Free math worksheets

- Locate where most observations are concentrated. - Trace the thinner tail away from that concentration; do not use the tallest bar alone.

1. Most observations are in the lower-value bins. 2. Frequencies decline as values increase, creating a long right tail. 3. The distribution is unimodal and skewed right.

The distribution is unimodal and skewed right.

- Compare the heights across all intervals. - Check whether any interval clearly dominates or whether the heights remain roughly level.

1. Every bin has the same frequency. 2. There is no prominent peak. 3. The distribution is approximately uniform.

Approximately uniform, because the bin frequencies are equal.

- Locate every observation on a number line. - Consider what happens to any measure based on distance from the center. - Skewness requires a distribution to extend away from its center.

1. All observations occupy a single value, so the distribution is concentrated entirely at \(7\). 2. The mean and median are both \(7\). 3. The range, interquartile range, and standard deviation are all \(0\). 4. The distribution has no left or right tail, so describing it as skewed is not useful.

The distribution is a single spike at \(7\), with center \(7\) and spread \(0\). It has no meaningful skew or unusual values.

- Organize the description around shape, center, spread, and unusual features. - Use the horizontal scale to estimate the balance point and the occupied range.

1. The frequencies mirror across the middle, so the distribution is approximately symmetric and unimodal. 2. The center is near \(15\). 3. Values span approximately \(0\) to \(30\). 4. There are no clear gaps or isolated outliers.

Approximately symmetric and unimodal, centered near \(15\), with a spread from about \(0\) to \(30\), and no clear gaps or outliers.

- Use the tail direction rather than the peak location. - A peak shows the mode; a tail shows skewness, and they can point in different directions.

1. Skewness is determined by the direction of the longer tail, not the location of the tallest bar. 2. Most values are high, while the frequencies taper toward lower values. 3. The long tail is on the left, so the distribution is skewed left.

The distribution is skewed left because its longer tail extends toward smaller values.

- Locate the stretches of x-values with many dots. - Name each occupied cluster and then state the exact empty range between them.

1. One cluster lies from \(2\) through \(4\). 2. A second cluster lies from \(8\) through \(10\). 3. There are no observations from \(5\) through \(7\), forming a gap between the clusters.

Clusters occur at \(2\)–\(4\) and \(8\)–\(10\), separated by a gap from \(5\) through \(7\).

- Count the prominent peaks rather than individual bars. - Two modes require two separated high regions with lower frequencies between them.

1. There are two prominent high regions separated by lower frequencies. 2. The first peak is in \([10,15)\), and the second is in \([25,30)\). 3. The distribution is bimodal.

Bimodal, with peaks near the intervals \([10,15)\) and \([25,30)\).

- Identify the main concentration of observations. - Compare the gap separating the distant point with the ordinary spacing among the remaining points.

1. Most values lie between \(10\) and \(14\). 2. The value \(25\) is separated far from the rest by a large gap. 3. Thus, \(25\) is a possible high outlier.

The value \(25\) is a possible high outlier because it is isolated well above the main cluster.

- Count prominent high regions, not merely individual tall bars. - Look for a low-frequency valley: adjacent tall bins without a valley can form one broad mode.

1. The heights mirror from left to right, so the distribution is symmetric. 2. The two adjacent central bars form one broad central peak rather than two separated peaks. 3. The distribution is symmetric and unimodal.

Symmetric and unimodal.

- Identify the smallest and largest plotted values for overall spread. - Compare all stack heights to decide whether one value is more frequent than the others.

1. Values extend from \(20\) to \(24\), giving a narrow spread of \(4\). 2. Each value occurs twice, so the frequencies are equal. 3. The shape is approximately uniform, with no clear single mode.

The spread is \(4\), the distribution is approximately uniform, and there is no unique mode.

- Use the shared center as a common reference point. - Compare how far each distribution extends from that center on both sides.

1. In a), nearly all observations lie in bins from about \(3\) to \(7\). 2. In b), observations extend from about \(1\) to \(10\). 3. Therefore, b) has greater variability or spread.

Histogram b) has greater variability because its values are distributed across a wider range.

- A complete description goes beyond center. - Add shape and spread, then check for gaps, clusters, or isolated values.

1. The shape is approximately symmetric and unimodal. 2. The spread runs from \(4\) to \(8\). 3. There are no clear gaps or outliers.

Add that the distribution is approximately symmetric and unimodal, spans \(4\) to \(8\), and has no clear gaps or outliers.

- Use grams when describing the center and overall spread. - Include shape and any gaps or unusual values rather than reporting only a center.

1. The distribution is approximately symmetric and unimodal. 2. It is centered near \(25\,\text{g}\). 3. Package masses range from about \(0\,\text{g}\) to \(50\,\text{g}\), with no clear gaps or outliers.

Package masses are approximately symmetric and unimodal, centered near \(25\,\text{g}\), and spread from about \(0\,\text{g}\) to \(50\,\text{g}\), with no clear unusual features.

- Compare the bar heights across all intervals. - Use the midpoint of the displayed range as a rough center. - Distinguish equal interval counts from equal raw values.

1. Equal bin frequencies give an approximately uniform histogram across the displayed range. 2. The range shown is from \(0\) to \(50\), and the midpoint gives an approximate center of \(25\). 3. There is no prominent peak, gap, or isolated interval. 4. Uniformity refers to similar frequencies across intervals; observations may take many different values within and across those intervals.

The distribution is approximately uniform, centered near \(25\), spread across \(0\) to \(50\), with no prominent unusual feature. Uniform does not mean identical observations.

- Compare the spacing within the main group with the distance to the largest value. - Separate unusualness from incorrectness. - Use the verification information when deciding whether to remove an observation.

1. Six values form a compact cluster from \(65\) to \(70\), while \(84\) is isolated to the right. 2. The distribution is right-skewed and has an unusual high observation at \(84\). 3. Because the value was verified and is plausible, it should be retained and reported rather than deleted merely for being unusual.

The distribution is right-skewed, with a cluster from \(65\) to \(70\) and a verified unusual value at \(84\). Retain and describe the value.

- Check the possible range of the measured variable. - Distinguish an extreme valid value from an impossible value. - Resolve data validity before using a value to characterize shape or spread.

1. Waiting time cannot be negative in this context, so \(-45\) is not merely an unusual plausible observation. 2. The original record and data-entry process should be checked for a sign error, coding value, or unit problem. 3. The distribution should be described only after the value is corrected or formally identified as missing or invalid; it should not be silently changed or included as a genuine wait.

The value \(-45\) is contextually impossible and must be investigated as a data error or code before the distribution is analyzed.

- Locate the proposed center on the horizontal scale. - Pair bins that are equally far to the left and right of that center. - Compare every paired frequency rather than relying on only one matching pair.

1. The paired bins equidistant from \(25\) have matching frequencies: \(1\) and \(1\), then \(4\) and \(4\). 2. The central bin has the highest frequency. 3. The claim is supported.

The claim is supported; the frequencies mirror around the central interval near \(25\).

- Identify the interval or neighboring intervals containing most of the frequency. - Use the balance of frequencies on both sides to estimate a center. - State why grouped bins do not reveal an exact mean or median.

1. The distribution is concentrated in \([20,30)\), so a reasonable center is near \(25\). 2. The histogram does not show exact values within bins, so the exact mean or median cannot be recovered. 3. The range midpoint uses only the outer bin boundaries and ignores frequencies within the distribution.

A reasonable center is near \(25\), but the exact center cannot be determined from grouped data.

- Include the variable’s meaning and units in each part of the description. - Describe where complaint counts concentrate and how far the values extend. - Use the isolated high count to discuss tail direction or an unusual day without overstating certainty.

1. The distribution is unimodal, with a center near \(3\) complaints. 2. Most days have \(1\) to \(5\) complaints. 3. The value \(9\) creates a right tail and may be an unusual high day. 4. The distribution is skewed right.

The daily complaint counts are unimodal and skewed right, centered near \(3\), mostly from \(1\) to \(5\), with a possible high outlier at \(9\).

- Describe the shape formed by the first five bins before considering the final bin. - Compare the left and right sides of that initial pattern. - Then determine how the separated high-value bin changes the overall tail.

1. Without the last bin, the frequencies rise toward \([20,30)\) and then fall symmetrically. 2. The additional high-value observations in \([50,60)\) extend the right side. 3. The full distribution has slight right skew rather than exact symmetry.

The final bin lengthens the right tail, making the distribution slightly right-skewed.

- Describe both the number of peaks and their arrangement. - Identify whether the two high regions are separated by distinctly lower frequencies. - Also compare the left and right halves to describe symmetry or asymmetry.

1. The distribution has two peaks, one at each end, so it is bimodal. 2. Frequencies are low in the middle, creating a central gap-like valley. 3. The shape is also approximately symmetric around \(15\).

It is approximately symmetric and bimodal, with two end clusters separated by a low-frequency middle region.

- Choose a plausible center from the dotplot. - Pair values that are equally far from that center. - Compare the paired frequencies to distinguish approximate symmetry from perfect symmetry.

1. A center near \(3\) would require similar frequencies at equal distances on both sides. 2. Frequencies at \(0\) and \(6\) match, and those at \(1\) and \(5\) match, but \(2\) and \(4\) do not. 3. The distribution is roughly but not perfectly symmetric.

It is roughly symmetric about \(3\), though the stacks at \(2\) and \(4\) are not equal.

- Identify the direction of the longer tail. - Recall that the mean is pulled toward relatively extreme values more than the median is. - Use those two observations to predict the mean–median order.

1. Most observations are at high values, with a long tail toward smaller values. 2. The distribution is skewed left. 3. In a left-skewed distribution, the mean is typically pulled below the median.

The distribution is skewed left, so the mean is likely less than the median.

- Compare bar heights at equal distances from the middle. - Look for where observations concentrate rather than assuming one central peak. - Describe horizontal extent separately from shape.

1. The bar pattern \(8,3,1,3,8\) is symmetric about the middle interval. 2. The distribution is U-shaped or bimodal, with concentrations near both ends and relatively few observations near the center. 3. An approximate center is \(25\), and the displayed spread extends across nearly the full range from \(0\) to \(50\). 4. The low middle frequency is a notable feature; the shape is not bell-shaped.

The distribution is symmetric and U-shaped or bimodal, centered near \(25\), spread across roughly \(0\) to \(50\), with a pronounced dip in the middle.

- Follow the direction in which the bars taper. - Use cumulative counts to locate the middle observation. - Distinguish a gradually thinning tail from a separated unusual value.

1. The frequencies decrease from left to right, producing a long right tail, so the distribution is right-skewed. 2. There are \(12+8+4+2+1=27\) observations, so the median is the fourteenth value. 3. The first interval contains positions \(1\) through \(12\), and the second contains positions \(13\) through \(20\), so the median lies in \([5,10)\). 4. The displayed times span from \(0\) up to less than \(25\) minutes. The sparse high-time intervals form a tail rather than a clearly isolated outlier.

The distribution is right-skewed, with its median in \([5,10)\). It spans roughly \(0\) to \(25\) minutes and has a sparse right tail but no clearly isolated outlier bin.

- Look for a regular numerical pattern in the locations of the tall stacks. - Consider how people commonly report approximate times. - Separate a feature of data collection from a feature of the underlying population.

1. The graph shows heaping at multiples of \(5\). 2. A plausible explanation is that respondents rounded their commute times to convenient five-minute values. 3. Because the pattern aligns with the reporting scale rather than with separated ranges of values, it may be a measurement artifact instead of evidence for five distinct subpopulations.

The distribution is heaped at multiples of \(5\), likely because of rounding. The stacks may reflect reporting behavior rather than five genuine clusters.

- Interpret the plus sign as part of the recorded category. - Ask whether values above the threshold remain distinguishable. - Do not treat a censored group as a single exact measurement.

1. The stack shows how many donations were at least \(\$100\). 2. Because all larger values are combined, the graph does not reveal their exact amounts, their maximum, or how far the upper tail extends. 3. The pile-up is caused by top-coding and should not be described as many donations exactly equal to \(\$100\).

The graph reveals the number of donations at least \(\$100\), but it hides their exact values and the shape and extent of the upper tail.

- Look for separated groups before choosing a single descriptive label. - Pair the smallest and largest values to examine symmetry. - Check whether the numerical center lies in a region containing data.

1. The values form two compact clusters, \(11\) through \(14\) and \(30\) through \(33\), separated by a large gap from \(14\) to \(30\). 2. The distribution is symmetric about \(22\), since paired values have average \(22\). 3. The range is \(33-11=22\). 4. A center near \(22\) lies in the empty gap and does not describe either cluster, so the two-group structure must be reported.

The distribution is symmetric with two distinct clusters at \(11\)–\(14\) and \(30\)–\(33\), center \(22\), range \(22\), and a large central gap. The center alone is misleading because no observations are near it.

- Compare frequencies at equal distances from the center. - Identify every interval that reaches the maximum height. - Describe a broad plateau differently from a single sharp peak.

1. Matching outer frequencies and equal middle frequencies make the distribution approximately symmetric about \(25\). 2. The three central intervals share the maximum frequency, creating a broad flat top rather than one distinct peak. 3. The displayed spread extends from \(0\) to \(50\), with no gap or isolated bar.

The distribution is approximately symmetric and flat-topped, centered near \(25\), and spread across \(0\) to \(50\). It has three tied modal intervals rather than one unique mode.

- Identify where most observations are concentrated and where the longer tail lies. - Use the ordered middle value for center. - Consider how a fixed maximum can affect possible values on one side.

1. Most observations are concentrated near the upper end, with a tail extending toward smaller percentages, so the distribution is left-skewed. 2. The median is the fifth value, \(93\%\). 3. The range is \(99-52=47\) percentage points. 4. The natural upper bound of \(100\%\) limits the right side and helps create the concentration near the top; it does not make the distribution symmetric.

The distribution is left-skewed, with median \(93\%\) and range \(47\) percentage points. The bound at \(100\%\) compresses the upper side.

- Look for a regular repeating pattern in the locations of tall stacks. - Consider how people round or prefer certain digits when estimating. - Distinguish a measurement artifact from naturally separated groups.

1. The graph has an alternating or sawtooth pattern, with heaping on even values. 2. A plausible explanation is that respondents rounded estimates to the nearest even unit or favored even-number responses. 3. The repeated spacing follows the recording preference, so it is evidence of digit preference rather than distinct underlying groups unless other information supports clusters.

The distribution shows even-number heaping. A reporting or rounding preference could create the pattern, so the high even-number stacks should not automatically be interpreted as separate real clusters.

- Count the distinct regions where frequencies peak. - Note the low-frequency region separating them. - Treat explanations for the pattern as hypotheses unless group labels are available.

1. The distribution is bimodal, with two clusters centered roughly near \(20\) and \(50\) minutes. 2. The low frequency between \(30\) and \(40\) minutes forms a dip or gap between the clusters. 3. The pattern may indicate a mixture of groups, such as commuters using different transportation modes, but the histogram alone does not identify the cause.

The distribution is bimodal, with clusters near \(20\) and \(50\) minutes and a sparse middle region. It may combine two groups, but the graph alone cannot establish why.

- Identify which observations were eligible to enter the graph. - Distinguish a data-selection boundary from a natural endpoint. - Avoid describing values that were never recorded in the displayed set.

1. For accepted applicants, the observed distribution has a lower boundary at \(60\) and is concentrated between \(60\) and \(75\). 2. The sharp boundary is created by the eligibility rule rather than necessarily by the natural shape of all scores. 3. No conclusion can be made about the number, shape, or spread of scores below \(60\) among all applicants because those values are excluded.

The accepted scores are truncated at \(60\) and concentrated from \(60\) to \(75\). The graph does not describe the lower part of the distribution for all applicants.

- Ask what locations inside the interval are still unknown. - Imagine different raw arrangements that would give the same bar height. - A histogram's resolution depends on bin width.

1. The histogram records only that \(20\) observations lie somewhere in \([40,60)\). 2. Those observations could be concentrated near one value, spread throughout the interval, or split into groups near \(41\) and \(59\). 3. A wide interval can merge separate raw-data features, so narrower bins or the original values are needed to justify a claim about one cluster.

The wide bar may hide multiple groups within \([40,60)\). Its height alone does not establish one raw-data cluster.

- Separate the frequency of each exact value from the spacing between values. - Identify whether any value repeats. - Use the ordered positions to find center and spread before assigning a shape label.

1. The median is \(9\), and the range is \(20-2=18\). 2. No value repeats, so the data have no mode. 3. Equal frequencies at five isolated values do not show that observations are evenly distributed across the full interval; the gaps \(3,4,5,6\) are not equal.

The median is \(9\), the range is \(18\), and there is no mode. The distribution is not uniform merely because each distinct value appears once; the observed positions are not evenly spaced across the range.

- Equal numbers of observations on each side of the median are expected from its positional definition. - Compare distances from the median, not only counts. - Use the isolated low value to identify the tail direction.

1. The median is \(12\), with four observations on each side, as occurs for any odd-sized ordered data set without median ties. 2. The upper values are close to the median, but \(0\) extends much farther to the left than \(16\) extends to the right. 3. The distribution is left-skewed, and its mean \(\frac{100}{9}\approx11.11\) is below the median.

The claim is false. Equal counts on the two sides of the median do not establish mirrored distances. The value \(0\) creates a long left tail, so the distribution is left-skewed.

- Pair values that are equally far from zero. - Locate the middle observation and compare opposite contributions to the mean. - Look for large gaps between the central cluster and the extremes.

1. Values occur in opposite pairs around \(0\), so the distribution is symmetric. 2. The mean and median are both \(0\). 3. The range is \(20-(-20)=40\). 4. The values \(-20\) and \(20\) are isolated on both sides, creating two unusual extremes while preserving symmetry.

The distribution is symmetric about \(0\), with mean and median \(0\), range \(40\), and isolated unusual values at \(-20\) and \(20\).

- Separate the measurement process from the phenomenon being measured. - Ask whether values in the empty region had a chance to appear in the data. - Describe only what the observed graph can support.

1. The observed graph has no stored values in \([5,10)\). 2. The sensor's recording failure would create that empty region even if underlying values occurred there. 3. Therefore it is a gap in the recorded data, but it cannot be interpreted as a genuine absence of underlying observations without additional information.

No. It is an observed gap caused by missing measurements, not established evidence of a true gap in the underlying distribution.

- Identify where the frequencies peak and how they taper. - Consider whether subtraction between neighboring x-values has meaning. - Bar spacing alone does not determine the variable type.

1. The distribution is unimodal at \(1\) call and tapers toward larger counts, so it is right-skewed. 2. Most days have \(0\), \(1\), or \(2\) calls, with a sparse right tail through \(4\). 3. The variable is quantitative because the values are numerical counts with meaningful differences; the spaces reflect discreteness, not category status.

The distribution is unimodal and right-skewed, concentrated from \(0\) to \(2\) calls with a tail to \(4\). It remains quantitative because the x-values are counts.

- Compare pairs of bars equally far from the center. - Identify the unique highest bar. - Inspect the full path from the center to the extremes rather than stopping after naming symmetry.

1. Matching frequencies at equal distances from the center make the distribution symmetric. 2. The center bin has a single pronounced peak, so the distribution is unimodal. 3. The frequencies rise again at both extreme bins after very low shoulder bins, producing unusually heavy or separated tails that should be reported.

The distribution is symmetric and unimodal with a sharp central peak, but it also has substantial observations in both extreme bins separated by sparse shoulders. Those tail features make the basic labels incomplete.

- Look for a value whose frequency is qualitatively different from nearby values. - Describe the positive observations separately from the zero group. - Include notable structure that a single skew label would hide.

1. The data have a separate concentration at \(0\), representing people with no volunteer hours. 2. The positive values form a right-skewed distribution extending to \(12\). 3. The overall distribution is zero-inflated: it combines a point mass at \(0\) with a positive right-skewed component. 4. Calling the whole graph merely right-skewed would omit the unusually large zero group.

The distribution is zero-inflated, with a large spike at \(0\) and a right-skewed positive part from \(1\) to \(12\).

- Verify the two centers, then inspect the full arrangement of values. - Equality of two summaries does not determine all features of shape. - Compare the distances and frequencies on both sides of the center.

1. The mean is \(\frac{0+0+0+4+5+6+13}{7}=4\), and the median is also \(4\). 2. Three observations are at \(0\), while the largest value \(13\) creates a long right side. 3. The distribution is not symmetric; it is right-skewed despite having equal mean and median.

No. The mean and median are both \(4\), but the distribution is right-skewed because of the high value \(13\) and the concentration at \(0\).

- Describe only features visible in the sample first. - Distinguish “bell-shaped” from a confirmed probability model. - Consider how sample size affects confidence in a shape claim.

1. The observed histogram can be described as approximately symmetric, unimodal, and bell-shaped. 2. With only \(18\) observations, bin choices and sampling variation can strongly affect the apparent shape. 3. A bell-shaped sample is consistent with a normal population but does not establish population normality.

Describe the sample as approximately symmetric, unimodal, and bell-shaped. Do not conclude that the population is normal from this small histogram alone.

- Describe the main group and the separated value before removing anything. - Reassess the full pattern after the sensitivity change. - Compare how much one observation controls the tail in a small data set.

1. With all observations, values \(1\) through \(6\) form a compact, evenly spaced group and \(20\) is isolated to the right, producing strong right skew. 2. Without \(20\), the remaining values \(1\) through \(6\) are evenly spread and symmetric about \(3.5\). 3. The comparison shows that one unusual observation can dominate a small sample's apparent shape.

With \(20\), the distribution is strongly right-skewed with an isolated high value. Without \(20\), it is symmetric and evenly spread from \(1\) to \(6\).

- Compare bars at equal distances from the middle. - Identify any interval containing no observations. - Include both global balance and local clustering in the description.

1. Matching bar heights on opposite sides make the distribution symmetric about roughly \(25\). 2. The empty interval \([20,30)\) creates a central gap that separates the data into a lower cluster and an upper cluster. 3. The distribution is therefore symmetric and bimodal or two-clustered, not a single-peaked symmetric distribution.

The distribution is symmetric with two clusters separated by the central gap \([20,30)\). Reporting only symmetry would hide its most important feature.

- A second mode needs a distinct rise after a noticeable dip. - Follow the bar heights from the highest bar toward the right. - Distinguish a broad shoulder from a separate cluster or peak.

1. The third bar is the unique highest bar, and the next two bars form a gradual shoulder rather than a second separated peak. 2. There is no intervening dip followed by a new rise, so the evidence does not support two modes. 3. The distribution is unimodal with a broad right shoulder and a thinning right tail.

The distribution is not clearly bimodal. It is unimodal with one peak, a broad shoulder to the right, and a short decreasing right tail.

- Identify the main concentration and any separated group of observations. - Reserve the term “outlier” for an isolated point or a value supported by a numerical rule. - If several adjacent high values form their own group, describe that structure before labeling individuals.

1. The main cluster lies from \(5\) to \(7\). 2. A gap separates that cluster from a smaller cluster at \(12\) to \(14\). 3. Without a numerical outlier rule, it is safer to describe the higher values as a second cluster rather than declare each one an outlier.

The graph shows a main cluster at \(5\)–\(7\), a gap, and a smaller cluster at \(12\)–\(14\); the graph alone does not require labeling all three as outliers.

- Translate the numerical endpoints back into clock times. - Ask what happens immediately after the largest time-of-day value. - Consider whether the variable naturally wraps around.

1. Times near \(24\) hours occur just before midnight, while times near \(0\) occur just after midnight. 2. On a clock, these times are close together, but a linear axis places them at opposite ends. 3. The two apparent edge clusters represent one cluster around midnight, so the circular nature of time-of-day must be considered.

The observations form one cluster around midnight. The linear scale splits it between the endpoints near \(0\) and \(24\).

- Identify what numerical quantity is actually spaced evenly on the axis. - A nonlinear transformation can change distribution shape. - Name the scale whenever shape depends on that transformation.

1. Equal distances on a logarithmic axis represent equal ratios, not equal differences, in the original values. 2. The transformation changes horizontal spacing and can turn a right-skewed distribution of original sizes into a more symmetric distribution of logarithms. 3. Therefore the observed symmetry describes the transformed values, not necessarily the original company-size distribution.

The symmetry applies to the logarithms of company size. The original sizes may still be strongly right-skewed.