interpret the word “times.” - Judge “about” by closeness to the claimed multiplier, not by exact equality.

1. The Text relative frequency is \(0.45\), and App is \(0.20\). 2. The ratio is \(\frac{0.45}{0.20}=2.25\). 3. Therefore, “about twice as many” is reasonable.

The statement is reasonable because \(0.45\div 0.20=2.25\), which is about \(2\).

- Read the Silver frequency from the graph and note its stated percentage. - Represent the frequency as the stated share of an unknown total. - Solve for the total and check that the remaining chart frequencies are compatible with it.

1. The Silver frequency is \(36\). 2. If \(36\) is \(20\%\) of the total \(n\), then \(36=0.20n\). 3. Thus, \(n=180\).

There are \(180\) observations.

- A larger percentage does not necessarily mean a larger count when totals differ. - Convert each approval percentage to a count using that chart’s own sample size. - Compare the resulting counts, not the slice angles alone.

1. Chart a) has \(0.60\cdot 80=48\) Approve responses. 2. Chart b) has \(0.52\cdot 150=78\) Approve responses. 3. Group b) has the greater count.

Group b), with \(78\) Approve responses compared with \(48\) in group a).

- Find the original total by combining all chart frequencies. - Increase both the Plastic frequency and the overall total by the added observations. - Form a new relative frequency rather than adjusting the old percentage directly.

1. The original total is \(40+25+15+40=120\). 2. The new Plastic count is \(40+6=46\), and the new total is \(126\). 3. The new relative frequency is \(\frac{46}{126}=\frac{23}{63}\approx 0.365\).

The new relative frequency is \(\frac{23}{63}\approx 0.365\), or about \(36.5\%\).

- Use one corresponding pair of bars to recover the common total. - For any matching category, frequency divided by relative frequency gives the same total. - Verify the recovered total with the remaining bar pairs.

1. Each relative frequency in b) equals the corresponding frequency in a) divided by the same total. 2. For category \(X\), \(50\div 0.50=100\). 3. The sample size is \(100\), and the other pairs also satisfy \(\frac{30}{100}=0.30\) and \(\frac{20}{100}=0.20\).

Each bar in b) is the corresponding bar in a) divided by \(100\): \(50\leftrightarrow0.50\), \(30\leftrightarrow0.30\), and \(20\leftrightarrow0.20\). The sample size is \(100\).

- Compare corresponding category heights on the common scale to find any exact match. - Compute the Yes–No gap separately in each panel. - Compare those two gaps and state which panel has the larger separation.

1. Unsure is \(0.15\) in both charts. 2. In a), the Yes–No gap is \(0.55-0.30=0.25\). 3. In b), the gap is \(0.50-0.35=0.15\). 4. The gap is larger in a) by \(0.10\).

Unsure is the same at \(0.15\). The Yes–No gap is \(0.25\) in a) and \(0.15\) in b), so it is \(0.10\) larger in a).

- Read the numerical values before judging the visual size of their difference. - Compare the value-axis baseline with zero. - Ask whether the bar lengths are proportional to the full magnitudes being compared.

1. The chart shows approval rates of \(48\%\) and \(52\%\), a difference of only \(4\) percentage points. 2. Because the vertical axis shows only the range from \(40\%\) to \(56\%\), the small difference occupies a large fraction of the plotted height and appears exaggerated. 3. A bar chart should begin its value axis at \(0\%\). Alternatively, a point plot could use a clearly labeled nonzero scale without encoding the values as bar lengths.

The truncated value axis exaggerates the \(4\)-percentage-point difference. Begin the bar-chart axis at \(0\%\), or use a non-bar display with a clearly labeled scale.

- Identify the unit used on each vertical axis. - Convert the count panel to a proportion using its sample size. - Compare the two results only after they use the same unit.

1. Panel a)'s support rate is \(\frac{48}{80}=0.60=60\%\). 2. Panel b)'s displayed support rate is \(55\%\). 3. Panel a) therefore has the greater support rate by \(5\) percentage points. 4. The bars use different vertical units, so their numerical heights must be converted before comparison.

Panel a) represents \(60\%\) support, compared with \(55\%\) in panel b). Panel a) is higher by \(5\) percentage points. The original bar heights are not directly comparable because one is a count and the other is a percentage.

- Ask whether the categories have a meaningful least-to-greatest order. - Imagine rearranging the category labels and track what happens to the cumulative heights. - Choose a graph whose values do not depend on an arbitrary ordering.

1. A cumulative value for Cherry includes Apple, Banana, and Cherry only because of the chosen alphabetical order. 2. Nominal categories have no natural order, so rearranging the labels would change every cumulative value without changing the data. 3. An ordinary frequency or relative-frequency bar chart should display each category separately.

Cumulative bars are inappropriate because their values depend on an arbitrary alphabetical order. Use an ordinary bar chart of the individual category frequencies or relative frequencies.

- In a bar chart, frequency is encoded by height rather than area. - Compare the two heights before reacting to the total shaded region. - Equal-width bars prevent width from introducing a second visual signal.

1. Both bar heights are \(30\), so the two categories have equal frequency. 2. The displayed widths are \(0.5\) and \(1.2\). Because the heights are equal, the area ratio is \(\frac{1.2}{0.5}=2.4\). 3. The wider bar has \(2.4\) times the visual area even though it represents the same frequency. Bars in one categorical chart should use a common width.

The categories are equally frequent, but the wider bar has \(2.4\) times the area. Unequal widths falsely make one category look more common.

- Recompute the category shares from the original frequencies. - Distinguish exact proportions from rounded labels. - Base sector angles on the exact fractions rather than on rounded percentages.

1. The exact relative frequencies are \(\frac{10}{60}=\frac{1}{6}\), \(\frac{1}{6}\), and \(\frac{40}{60}=\frac{2}{3}\). 2. The displayed percentages are each rounded to the nearest whole percent, so their sum of \(101\%\) can result from rounding. 3. The exact angles are \(\frac{1}{6}\cdot360^\circ=60^\circ\), \(60^\circ\), and \(\frac{2}{3}\cdot360^\circ=240^\circ\).

The graph is not necessarily wrong; the \(101\%\) total is caused by rounding. The exact sector angles are \(60^\circ\), \(60^\circ\), and \(240^\circ\).

- Identify exactly which people are included in each graph label. - Keep the category count fixed while changing the reference total. - Express the revised ratio as a percentage.

1. The original bar height is \(\frac{72}{180}=0.40\), using only valid responses. 2. Using all invitees gives \(\frac{72}{200}=0.36\). 3. The bar must change because the numerator stays \(72\) while the denominator increases from \(180\) to \(200\).

The revised bar height should be \(36\%\). The original \(40\%\) used valid responses as the denominator, while the revised graph uses all \(200\) invitees.

- Decide what one unit in each bar represents. - Check whether the same product can contribute to several bars. - Distinguish category occurrences from distinct observational units.

1. The bar heights sum to \(30+22+18=70\) defect-type occurrences. 2. Because one product can appear in several categories, the categories are not mutually exclusive. 3. The sum counts recorded defects, not distinct defective products. The number of defective products could be at most \(50\) and cannot be recovered without product-level overlap information.

The sum \(70\) is the number of recorded defect occurrences, not the number of distinct products. Overlap among categories prevents the product count from being determined.

- Treat each labeled endpoint as a cumulative total from the left edge. - Find a segment’s width by subtracting its two boundary positions. - Check that the four segment widths sum to \(1\).

1. The first segment has relative frequency \(0.18\). 2. Subtract consecutive endpoints: \(0.47-0.18=0.29\), \(0.76-0.47=0.29\), and \(1.00-0.76=0.24\). 3. The endpoint of a later segment includes all segments before it, so it is cumulative rather than an individual share.

The category relative frequencies are \(0.18\), \(0.29\), \(0.29\), and \(0.24\).

- Compare the two sample sizes first. - Recall how a circle's area changes when its radius is scaled. - Solve for the radius factor that produces the desired area factor.

1. The actual sample-size ratio is \(\frac{400}{100}=4\). 2. Circle area is proportional to the square of the radius, so a radius ratio of \(4\) creates an area ratio of \(4^2=16\). 3. To obtain an area ratio of \(4\), the radius ratio must be \(\sqrt{4}=2\).

The design shows an area ratio of \(16:1\), although the sample-size ratio is only \(4:1\). The second chart's radius should be \(2\) times the first if area is intended to represent sample size.

- Distinguish a visual position relative to zero from the underlying count. - Use bar magnitudes when recovering frequencies. - Combine the two disagreement categories after interpreting the encoding.

1. The negative signs indicate plotting direction, not negative category counts. 2. The frequencies are the magnitudes \(18\) and \(27\). 3. The total number of disagreement responses is \(18+27=45\).

The graph does not show negative frequencies; the signs encode the left side of the diverging display. There are \(45\) disagreement responses.

- Compare both the differences and the ratios between consecutive counts. - Identify what a logarithmic scale preserves as equal spacing. - Read the axis values rather than treating bar height as a linear measurement.

1. The numerical increases are \(100-10=90\) and \(1000-100=900\), so the counts do not increase by equal amounts. 2. On a logarithmic axis, equal vertical spacing represents equal multiplicative factors. 3. Each count is \(10\) times the preceding count, so the chart shows equal ratios rather than equal differences.

The claim is false. Equal spacing on the logarithmic axis means each count is multiplied by \(10\), not increased by the same amount.