Why Examine Concentrations Within a Sampling Window?
A concentration-time profile shows how recorded concentrations relate to sampling time. A complementary question is what the distribution of concentrations looks like among observations collected during a particular part of that profile. For example, a team may want to inspect concentrations in a prespecified interval after a dose, or during a defined visit window. A histogram, density plot, or boxplot can summarize the observations in that interval without drawing a separate line for every record.
The window must be defined before interpreting the display. A window might be written as \(W=[a,b)\), meaning that observations with sampling times \(t\) satisfying \(a\leq t<b\) are included. The interval’s boundaries and time units should be stated, along with the concentration units. Changing the window can change the distribution, so a plot without a clear window definition is hard to interpret.
If each subject contributes multiple observations to a window, the plot gives each included record weight, not each subject equal weight. Some subjects may therefore contribute more observations than others. This distinction matters when describing the display: it summarizes the observed records in the window, and should not automatically be described as a distribution of independent subjects.
Histograms and Density Plots
A histogram divides the concentration axis into bins and displays how many observations fall in each bin. Its appearance depends on the bin boundaries and bin width, so those choices should be reasonable for the data and kept consistent when comparing windows. Very wide bins can hide structure; very narrow bins can make random fluctuations look like meaningful features.
When bins have equal width, a histogram may display counts or relative frequencies. Counts depend on the number of observations, making them awkward to compare across windows with different sample sizes. A density histogram instead divides each bin’s count by the total count and its width. Its vertical scale is concentration to the power of negative one, and the total area of the bars is one.
Here \(n_j\) is the number of observations in bin \(j\), \(n\) is the number in the window, and \(w_j\) is the bin width in concentration units. Multiplying a bar’s height by its width gives the proportion of observations in that bin. For unequal-width bins, heights must reflect each bin’s own width for area to represent proportion correctly.
A density plot is a smoothed estimate of the distribution rather than a set of bars. A common construction places a smooth kernel around each observed concentration and averages the contributions. The bandwidth controls the smoothing: a small bandwidth can show a ragged curve with local peaks, while a large bandwidth can blur distinct features. A smooth curve is not proof that the underlying distribution is smooth, and apparent bumps should be checked against the sample size and the histogram.
Worked Example: Calculating Histogram Densities
Suppose a hypothetical prespecified window contains eight recorded concentrations, in mg/L: \(1.0,\ 1.2,\ 1.5,\ 1.8,\ 2.1,\ 2.5,\ 3.0,\ 8.0\). Use bins \([0,2),[2,4),[4,6),[6,8),[8,10)\), each 2 mg/L wide.
The bin counts are \(4,3,0,0,1\), respectively. For the first bin, the density is \(4/(8\times2)=0.25\) per mg/L. For the second it is \(3/(8\times2)=0.1875\) per mg/L. The third and fourth densities are zero, and the fifth is \(1/(8\times2)=0.0625\) per mg/L.
The bar areas are \(0.25(2)=0.50\), \(0.1875(2)=0.375\), zero, zero, and \(0.0625(2)=0.125\). They sum to \(1.00\), as required. The display shows that half the recorded concentrations are below 2 mg/L and one observation is in the 8-to-10 mg/L bin. The isolated high observation suggests right-skew in this small sample, but eight records are not enough to establish a stable distributional shape.
Boxplots: Center, Spread, and Potentially Extreme Values
A boxplot gives a compact summary based on the median and quartiles. The box extends from the first quartile \(Q_1\) to the third quartile \(Q_3\); its length is the interquartile range, \(IQR=Q_3-Q_1\). The median is marked inside the box. Whiskers commonly extend to the most extreme observations no farther than \(1.5\,IQR\) below \(Q_1\) or above \(Q_3\). Observations beyond these fences are plotted individually and are often called potential outliers.
Quartile conventions can differ across software, especially for small samples. State or check the convention when exact values matter. A point beyond a fence is a flag for investigation, not a finding that the value is erroneous. It may be a valid concentration, a data issue, or a consequence of the shape of the distribution.
Worked Example: Interpreting a Boxplot Summary
For a hypothetical window, the sorted concentrations in mg/L are \(0.4,\ 0.5,\ 0.6,\ 0.7,\ 0.9,\ 1.1,\ 1.4,\ 2.0,\ 5.0\). Use the convention that, for an odd sample size, the overall median is excluded when finding the medians of the lower and upper halves.
The overall median is the fifth value, \(0.9\) mg/L. The lower half is \(0.4,0.5,0.6,0.7\), so \(Q_1=(0.5+0.6)/2=0.55\) mg/L. The upper half is \(1.1,1.4,2.0,5.0\), so \(Q_3=(1.4+2.0)/2=1.70\) mg/L. Thus \(IQR=1.70-0.55=1.15\) mg/L.
The lower and upper fences are \(0.55-1.5(1.15)=-1.175\) mg/L and \(1.70+1.5(1.15)=3.425\) mg/L. The smallest observation, \(0.4\) mg/L, is within the lower fence; \(5.0\) mg/L is above the upper fence and would be plotted separately under this convention. The upper whisker reaches \(2.0\) mg/L, the largest observation still within the upper fence. The flagged value should be checked against source records and context, not automatically deleted.
Choosing and Comparing Displays
Each display emphasizes different features. A histogram makes gaps, clusters, and tail shape visible, but depends on binning. A density plot gives a smooth shape but depends on bandwidth. A boxplot is compact and supports quick comparisons of median and middle-half spread, but hides much of the distribution’s shape. They are complementary summaries, not competing tests.
| Display | Useful for | Check before interpreting |
|---|---|---|
| Histogram | Shape, gaps, clusters, and counts or proportions by range | Bin width, bin boundaries, and whether the vertical scale is count or density |
| Density plot | A smoothed view of concentration shape | Bandwidth, sample size, and whether apparent peaks persist under reasonable smoothing choices |
| Boxplot | Median, middle-half spread, and points beyond conventional fences | Quartile convention and the fact that flagged points are not necessarily errors |
For a comparison between windows, state both windows and the number of records in each. If counts differ, density histograms or normalized density plots are generally more informative than raw counts for comparing shape. Keep concentration units and axis limits consistent when the goal is visual comparison. A different vertical scale can make similar spreads look quite different.
Worked Example: Comparing Different Window Sample Sizes
Consider two hypothetical windows with concentrations in mg/L. Window A has four records: \(1,2,3,4\). Window B has eight: \(1,1,2,2,3,3,4,4\). For both, use bins \([0,2),[2,4),[4,6)\), each 2 mg/L wide.
Window A has counts \(1,2,1\). Its density heights are \(1/(4\times2)=0.125\), \(2/(4\times2)=0.25\), and \(1/(4\times2)=0.125\) per mg/L. Window B has counts \(2,4,2\), giving \(2/(8\times2)=0.125\), \(4/(8\times2)=0.25\), and \(2/(8\times2)=0.125\) per mg/L. The density histograms match, because the proportions in each bin match even though Window B has twice as many records.
This comparison concerns the observed concentration distributions across records. It does not show that the windows represent the same subjects or that their records are independent. If subjects contributed repeatedly, that sampling structure should be reported; a simple histogram does not adjust for it.
Assumptions, Diagnostics, and Common Pitfalls
These plots are descriptive and do not require normally distributed concentrations. They do require that the plotted records are correctly identified and that the window, units, and treatment of unusual or censored values are understood. Before interpreting a shape, check sample size, data range, missingness, and whether repeated records from the same subjects are present. Compare plots with the underlying records when a value appears isolated or implausible.
Concentrations are often right-skewed: most values may be relatively low while a smaller number are much higher. A long right tail can pull the mean above the median, but a boxplot alone cannot show how that tail is shaped. A logarithmic concentration axis or a display of log concentrations can make multiplicative ranges easier to inspect, but the interpretation changes: a boxplot of log concentrations summarizes log-scale values, not the original-scale quartile distances. State which scale is shown and retain the original concentration units in the description.
Below-quantification observations need special care. They are not automatically zero or equal to the quantification limit. Replacing them with a convenient number changes the displayed distribution, potentially creating artificial spikes or distorting its lower tail. Follow the data’s documented censoring convention, show the handling clearly, and avoid implying exact values where only a bound is known.
- Using raw counts to compare windows of unequal size. Larger samples tend to produce taller count bars even when proportions are alike. Use normalized densities when comparing shape.
- Treating every record as an independent subject. Repeated observations may give some subjects more influence. Describe the plot as record-weighted unless the construction gives subjects equal weight.
- Calling every boxplot flag an error. The fence rule identifies unusual values relative to the middle spread; it does not diagnose data quality.
- Reading a smoothed curve as the data itself. Density estimates depend on bandwidth. Check the observations and a histogram before treating a small peak or dip as meaningful.
A Practical Review Sequence
Write the time window and boundary rule, then state the concentration units and the number of included records.
Check missing, censored, repeated, and unusually large or small records before choosing bins or drawing a summary.
Use a histogram or density plot for shape and a boxplot for a concise summary of median and spread; consider more than one view.
Describe skew, spread, gaps, and flagged values as features of the plotted records, noting limitations such as repeated observations or small sample size.
Check Your Understanding
Answer each question, then check the brief solutions below.
- A window has 20 observations. A bin 4 mg/L wide contains 6 observations. What is its density height?
- Why can’t the height of a boxplot’s flagged point alone establish that the concentration is erroneous?
- Two windows have the same proportions in every bin, but one contains twice as many records. What should their density histograms show, and what would raw-count histograms show?
- What should be checked before interpreting a right-skewed distribution that includes below-quantification records?
Solutions. (1) \(6/(20\times4)=0.075\) per mg/L. (2) The \(1.5\,IQR\) fence is a descriptive flag, not a data-validation rule; verify the record and context. (3) Their density heights should match; raw-count bars in the larger window should be twice as high for corresponding bins. (4) Check the censoring convention and how those records were handled, along with sample size, units, and possible repeated observations; a censored value is not automatically zero or the quantification limit.