The Mean, Median and Mode

A chemist titrates the same acetic acid solution nine times. She gets nine slightly different endpoint volumes. Which single number should go in the lab report? That question matters. It asks how to compress many numbers into one representative value. This is the entire subject of central tendency. It matters every time an instrument is read more than once. Analytical chemistry is built on repeated measurement. No titration is trusted after a single trial. No spectrophotometric absorbance reading is either. No mass on a balance is either. This is because Accuracy and Precision teaches something important. Random error scatters replicate readings around some "true" center. The mean, median, and mode are three different mathematical recipes for locating that center. They are not interchangeable. A data set with one wild outlier can have a badly biased mean. Its median, though, stays put. A bimodal data set mixes two real populations together. Such a set can have no meaningful single "center" at all. The big picture is this: central tendency is the first of two questions every data set answers. The second is spread, covered in Standard Deviation and Variance. You cannot report a measurement meaningfully with a center alone. "25.00 mL" means nothing on its own. You must know whether replicate values ranged from 24.98 to 25.02 mL. Or whether they ranged from 23.5 to 26.5 mL instead. But the center must be chosen correctly first. Only then can the spread be calculated around it. Choosing the wrong measure of center is a common error. So is applying it uncritically to contaminated data. These are among the most frequent quiet errors in student lab reports. This article works through the definitions and the arithmetic. It covers the conditions under which each measure is preferred. It also covers the traps that trip up real analytical work. These include particularly the outlier problem and the discrete-instrument-reading problem. Every example uses actual titration, gravimetric, and spectrophotometric numbers. These are carried to the correct number of significant figures. A statistic is only as trustworthy as the digits that go into it.

Key takeaways

The mean ( $\bar{x}$ ) is the arithmetic average; it uses every data point and is the standard choice for symmetric, outlier-free replicate data.
The median is the middle value of ranked data; it is robust to outliers and is preferred when a data set is skewed or contains a suspect point.
The mode is the most frequent value; it is most useful for discrete or categorical data (e.g., the most common integer absorbance reading on a coarse scale) and is rarely used alone in quantitative analytical work.
For symmetric, well-behaved (approximately normal) replicate data, mean = median ≈ mode; divergence among the three is itself diagnostic of skew or contamination.
A single outlier can shift a mean substantially but shifts a median only slightly or not at all — this is the basis for preferring the median before applying outlier tests like the Q-test.
Reporting a mean with the correct number of significant figures requires knowing the precision of the individual measurements, not just averaging raw digits blindly.
Key idea: the mean, median, and mode are three different definitions of "center," and choosing correctly — based on the shape and cleanliness of the data — matters as much as computing them correctly.

Where this lives today

Modern analytical instruments report a mean automatically — an ICP-OES or HPLC data system averages replicate injections and displays a single number with an automatically computed relative standard deviation — but the underlying choice of which statistic to trust is still a human judgment built into every validated method. Pharmacopeial assay methods (USP, EP) specify not just how many replicates to run but explicit criteria for rejecting outliers before averaging, precisely because an automatically computed mean is only trustworthy once contamination has been ruled out. In clinical and environmental chemistry, median values are often legally mandated reporting statistics specifically because real-world sample populations are skewed: median blood lead level across a population, or median contaminant concentration across monitoring wells, is far less sensitive to a single anomalously high (or low, near a detection limit) reading than the mean would be, and regulatory limits are often written in terms of medians for exactly this reason. In pharmaceutical manufacturing quality control, control charts (Shewhart charts) track a running mean of process measurements (tablet mass, dissolution rate) against statistically derived control limits; a shift in the process mean, or a jump in variability that later feeds into Standard Deviation and Variance, triggers investigation long before a batch would fail a final specification test. Bimodality in such charts — two clustered populations rather than one — is itself a diagnostic red flag for equipment drift or a raw-material lot change, exactly analogous to the mixed-population absorbance example above. In mass spectrometry and chromatographic peak integration, the mode resurfaces in an unexpected way: when software bins retention times or m/z values into discrete channels for a histogram-based analysis, the modal channel — the single most heavily populated bin — is used to assign the nominal peak position, distinct from a mean-based centroid calculation used for higher-precision mass assignment. Choosing between mode-based and mean-based peak-picking algorithms is an active area of software design in modern proteomics and metabolomics pipelines.

Ask the tutor: If I have 15 replicate absorbance readings and one looks unusually high, how do I decide whether to reject it before computing my reported mean?

The physical picture — thorough prose

Picture a burette reading recorded to the nearest 0.01 mL, repeated nine times by a careful analyst titrating the same solution under the same conditions. Random fluctuations — a slightly different rate of drop formation, a marginally different judgment of the endpoint color, tiny parallax differences in reading the meniscus — scatter the nine volumes around some central value. If you plot those nine numbers as dots on a number line, they cluster, with more points near the center and fewer at the extremes. The whole enterprise of "central tendency" is about picking the single number that best represents where that cluster is centered. The mean treats every data point as equally informative and folds all of them into the calculation: add everything up, divide by how many there are. This is optimal — in the precise statistical sense of minimizing the sum of squared deviations — when the underlying errors are random and symmetric, which is the condition described by the Normal Distribution. But because every point contributes, a single bad point (a stuck buret tip, a transcription error, a contaminated aliquot) can drag the mean noticeably off target. The median ignores the actual magnitude of most points and asks only: rank the data, what sits in the middle? Because it depends only on rank order, not on distance, a single wildly erroneous point can only move the median by one rank position, not by its full deviation. This robustness is the median's whole appeal, and it is why national statistical agencies report median household income (dominated by a small number of very high earners) rather than mean household income. The mode asks a different question entirely: which value occurs most often? For continuous analytical measurements (volumes, absorbances, masses) recorded to many decimal places, no two readings are usually identical, so the mode is often undefined or trivial. The mode becomes meaningful when data are discretized — a pH meter reading only to the nearest 0.1 pH unit across many replicate samples, or a survey of the most common oxidation state observed across a set of compounds. It is also the natural descriptor for categorical data, such as the most frequently identified unknown in a class of 40 students.

Figure

Three data sets: symmetric, right-skewed, and outlier-contaminated, showing how mean, median, and mode respond differently

The instrument itself matters too. A digital pH meter reading to 0.01 units on truly continuous chemistry produces essentially unique replicate values — mode is meaningless. A student manually reading a analog burette to the nearest 0.05 mL across 20 trials might see the same value repeat three or four times — now the mode carries real information about instrument resolution, though it still should not replace the mean or median as the reported "result."

Figure

Mean vs. median location for a small right-skewed titration data set (mL), with a stem-and-leaf-style dot plot

The core ideas — the governing equations/relationships

For a set of $n$ replicate measurements $x_1, x_2, \dots, x_n$ :

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

where: $\bar{x}$ is the sample mean, $x_i$ is the $i$ -th measurement, and $n$ is the number of replicates. Reading: sum every value and divide by how many there are — every measurement contributes equally. For the median, first sort the data into ascending order to get order statistics $x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}$ , then:

\tilde{x} = \begin{cases} x_{\left(\frac{n+1}{2}\right)} & n \text{ odd} \\[2mm] \dfrac{1}{2}\left[x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}\right] & n \text{ even} \end{cases}

where: $\tilde{x}$ is the median, $x_{(k)}$ is the $k$ -th smallest value after sorting. Reading: for odd $n$ there is a single middle value; for even $n$ the median is the average of the two central values, so it can fall between two actual data points. The mode has no algebraic formula — it is defined operationally as

\text{mode} = \arg\max_{v} \; f(v)

where: $f(v)$ is the frequency (count) with which value $v$ occurs in the data set, and $\arg\max_v$ selects the value(s) $v$ achieving that maximum count. Reading: find whichever value shows up most often; a data set can have one mode (unimodal), two (bimodal, often signaling two mixed populations), or none (all values distinct). A useful diagnostic relating the three, valid only as a rough rule-of-thumb for moderately skewed unimodal distributions (not derivable exactly, but widely used for a first read on skew direction):

\text{mean} - \text{mode} \approx 3\,(\text{mean} - \text{median})

where: this is Karl Pearson's empirical skew relation. Reading: if the mean sits noticeably above the median, the mode sits even further below the mean, signaling a right-skewed (long upper tail) data set — exactly the titration example with one abnormally high volume.

Derivation AWhy the mean minimizes total squared deviation

Goal. Show that the arithmetic mean $\bar{x}$ is the unique value $c$ that minimizes the sum of squared deviations $S(c) = \sum_{i=1}^n (x_i - c)^2$ , which is the formal justification for using the mean as "the center" under random, symmetric error.

Step 1. Write the sum of squared deviations from an arbitrary trial center $c$ :

S(c) = \sum_{i=1}^{n} (x_i - c)^2

This is a smooth, upward-opening (convex) quadratic function of $c$ , since expanding gives a positive coefficient on $c^2$ : $S(c) = \sum x_i^2 - 2c\sum x_i + nc^2$ .

Step 2. Because $S(c)$ is convex, its unique minimum occurs where the derivative with respect to $c$ vanishes. Differentiate term by term:

\frac{dS}{dc} = \sum_{i=1}^{n} \frac{d}{dc}(x_i - c)^2 = \sum_{i=1}^{n} 2(x_i - c)(-1) = -2\sum_{i=1}^{n}(x_i - c)

Step 3. Set the derivative to zero and solve for $c$ :

-2\sum_{i=1}^{n}(x_i - c) = 0 \quad\Rightarrow\quad \sum_{i=1}^{n} x_i - nc = 0

c = \frac{1}{n}\sum_{i=1}^{n} x_i = \bar{x}

Step 4. Confirm it is a minimum, not a maximum, by checking the second derivative: $\dfrac{d^2 S}{dc^2} = 2n > 0$ for any $n \geq 1$ , confirming convexity and that $c = \bar{x}$ is indeed the minimizer. This is why least-squares regression, propagation of random error, and the definition of variance itself (built on squared deviations from the mean, in Standard Deviation and Variance) are all anchored to the mean rather than the median — squared-error minimization is what singles the mean out mathematically. The median instead minimizes the sum of absolute deviations $\sum |x_i - c|$ , a different and non-differentiable optimization that favors robustness over the smooth calculus above.

Go deeper · Derivation B · 1 minWhy the median is unaffected by the magnitude of an outlier

Derivation BWhy the median is unaffected by the magnitude of an outlier

Goal. Show explicitly, using order statistics, that replacing the largest value in a sorted data set with an arbitrarily larger value leaves the median completely unchanged (for odd $n$ ), while it necessarily changes the mean.

Step 1. Take a sorted odd-sized data set of $n = 2m+1$ values, $x_{(1)} \le x_{(2)} \le \dots \le x_{(2m+1)}$ . By definition the median is the middle order statistic:

\tilde{x} = x_{(m+1)}

Step 2. Now replace the largest value $x_{(2m+1)}$ with some much larger value $x_{(2m+1)}' \gg x_{(2m+1)}$ (an outlier), while keeping every other value fixed. Because $x_{(2m+1)}'$ is still the largest value in the set, the sorted order of all other points is completely unchanged — the rank positions $x_{(1)}$ through $x_{(2m)}$ are identical to before.

Step 3. Since the median depends only on which value occupies rank $m+1$ , and that rank is still occupied by the same $x_{(m+1)}$ as before (the outlier only affects rank $2m+1$ , the last position), the median is literally unchanged:

\tilde{x}' = x_{(m+1)} = \tilde{x}

This holds no matter how large $x_{(2m+1)}'$ becomes — the median is completely insensitive to the magnitude of the largest point, only ever "seeing" that it is the largest.

Step 4. Contrast with the mean. Because the mean sums the actual values, replacing $x_{(2m+1)}$ with $x_{(2m+1)}'$ changes the mean by exactly:

\bar{x}' - \bar{x} = \frac{x_{(2m+1)}' - x_{(2m+1)}}{n}

This difference grows without bound as $x_{(2m+1)}' \to \infty$ , while the median's shift stays exactly zero. This is the rigorous statement behind "the median is robust to outliers": the median has what statisticians call a breakdown point of 50% (roughly half the data would need to be corrupted before the median itself moves into the corrupted region), whereas the mean's breakdown point is $1/n$ — a single bad point can move it arbitrarily far.

Worked examples

Example 1 — Foundational: mean, median, and mode of a clean replicate titration series. A student titrates 25.00 mL aliquots of a hydrochloric acid solution against standardized 0.1024 M NaOH, six times, recording the volume of NaOH delivered at the phenolphthalein endpoint:

24.98,\ 25.02,\ 24.95,\ 25.06,\ 25.00,\ 24.99 \ \text{mL}

Mean. Sum the six values:

\sum x_i = 24.98 + 25.02 + 24.95 + 25.06 + 25.00 + 24.99 = 150.00 \text{ mL}

\bar{x} = \frac{150.00}{6} = 25.000 \text{ mL}

Since the individual readings are good to the nearest 0.01 mL (four significant figures each), and averaging six values improves the precision of the mean by roughly $\sqrt{n} = \sqrt{6} \approx 2.4$ , it is defensible to report the mean to one more digit than a single reading, i.e., $\bar{x} = 25.00_0$ mL — commonly written 25.00 mL (the trailing digit understood as uncertain). Median. Sort the six values ascending:

24.95,\ 24.98,\ 24.99,\ 25.00,\ 25.02,\ 25.06

Since $n = 6$ is even, the median is the average of the 3rd and 4th values:

\tilde{x} = \frac{24.99 + 25.00}{2} = \frac{49.99}{2} = 24.995 \text{ mL}

Mode. Every value is distinct — no repeats — so this data set has no mode. This is expected: with continuous burette readings to 0.01 mL and only six trials, exact repeats are statistically unlikely even with perfectly random error. Interpretation. Mean (25.00 mL) and median (24.995 mL) agree to within 0.005 mL, well inside the reading uncertainty of ±0.01 mL on a single measurement — this agreement is exactly what is expected for clean, symmetric, outlier-free replicate data, and it justifies reporting the mean as the representative volume, since the mean is the statistically efficient choice (Standard Deviation and Variance will use this same mean as the center from which to compute the standard deviation).

Example 2 — Applied: an outlier changes the story in a gravimetric chloride determination. A gravimetric analysis of chloride as AgCl is performed in quintuplicate on the same unknown, and the %Cl by mass is calculated for each trial:

35.11,\ 35.08,\ 35.14,\ 35.09,\ 34.42\ \%

Mean.

\sum x_i = 35.11 + 35.08 + 35.14 + 35.09 + 34.42 = 174.84

\bar{x} = \frac{174.84}{5} = 34.968 \approx 34.97\%

Notice the mean, 34.97%, sits noticeably below four of the five actual results — dragged down by the 34.42% trial, which is 0.6–0.7 percentage points below the other four (themselves clustered within 0.06% of each other). This is a strong visual and numerical signal of a potential outlier — perhaps a partial loss of precipitate during filtration. Median. Sort:

34.42,\ 35.08,\ 35.09,\ 35.11,\ 35.14

With $n = 5$ (odd), the median is the 3rd value:

\tilde{x} = 35.09\%

The median, 35.09%, lies right in the tight cluster of four consistent results and is essentially unaffected by the low outlier — exactly the robustness proven algebraically above. Applying a formal outlier test before reporting. Rather than simply eyeballing which statistic "looks right," an analyst should apply a quantitative outlier test — commonly the Q-test — before discarding a point. For the suspect value 34.42%, the gap to its nearest neighbor is $35.08 - 34.42 = 0.66$ , and the full range of the data is $35.14 - 34.42 = 0.72$ :

Q_{\text{exp}} = \frac{\text{gap}}{\text{range}} = \frac{0.66}{0.72} = 0.917

For $n = 5$ at 90% confidence, the critical value is $Q_{\text{crit}} = 0.64$ . Since $Q_{\text{exp}} = 0.917 > 0.64$ , the point is rejected as a statistical outlier. Recomputing the mean on the remaining four values:

\bar{x}_{\text{corrected}} = \frac{35.11 + 35.08 + 35.14 + 35.09}{4} = \frac{140.42}{4} = 35.105 \approx 35.11\%

This corrected mean (35.11%) is much closer to the median of the full five-point data set (35.09%) than the uncorrected five-point mean (34.97%) ever was — illustrating that the median served as an early warning that something in the mean was off, even before the formal test confirmed it.

Example 3 — Exam/competition level: mixed-population data and the failure of a single "center." An instructor pools spectrophotometric absorbance readings (at 508 nm, for an iron(II)–phenanthroline complex) submitted by two lab sections measuring what should be identical standard solutions. The 12 pooled absorbance values are:

0.412,\ 0.415,\ 0.409,\ 0.418,\ 0.411,\ 0.414,\ 0.512,\ 0.508,\ 0.515,\ 0.509,\ 0.511,\ 0.513

Step 1 — compute mean and median blindly. Sum:

\sum x_i = (0.412+0.415+0.409+0.418+0.411+0.414) + (0.512+0.508+0.515+0.509+0.511+0.513)

= 2.479 + 3.068 = 5.547

\bar{x} = \frac{5.547}{12} = 0.46225 \approx 0.4623

Sorted, the middle two values (6th and 7th of 12) are $0.415$ and $0.508$ :

\tilde{x} = \frac{0.415 + 0.508}{2} = \frac{0.923}{2} = 0.4615

Step 2 — recognize the trap. Neither 0.4623 (mean) nor 0.4615 (median) is close to any actual measurement in the data set — every real reading is either near 0.41 or near 0.51, and the "center" falls in a gap where no data point exists at all. Both the mean and the median are answering the question correctly on paper, but the question itself was wrong: this is not one population with random scatter, it is two distinct populations (evidently one lab section's dilution or cuvette blank differed systematically from the other's — a classic bimodal data set from a systematic, not random, error). A histogram would show two separate clusters. Step 3 — mode identification confirms it. If the readings are grouped in bins of width 0.01 (0.405–0.415, 0.505–0515, etc.), the bin 0.409–0.418 contains six points and the bin 0.508–0.515 contains six points — a bimodal distribution, with two local maxima of frequency and no single dominant mode. This is precisely the case where reporting a mean or median as "the result" is statistically meaningless and analytically misleading; the correct action is to separate the two sub-populations (identify which lab section produced which cluster, find and fix the systematic error — likely a blank or calibration issue), and then compute mean, median for each cluster separately:

\bar{x}_1 = \frac{0.412+0.415+0.409+0.418+0.411+0.414}{6} = \frac{2.479}{6} = 0.4132

\bar{x}_2 = \frac{0.512+0.508+0.515+0.509+0.511+0.513}{6} = \frac{3.068}{6} = 0.5113

Only after separating the populations do the mean and median within each group become meaningful descriptors — a step that requires recognizing, from the shape of the data (visualized as a histogram or dot plot), that a single number was never appropriate for the pooled set in the first place.

Limiting cases, scope & common traps

Situation	Best measure	Why
Symmetric, unimodal, outlier-free replicate data (typical clean titration/gravimetric series)	Mean	Uses all data, statistically efficient, minimizes squared error; matches median and mode closely
Data with one or a few extreme values (contaminated, or a genuine but rare skew)	Median (after a formal outlier test, e.g. Q-test, if rejecting a point)	Robust — insensitive to the magnitude of extreme points
Skewed data (long tail on one side, e.g. trace-contaminant concentrations near a detection limit)	Median, or mean of $\log$ -transformed data	Mean is biased toward the tail; median reflects the "typical" value better
Discrete/coarse instrument readings, categorical data (most common integer pH reading, most frequent identified unknown)	Mode	Mean/median may fall between possible readings and be physically unattainable
Bimodal or multimodal pooled data (two populations mixed, systematic error between groups)	None — separate the populations first	A single mean/median can fall in a gap where no data exist and represents neither group
$n = 1$ (single measurement, no replicates)	Trivially, mean = median = the value itself; mode undefined	With one data point there is no distribution to summarize — central tendency requires $n \geq 2$
As $n \to \infty$ for a true normal population	Mean → true population mean $\mu$ ; median → $\mu$ as well (symmetric distribution)	For a genuinely normal population, mean and median coincide exactly in the limit — divergence in finite samples is either random sampling noise or evidence the population isn't normal

The chemist behind it

The formal vocabulary of mean, median, and mode as the three measures of central tendency is a product of the statistical revolution of the late 19th century, not of chemistry itself — but its adoption into quantitative chemical analysis was driven by chemists and statisticians working side by side on the problem of how to trust an instrument reading. Karl Pearson, working at University College London in the 1890s, formalized and named much of the vocabulary still used today — he coined the term "standard deviation," systematized moment-based measures of skewness (the mean–median–mode relation used above is his empirical rule), and pushed the discipline of biometry to demand quantitative rigor in place of qualitative description. Pearson's insistence that a "typical value" must be defined precisely, and that its choice must be justified by the shape of the data rather than assumed, is the direct ancestor of the modern analytical-chemistry practice of testing data for normality and outliers before reporting a mean. Francis Galton, Pearson's mentor, had earlier introduced the median as a robust alternative to the mean while studying inherited traits, explicitly motivated by data sets — like human height or intelligence test scores — that are not symmetric and are easily distorted by a few extreme individuals. The same logic transfers directly onto a chemist's bench: a burette reading contaminated by a stuck stopcock is no different, statistically, from an outlying data point in a biometric survey.

Origins · 1 minKarl Pearson and how this was discovered

Karl Pearson

1857–1936 · founder of mathematical statistics, University College London

Pearson trained first in law and mathematics at Cambridge before turning to statistics, founding the journal Biometrika in 1901 and the world's first university statistics department at UCL in 1911. Beyond naming the standard deviation and formalizing skewness, he developed the chi-squared goodness-of-fit test — still the standard tool for testing whether experimental data (including analytical chemistry replicate sets) follow an assumed distribution. His rigorous, data-first insistence on quantifying "how typical is typical" reshaped every quantitative science that depends on replicate measurement, chemistry included.

History

- 1846 — Adolphe Quetelet applies the "average man" concept and the normal curve of errors to large human measurement data sets, an early precursor to modern descriptive statistics.

- 1874 — Francis Galton popularizes the median as a robust summary statistic while studying skewed biometric data (e.g., human stature).

- 1893 — Karl Pearson introduces the term "standard deviation" and systematizes skewness measures relating mean, median, and mode.

- 1908 — William Sealy Gosset ("Student") develops small-sample statistics (the t-distribution) directly motivated by quality control at the Guinness brewery, extending these ideas to the small-

n

replicate measurements typical of a chemistry lab.

- 1920s–1930s — Statistical quality control and outlier-rejection tests (precursors to the modern Q-test and Grubbs' test) are formalized for industrial and analytical measurement, cementing mean/median/outlier logic into standard laboratory practice.

Are mean, median, and mode always close together for real lab data?

No — they coincide closely only for data that are roughly symmetric and free of outliers, which describes many clean replicate titration or gravimetric series but by no means all real data. Skewed data (a long tail on one side, common near a detection limit or with trace contaminants) pulls the mean toward the tail while the median stays near the bulk of the data. A single contaminated point can shift the mean substantially while barely moving the median. When the three measures diverge noticeably, that divergence is itself diagnostic information: it signals skew, contamination, or — in the most dramatic case — a bimodal data set representing two different underlying populations that should never have been pooled and averaged together in the first place.

Why is the mean usually preferred over the median in analytical chemistry, if the median is more robust?

Because the mean is the mathematically efficient choice — the one derivable formula shown above proves it minimizes squared deviation — and virtually every downstream statistical tool used in analytical chemistry, from standard deviation to confidence intervals to propagation of error, is built around the mean as the reference center. The median's robustness is a safety feature for contaminated or skewed data, not a general-purpose replacement; using the median routinely when data are actually clean throws away real statistical information contained in the exact magnitudes of every point, since the median only uses rank order. The correct workflow is to default to the mean, but to actively check for outliers and skew (visually, or with a formal test) before trusting it.

Can a data set have more than one mode?

Yes. A data set with two values tied for highest frequency is called bimodal, and one with three or more tied maxima is multimodal. In chemistry, a genuinely bimodal frequency distribution most often signals that two different populations have been mixed together — two lab sections with a systematic calibration difference, two polymorphic forms of a crystalline solid with different melting points pooled in one DSC data set, or particle sizes from two different sieving fractions accidentally combined. Recognizing bimodality (usually easiest from a histogram or dot plot) is itself valuable diagnostic information, often more useful analytically than any single "central" number computed from the pooled data.

How does the choice of central tendency connect to significant figures and reported precision?

The number of significant figures carried in a reported mean should reflect the actual precision achieved, not simply how many digits a calculator displays. As a working rule, averaging $n$ replicate readings improves the precision of the mean by roughly a factor of $\sqrt{n}$ relative to a single reading (this becomes rigorous once Standard Deviation and Variance defines the standard error of the mean), which occasionally justifies reporting one extra digit in $\bar{x}$ beyond what a single measurement supports — but never more than that, and never simply because the raw sum happened to produce more digits.

Is the median always a value that actually appears in the data set?

Only when the number of data points $n$ is odd — then the median is literally the middle-ranked data point itself. When $n$ is even, the median is defined as the average of the two central order statistics, and this average frequently does not equal any real measurement in the set (as seen with the 24.995 mL median in Example 1, which was never actually measured). This is an important practical distinction: an odd-numbered median is a genuine observed value with all the physical meaning that implies, while an even-numbered median is a computed midpoint that may fall in a region where the instrument or method was never actually tested.

Why doesn't the empirical relation mean − mode ≈ 3(mean − median) always hold exactly?

Because it is Pearson's empirical approximation, derived from observing many real skewed distributions, not an exact algebraic identity following from the definitions of the three statistics. It works reasonably well for moderately skewed, unimodal distributions and gives a fast diagnostic — a rough numerical check on the direction and rough size of skew — without requiring the full machinery of a skewness coefficient computed from third moments. It should never be treated as a rigorous formula the way the least-squares derivation of the mean is; it is a rule of thumb, useful for a first pass, not a proof.

Should the mode ever be reported alongside the mean and standard deviation in a formal analytical report?

Rarely, and only when the data are genuinely discrete or categorical in nature — for example, reporting that the modal number of significant figures used across a class's lab reports was three, or that the modal identified unknown compound in a qualitative analysis exercise was compound X. For continuous instrumental measurements reported to appropriate precision, the mode is usually either undefined (no repeats) or an artifact of instrument resolution rather than a meaningful chemical result, so standard reporting practice centers on the mean (or median, if outliers or skew are documented) together with a measure of spread, leaving the mode as a supplementary observation at most.