Why Look for Patterns in Recorded Values?
A numeric data field can look orderly in a summary while containing clues about how it was recorded. If many reported travel times end in zero or five, or an unusually large number of records share exactly the same value, people may have rounded, selected from a short list, or entered data using a convention. A sudden gap in the values can also be worth investigating. These patterns do not, by themselves, establish that any record is wrong. They help an analyst ask focused questions about how the data were collected and entered.
Histograms give a broad view of the distribution, while dotplots make repeated values easy to see in smaller data sets. The choice of histogram bin width and boundaries matters: a broad bin can combine distinct spikes, and a poorly aligned set of bins can make a regular pattern seem irregular. The aim is not to diagnose an error from a graph alone; it is to describe a pattern clearly enough to decide what should be checked.
These descriptions are exploratory, not verdicts. A spike could reflect a genuine common value, a standard service option, a recording convention, or an error. A gap could reflect a rule that excludes values, a restricted range in the population, or chance variation in a small sample. Record the pattern, its size, and the follow-up information needed to interpret it.
Pair a Dotplot With a Carefully Set Histogram
A dotplot displays individual observations along a number line, stacking dots when values repeat. It is particularly useful for a small or moderate number of observations because the exact repeated values remain visible. In R, a simple stacked display can be made with `stripchart(x, method = "stack", pch = 19)`, where `x` is a numeric vector. A dotplot is less convenient when there are many observations or a wide range of values, because the display can become crowded.
A histogram groups observations into intervals, or bins. It is useful for seeing broad distributional shape and locating concentrations or empty regions, but it does not show each exact value. In R, `hist(x, breaks = ...)` lets the analyst specify bin boundaries. Inspecting more than one sensible bin width can help distinguish a persistent feature from a feature created by the bin choices. For rounded integer data, breaks at integer boundaries can make adjacent values easier to compare; breaks that group many values together may conceal digit preference.
A frequency table is a useful companion to either plot. It provides exact counts at repeated values or within specified intervals. For example, a plot may make a peak look striking, but the table lets the analyst state whether that peak represents three observations or thirty. Always state the denominator when reporting a proportion.
Worked Example: Checking for Values Ending in Zero or Five
A hypothetical survey records one-way commute times, in whole minutes, for 28 workers. The values and their frequencies are: 12: 1, 13: 1, 14: 2, 15: 1, 16: 1, 17: 1, 18: 2, 19: 1, 20: 5, 21: 1, 22: 1, 23: 1, 24: 1, 25: 3, 26: 1, 27: 1, 28: 1, 29: 1, and 30: 2. Use a dotplot and the frequencies to assess whether values ending in zero or five merit a data check. No probability distribution is assumed for these hypothetical observations.
Method. First identify the values with the specified last digit, then add their frequencies and divide by the total number of observations. A dotplot can show whether the same values are also local peaks.
The relevant values are 15, 20, 25, and 30. Their frequencies are 1, 5, 3, and 2. Thus the number of observations at those values is \(1+5+3+2=11\), and the proportion is \(11/28=0.392857\ldots\), or \(39.3\%\) when rounded to one decimal place.
The dotplot would show a pile of five observations at 20, compared with one at 19 and one at 21. It would also show three at 25, compared with one at 24 and one at 26. These local contrasts make the values at 20 and 25 worth checking in the recording process. The calculation also includes 15 and 30, but their frequencies alone are not as striking.
Interpretation and check. The observed pattern warrants a question such as whether respondents were asked to estimate or round commute times. It does not prove that any reported time is inaccurate. The check is consistent: the frequencies in the data sum to 28, and 11 of those 28 observations are at the selected values. The common mistake this method avoids is calling a pattern “common” without calculating how many observations it affects.
Interpreting Spikes and Gaps
An abrupt spike is a sharp frequency increase at one value relative to nearby values. A gap is a range with no or few observations, especially when observations occur on both sides. Both can be identified visually, but the interpretation depends on context and on the scale of the plot. For example, a gap in values recorded in whole units may mean something different from a gap in values recorded to the nearest hundredth.
To describe a spike, report the value, its frequency, the total sample size, and—when useful—the frequencies immediately to either side. To describe a gap, give its interval and say whether it is truly empty or merely sparse. These details make a plot-based observation precise and reproducible. They also help prevent a reader from mistaking the visual height of a bin for a count when bins have different widths.
Worked Example: A Spike in Recorded Service Times
A hypothetical administrator reviews 30 service times recorded in whole hours. The frequency table is: 1 hour: 2; 2 hours: 3; 3 hours: 2; 4 hours: 1; 5 hours: 12; 6 hours: 1; 7 hours: 2; 8 hours: 2; 9 hours: 1; 10 hours: 2; 11 hours: 1; and 12 hours: 1. Assess the concentration at 5 hours and state an appropriate follow-up. No probability distribution is assumed.
Method. Quantify the peak as a fraction of the sample, then compare its frequency with the adjacent values. A histogram with one-hour bins or a dotplot can show the shape, but the frequency table gives the exact counts.
The total is \(2+3+2+1+12+1+2+2+1+2+1+1=30\). The fraction recorded as exactly 5 hours is \(12/30=0.40\), or \(40\%\). The neighboring values have frequencies 1 at 4 hours and 1 at 6 hours. The count at 5 hours is therefore 12 times either adjacent count.
Interpretation and check. This is a sharp concentration worth investigating. The administrator could check whether 5 hours is a standard category, a typical scheduled duration, or a value that data-entry staff use when an exact time is unavailable. The count at 5 hours is not evidence on its own that the records are erroneous: a real operational process could generate the same pattern. The common mistake avoided here is treating a high bar as proof of inaccurate recording rather than as a reason to check how the value was produced.
A Short Justification for Rounding Patterns
A rounded value can accumulate observations from a range of underlying values. To see why, suppose a continuous quantity \(X\) has a smooth density \(f_X(x)\), and recorded values are rounded to the nearest multiple of \(h\). Assume ties occur with probability zero. Then a recorded value of \(kh\) corresponds to an underlying value between \(kh-h/2\) and \(kh+h/2\). Under those assumptions, the probability of that recorded value is:
where \(Y\) is the rounded value and \(k\) is an integer. If the density changes little across that interval, the integral is approximately \(h f_X(kh)\). Thus rounding can place many nearby underlying values at the same recorded value, creating a spike. This explains why spikes at multiples of a rounding unit can occur even when the underlying quantity varies smoothly.
This justification is not a test that every spike must follow a rounding rule. The smooth-density and rounding assumptions may not hold, and the true process may itself generate repeated values. It also does not imply that equal counts should occur at all endings: the underlying density can vary across the scale. Treat the formula as an explanation for one possible mechanism, not as a universal benchmark for the expected frequencies.
Check the variable definition, units, allowed values, and recorded precision before interpreting a pattern.
Use a dotplot for repeated exact values in a manageable sample; use a histogram for a broader view and inspect sensible bin widths and boundaries.
Count observations at preferred digits or in a spike, calculate the proportion, and record adjacent counts or empty intervals.
Ask whether the pattern fits the data-collection process, coding rules, or source records. Preserve the distinction between a pattern and evidence of an error.
Worked Example: A Gap and a Repeated Value
A hypothetical set of 20 recorded service durations, in minutes, is 31, 34, 36, 39, 41, 43, 44, 45, 45, 45, 45, 55, 56, 58, 61, 64, 66, 69, 71, and 74. An analyst makes a dotplot and a histogram using breaks at 30, 40, 50, 60, 70, and 80 minutes. Identify the spike and gap, calculate the spike's share, and state one limitation of the plot-based conclusion. No probability distribution is assumed.
Method. Read the exact repeated values from the dotplot, count observations in each histogram interval, and distinguish the observed absence of values from a claim about why they are absent.
There are four observations at 45 minutes, so the spike's share is \(4/20=0.20\), or \(20\%\). With the specified right-closed histogram intervals, the counts are 4 in \((30,40]\), 7 in \((40,50]\), 3 in \((50,60]\), 4 in \((60,70]\), and 2 in \((70,80]\). The interval counts sum to \(4+7+3+4+2=20\), as required.
The values from 46 through 54 minutes are absent, producing a gap between the repeated 45-minute value and the next observation at 55 minutes. The plot establishes that this gap is present in these 20 records; it cannot establish whether an entry rule, a genuine feature of the process, or some other mechanism caused it. A focused follow-up would check the recording instructions and source records around that range.
Interpretation and check. The common mistake this method avoids is treating an empty interval as proof that values were deleted or mistyped. The spike and gap justify investigation, while the data alone do not settle their cause. The histogram counts add to the sample size, and the repeated-value proportion uses the same denominator of 20.
Common Plotting and Interpretation Traps
- Changing the bins and treating the picture as new evidence. A broad bin can hide a narrow spike; a narrow bin can make random variation look dramatic. Keep track of the boundaries and compare a small number of meaningful choices.
- Confusing bin counts with exact-value counts. A histogram bar generally combines a range of values. Use a dotplot or frequency table to establish whether one exact value is repeated.
- Reporting only a percentage. “Forty percent” is more informative when paired with the count and sample size, such as 12 of 30. The denominator matters when comparing samples.
- Assuming selected ending digits must be equally frequent. A variable's distribution may not be uniform, and the data-collection process may favor some values. Describe the observed concentration rather than comparing it with an unjustified equal-frequency expectation.
- Calling a pattern an error. A graph can flag a feature for investigation; source records and collection rules are needed to determine whether it reflects an error or a legitimate process.
Check Your Understanding
For each situation, calculate what is requested and distinguish the observed pattern from its possible cause.
- A hypothetical set of 20 reported ages has frequencies 10: 4, 11: 2, 12: 1, 13: 2, 14: 1, 15: 3, 16: 2, 17: 1, 18: 2, and 19: 2. What proportion of records end in zero or five, and what can that proportion establish? (Answer: 7/20 = 0.35, or 35%; it establishes the observed frequency, not that rounding or error caused it.)
- In 17 hypothetical inspections, recorded durations of 2, 3, 4, 5, 6, and 7 hours have frequencies 1, 2, 2, 9, 2, and 1, respectively. Calculate the share at 5 hours and compare it with each adjacent value. (Answer: 9/17 = 0.5294, or 52.94%; the frequency at 5 hours is 4.5 times the frequency at either 4 or 6 hours.)
- A hypothetical data set contains 101, 104, 109, 111, 115, 119, 141, 145, 149, 151, 155, 159, 161, and 166. With histogram breaks at 100, 110, 120, 130, 140, 150, 160, and 170, identify the empty intervals and their total count. (Answer: \((120,130]\) and \((130,140]\) are empty; together they contain 0 of the 14 observations.)
- Eight hypothetical records, all labelled as minutes, are 15, 20, 25, 30, 0.5, 0.5, 0.5, and 35. What fraction is recorded as 0.5, and what should be checked before concluding that the values are wrong? (Answer: 3/8 = 0.375, or 37.5%; check the unit and source records, since the plot alone cannot establish whether these are unit-conversion errors.)
- A dotplot shows 6 observations at 40 units, 2 at 39, and 1 at 41, in a sample of 25. Calculate the share at 40 and give a cautious interpretation. (Answer: 6/25 = 0.24, or 24%; it is a concentration worth checking, but it does not by itself prove a recording error.)