Aimathic – Describe a quantitative distribution

Describe a quantitative distribution

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Difficulty: 1 – 5

53970012

Describe the shape of the histogram and identify the direction of skewness.

Hints

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

Solution

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.

Answer

The distribution is unimodal and skewed right.

53970312

Classify the histogram’s shape and justify the classification.

Hints

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

Solution

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

Answer

Approximately uniform, because the bin frequencies are equal.

54875112

Every observation in a data set equals $7$. Describe the distribution's shape, center, spread, and unusual features.

Hints

- 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.

Solution

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.

Answer

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

53969912

Describe the distribution shown in the histogram in terms of shape, approximate center, spread, and unusual features.

Hints

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

Solution

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.

Answer

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

53970112

Eimear says the distribution is skewed right because the tallest bar is on the right. Correct the description.

Hints

- 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.

Solution

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.

Answer

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

53970212

Describe the dotplot's clusters and gap.

Hints

- Locate stretches of x-values with concentrated stacks of dots. - Identify the empty run of x-values between the occupied groups.

Solution

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.

Answer

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

53970412

Is the displayed distribution unimodal or bimodal? Identify the approximate locations of the modes.

Hints

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

Solution

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.

Answer

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

53970512

Describe the unusual feature in the dotplot and explain why it is unusual in context-free statistical language.

Hints

- Identify the main concentration of observations. - Compare the gap before the isolated point with the spacing inside the main cluster.

Solution

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.

Answer

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

53970912

Which pair of terms best describes the histogram: symmetric and bimodal, symmetric and unimodal, or right-skewed and uniform? Justify.

Hints

- 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.

Solution

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.

Answer

Symmetric and unimodal.

53971012

Describe the spread and shape of the dotplot. Does it show a clear single mode?

Hints

- Read the smallest and largest occupied x-values to describe overall spread. - Compare the stack heights across all occupied values. - A unique mode would require one value to occur more often than every other value.

Solution

1. Values extend from $20$ to $24$, giving a range of $4$. 2. Each value occurs twice, so the frequencies are equal across the displayed values. 3. The shape is approximately uniform, with no unique mode.

Answer

The range is $4$, the distribution is approximately uniform, and there is no unique mode.

53971212

Both histograms are centered near $5$. Describe how their variability differs.

Hints

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

Solution

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.

Answer

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

53971312

Yannis describes the dotplot only as “centered at $6$.” State at least two additional features needed for a fuller distribution description.

Hints

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

Solution

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

Answer

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

53971812

The variable is package mass in grams. Write a concise distribution description that uses units and context.

Hints

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

Solution

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.

Answer

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.

54873312

Describe the histogram's shape, approximate center, spread, and unusual features. Explain why calling the distribution “uniform” does not mean that all observations are equal.

Hints

- Compare all bar heights rather than focusing on one interval. - Use the midpoint of the displayed range as a rough center. - Distinguish equal interval frequencies from equal raw observations.

Solution

1. Equal bar heights across the five intervals give an approximately uniform histogram. 2. The displayed range is from $0$ to $50$, and its midpoint gives an approximate center of $25$. 3. There is no prominent peak, gap, or isolated interval. 4. Uniformity here refers to similar frequencies across equal-width intervals; observations may take many different values within and across those intervals.

Answer

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

54873412

The dotplot shows measured heights in inches. The isolated high plotted value was checked against the original record and is correct. Describe the distribution and explain how that verified value should be handled.

Hints

- Compare the spacing within the main cluster to the distance from the cluster to the isolated high point. - Separate a genuine unusual observation from a recording error. - Use the verification information when deciding whether the observation should stay in the data set.

Solution

1. Six values form a cluster from $64$ to $74$, 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.

Answer

The distribution is right-skewed, with a cluster from $64$ to $74$ and a verified unusual value at $84$. Retain and describe the value.

54874512

The dotplot shows recorded waiting times in minutes. Unai describes the isolated low value as an outlier and continues the analysis. Explain why that value requires a different response before the distribution is described.

Hints

- Check whether every plotted value is possible for the measured variable. - Distinguish an extreme valid observation from an impossible one. - Resolve data validity before using a questionable value to characterize shape or spread.

Solution

1. The isolated value is $-45$ minutes, but waiting time cannot be negative in this context. 2. The value is therefore not merely an unusual plausible observation; 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 analyzed as a genuine wait.

Answer

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

55621712

The stem-and-leaf plot shows customer checkout times in minutes. <table><thead><tr><th>Stem</th><th>Leaves</th></tr></thead><tbody><tr><td>$1$</td><td>$8\ 9$</td></tr><tr><td>$2$</td><td>$0\ 0\ 1\ 1\ 2\ 2\ 2\ 3\ 3\ 4\ 4\ 5\ 6$</td></tr></tbody></table> Key: $2\mid4=24$ minutes. Describe the distribution's shape, center, spread, and any obvious unusual features.

Hints

- Use the key to translate stems and leaves into actual checkout times. - Count the observations to locate the median position. - Compare frequencies at values equally far from the center and scan for missing or isolated values.

Solution

1. The values form matching pairs around $22$ minutes, with the greatest concentration at $22$, so the distribution is approximately symmetric and unimodal. 2. There are $15$ observations, so the median is the eighth value, $22$ minutes. 3. The times range from $18$ to $26$ minutes, giving a range of $8$ minutes. 4. Every whole-minute value from $18$ through $26$ appears, so there are no gaps or clearly isolated observations.

Answer

The checkout times are approximately symmetric and unimodal, centered at a median of $22$ minutes, spread from $18$ to $26$ minutes, with no obvious gaps or isolated outliers.

55621812

The table summarizes the number of service requests received during an hour over $22$ observed hours. <table><thead><tr><th>Requests</th><th>Frequency</th></tr></thead><tbody><tr><td>$0$</td><td>$6$</td></tr><tr><td>$1$</td><td>$5$</td></tr><tr><td>$2$</td><td>$4$</td></tr><tr><td>$3$</td><td>$3$</td></tr><tr><td>$4$</td><td>$2$</td></tr><tr><td>$5$</td><td>$1$</td></tr><tr><td>$6$</td><td>$1$</td></tr></tbody></table> Describe the distribution's shape, center, spread, and any obvious unusual features.

Hints

- Read the frequency pattern from low request counts to high request counts. - Use cumulative frequencies to locate the two middle ordered positions. - Distinguish a gradual tail from a gap or isolated observation.

Solution

1. Frequencies decrease as the request count increases, so the distribution has a long right tail and is skewed right. 2. With $22$ observations, the median is the average of positions $11$ and $12$. Position $11$ is $1$, and position $12$ is $2$, so the median is $1.5$ requests. 3. The observed counts range from $0$ to $6$, so the range is $6$ requests. 4. Every integer from $0$ through $6$ occurs, so there is no gap; the high counts form a gradual tail rather than an isolated point.

Answer

The distribution is right-skewed, centered at a median of $1.5$ requests, spread from $0$ to $6$ requests, with no gaps or clearly isolated outliers.

53970612

A claim states that the distribution is approximately symmetric about $25$. Use the histogram to justify or reject the claim.

Hints

- 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.

Solution

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.

Answer

The claim is supported; the frequencies mirror around the central interval near $25$.

53970712

Estimate a reasonable center for the distribution and explain why the midpoint of the full range is only an approximation.

Hints

- 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.

Solution

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.

Answer

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

53970812

Describe the distribution in context if the variable is the number of customer complaints received per day.

Hints

- Describe the concentration and tail using the complaint-count scale. - Include a center, spread, shape, and any visibly unusual value. - Use cautious language when describing an isolated high count without a formal outlier rule.

Solution

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.

Answer

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

53971112

The final bin $[50,60)$ has frequency $2$, while the first five bins are roughly symmetric. Describe how that final bin affects the overall shape.

Hints

- 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.

Solution

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.

Answer

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

53971412

Describe the histogram’s shape. Is “bimodal” sufficient, or is another feature important?

Hints

- 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.

Solution

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 $15$.

Answer

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

53971512

Is the dotplot approximately symmetric? Use matching positions around its center to justify your answer.

Hints

- Choose a plausible center from the display. - Pair x-values that are equally far from that center. - Compare paired stack heights to distinguish approximate symmetry from perfect symmetry.

Solution

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 the frequencies at $2$ and $4$ do not. 3. The distribution is roughly but not perfectly symmetric.

Answer

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

53971612

Describe the distribution and predict whether its mean is likely less than or greater than its median.

Hints

- 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.

Solution

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.

Answer

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

54872412

The histogram shows a quantitative distribution in five equal-width intervals from $0$ to $50$. Describe the distribution's shape, approximate center, spread, and any notable features. Do not call it bell-shaped merely because it is symmetric.

Hints

- 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.

Solution

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.

Answer

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.

54872512

The histogram shows customer waiting times in minutes. Describe the distribution's shape, locate the median interval, describe its spread, and comment on unusual features.

Hints

- 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.

Solution

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.

Answer

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.

54872612

A dotplot of self-reported commute times has unusually tall stacks at $20$, $25$, $30$, $35$, and $40$ minutes, with few observations at nearby whole-number values. Describe the notable pattern and give a plausible measurement explanation. Why should the stacks not automatically be interpreted as five real clusters?

Hints

- 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.

Solution

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.

Answer

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

54872712

Annual donations are recorded in dollars, but every donation of $\$100$ or more is stored as “$100+$.” A graph shows a large stack at $100+$. Describe what the graph reveals and what it hides about the upper tail.

Hints

- 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.

Solution

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 $\$100$.

Answer

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.

54872812

Describe the dotplot's shape, center, spread, and notable features. Explain why reporting a single center alone would give an incomplete summary.

Hints

- Look for separated groups before summarizing the center. - Check whether values can be paired at equal distances from a central point. - Compare where the numerical center lies with where the observations actually occur.

Solution

1. The values form two compact clusters, $10$ through $16$ and $28$ through $34$, separated by a large gap. 2. The distribution is symmetric about $22$, since paired values have average $22$: $(10,34)$, $(12,32)$, $(14,30)$, and $(16,28)$. 3. The range is $34-10=24$. 4. A center near $22$ lies in the empty gap and does not describe either cluster, so the two-cluster structure must be reported.

Answer

The distribution is symmetric with two distinct clusters at $10$–$16$ and $28$–$34$, center $22$, range $24$, and a large central gap. The center alone is misleading because no observations are near it.

54872912

Describe the histogram's shape, approximate center, spread, and modal structure. Explain why naming one modal interval would be misleading.

Hints

- Compare bar heights at equal distances from the middle of the displayed range. - Identify every interval that reaches the maximum height. - Distinguish a broad plateau from one sharp modal interval.

Solution

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.

Answer

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.

54873012

The dotplot shows completion percentages. Describe the distribution's shape, center, spread, and the role of the natural upper bound of $100\%$.

Hints

- Identify where most observations concentrate and which direction has the longer tail. - Use the middle plotted observation for the median and the two extreme plotted observations for the range. - Consider how a fixed maximum constrains only one side of the distribution.

Solution

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 ordered value, $95\%$. 3. The range is $100-50=50$ percentage points. 4. The natural upper bound of $100\%$ limits the right side and contributes to the pileup near the top; it does not make the distribution symmetric.

Answer

The distribution is left-skewed, with median $95\%$ and range $50$ percentage points. The bound at $100\%$ compresses the upper side and contributes to the pileup near the top.

54873112

A dotplot of self-estimated distances has high stacks at every even whole number and low stacks at neighboring odd numbers. Describe the pattern and give a plausible measurement explanation. Why should the alternating stacks not automatically be reported as many separate clusters?

Hints

- 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.

Solution

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.

Answer

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.

54873212

A histogram of commute times has one peak near $20$ minutes and another near $50$ minutes, with relatively few observations from $30$ to $40$ minutes. Describe the distribution and give one cautious interpretation of the two peaks.

Hints

- 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.

Solution

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.

Answer

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.

54873512

A program accepts only applicants with test scores of at least $60$. A histogram of accepted applicants begins sharply at $60$ and is concentrated from $60$ to $75$. What can be described about the accepted group, and what cannot be inferred about the score distribution of all applicants?

Hints

- 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.

Solution

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.

Answer

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.

54873612

A histogram has one wide interval $[40,60)$ containing $20$ observations, and that bar is the tallest. An analyst calls the distribution unimodal. Explain why the wide bar does not prove that the raw data have one cluster.

Hints

- 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.

Solution

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.

Answer

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

54873712

Shiori says the dotplot is “uniform across the displayed interval” because every observed value occurs once. Evaluate that statement. Also give the median, range, and modal structure.

Hints

- Separate the frequency at each occupied x-value from the spacing between occupied values. - Use the middle plotted point for the median and the extreme points for the range. - Interpret “uniform across an interval” as a statement about how observations are spread over that interval, not just whether exact values repeat.

Solution

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 among the five observed values do not show approximately uniform coverage of the full interval from $2$ to $20$; the occupied positions have unequal gaps and many values in the interval are unobserved.

Answer

The median is $9$, the range is $18$, and there is no mode. The dotplot does not support calling the distribution uniform across the full interval merely because each observed value appears once.

54873812

Mateus says the dotplot is symmetric because four observations lie below the median and four lie above it. Evaluate the claim and describe the actual shape.

Hints

- Symmetry requires more than equal counts on the two sides of the median. - Compare distances and spacing on opposite sides of the center. - Identify which direction contains the longer tail.

Solution

1. Having the same number of observations on the two sides of the median is a positional feature of this ordered data set, not evidence of mirror symmetry. 2. The upper observations lie fairly close to the median, while the isolated low value extends much farther to the left. 3. The distribution therefore has a long left tail and is left-skewed.

Answer

Mateus's claim is false. Equal counts on the two sides of the median do not establish mirrored distances. The isolated low value creates a long left tail, so the distribution is left-skewed.

54873912

Describe the dotplot's shape, center, spread, and unusual features.

Hints

- Look for pairs of plotted values equally far from zero. - Use the central plotted value and the cancellation of opposite pairs to reason about center. - Compare the central cluster with the two distant extremes.

Solution

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.

Answer

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

54874012

A sensor records values below $5$ and values at least $10$, but it fails to store readings from $5$ up to $10$. The observed histogram has an empty region from $5$ to $10$. Should the empty region be described as a true gap in the underlying distribution? Explain.

Hints

- 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.

Solution

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.

Answer

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

54874112

The graph shows the number of service calls received per day. Describe the distribution and explain why the separated bars do not make the variable categorical.

Hints

- Identify where the graph peaks and how frequencies change toward larger counts. - Decide whether differences such as one additional service call have quantitative meaning. - Do not classify a variable solely from whether displayed bars touch.

Solution

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 x-values are numerical counts with meaningful differences; the spaces between bars reflect discreteness, not category status.

Answer

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.

54874212

Milagros describes the histogram only as “symmetric and unimodal.” Explain why those labels are accurate but incomplete, and describe the peak and tails more fully.

Hints

- Compare pairs of bars equally far from the center. - Identify the unique highest bar. - Follow the bar heights all the way from the central peak to the two extremes.

Solution

1. Matching bar heights at equal distances from the center make the distribution symmetric. 2. The center bar has the unique highest frequency, so the distribution is unimodal with a sharp central peak. 3. Frequencies fall to very low levels beside the center but rise again at both extreme bins, producing substantial separated tail mass that should also be reported.

Answer

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

54874312

A distribution of weekly volunteer hours has a large stack at $0$, followed by positive values spread from $1$ to $12$ with decreasing frequency. Describe the distribution and explain why one smooth shape label does not tell the whole story.

Hints

- 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.

Solution

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.

Answer

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

54874612

For the distribution shown, the mean and median are equal. Does that equality prove the distribution is symmetric? Use the dotplot to describe the actual shape.

Hints

- Treat the stated equality of center measures as only one piece of information. - Compare the plotted frequencies and distances on both sides of the center. - Identify which side extends farther from the main concentration.

Solution

1. The mean and median are both $4$, but equality of these two summaries does not determine the complete shape. 2. Three observations are at $0$, while the largest value $13$ extends much farther to the right. 3. The distribution is therefore right-skewed despite having equal mean and median.

Answer

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

54874712

A histogram is roughly symmetric, has one central peak, and tapers on both sides, but it contains only $18$ observations. Give an appropriate description of the observed shape and explain why the graph does not prove that the population is normally distributed.

Hints

- 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.

Solution

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.

Answer

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

54874812

The dotplot includes one verified isolated high observation. Describe the shape using all observations, then describe how the shape changes if that verified observation is removed for a separate sensitivity analysis.

Hints

- Describe the full display before mentally removing any observation. - Identify the isolated high point and then reassess the remaining pattern without it. - Compare how much the tail direction depends on that single observation.

Solution

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.

Answer

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

54875012

Describe the histogram's shape. Explain why the empty central interval is more informative than calling the graph merely symmetric.

Hints

- Compare bars at equal distances from the center of the displayed range. - Identify any interval with zero frequency between occupied groups. - Include both overall balance and local clustering in the description.

Solution

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 two-clustered, rather than a single-peaked symmetric distribution.

Answer

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

54875212

Ece calls the displayed distribution bimodal because several bars remain fairly high after the tallest bar. Evaluate the claim and give a more precise shape description.

Hints

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

Solution

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.

Answer

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

53971712

A report calls the values $12$, $13$, and $14$ three separate outliers. Give a more cautious description based only on the dotplot.

Hints

- Identify the main concentration and any separated group of observations. - Distinguish a cluster of adjacent values from one isolated point. - Use cautious language when no numerical outlier criterion is supplied.

Solution

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.

Answer

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.

54874412

Event times are converted to hours after midnight. A histogram has one cluster near $0$ hours and another near $24$ hours. Most observations correspond to times from about $11{:}45$ p.m. to $12{:}15$ a.m. Explain why the linear histogram makes one time cluster look like two.

Hints

- 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.

Solution

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.

Answer

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

54874912

A histogram of company sizes uses a logarithmic horizontal axis. The bars look approximately symmetric on that scale. Explain why the report should say “symmetric on the log scale” rather than claiming that the original company sizes are symmetric.

Hints

- 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.

Solution

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.

Answer

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

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