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Incoming links: Arithmetic mean
Past exam of the mathematics course of the University of Cambridge 2019 iii Paper 105 1 a Solution 2026-10-03
Write . Since has arithmetic mean zero,and the pairwise-difference identity givesFirst take smooth. Join to by changing one coordinate at a time and apply the Cauchy-Schwarz inequality:For a one-dimensional slice , the fundamental theorem of calculus yieldsIntegrating the th summand over therefore gives at most . HenceThe density of smooth functions in a Sobolev space extends the estimate to every . Thus
Past exam of the mathematics course of the University of Cambridge 2025 iii Paper 218 1 f Solution Created 2026-09-24 Updated 2026-09-25
Among all depth-one regression tree splits, the best split separates the second observation from the first and third by cutting advert1 between and . The test point with advert1 equal to zero reaches the leaf containing responses and , so the output is their arithmetic mean,This decision stump makes a piecewise-constant prediction far outside the observed predictor range and cannot extrapolate the spending trend towards the origin. Its shallow structure and leaf averaging keep its variance of an estimator modest, while that extrapolation failure can produce substantial bias of an estimator.
Write for row of the data matrix. The k-means clustering objective is the within-cluster sum of squared Euclidean distances:The nonempty form a partition of a set of observation indices. For fixed clusters, part (a) shows that the best is their centroid.
Initialize distinct centres, for example by choosing distinct observations. Then repeat the following steps:
- Assign each observation to a nearest centre in Euclidean distance. Keep its current assignment on an exact tie; for a previously unassigned observation use a fixed ordering to resolve ties.
- If a cluster is empty, choose an observation from a cluster with at least two members and make it the empty cluster's sole member, placing its centre at that observation. A definite rule is to choose an eligible observation farthest from its currently assigned centre, breaking ties by observation index. Repeat until all clusters are nonempty.
- Replace each centre by the arithmetic mean of its assigned observations.
- After updating, check whether every observation is still assigned to a nearest updated centre, retaining its assignment on ties. If so, stop; otherwise repeat.
The empty-cluster repair is possible for and cannot increase : the transferred observation now has squared distance zero, while all other squared distances stay unchanged before the centre update.
Assignment cannot increase , and any changed assignment strictly decreases it under the tie rule. Updating centres cannot increase by the centroid sum-of-squares identity. After an update, the centres are determined by the partition of a set. There are only finitely many possible assignments. Unless this is the initial round, any nonterminal round starts with an assignment that strictly reduces ; repairing empty clusters and updating cannot undo that decrease. Thus a partition cannot recur, proving finite termination of K-means clustering. The final result need not be a global minimum; different initial centres can lead to different final clusterings.
Sort the distinct observations as . For ordered centres , a point is closer to than to precisely when it lies to the left of their midpoint. Thus each Voronoi cell is an interval, with either choice allowed for an endpoint tie. Consequently, after assignment, each nonempty cluster is a contiguous block of the sorted observations.
The arithmetic means of disjoint ordered blocks are strictly ordered, so updating centres preserves this order. The empty-cluster repair in part (b) preserves contiguous blocks too: an observation farthest from its assigned centre lies at an endpoint of its block, and removing that endpoint makes a new singleton block. In one dimension with distinct observations, choosing distinct observations as initial centres makes the first assignment nonempty.
The one-dimensional K-means iteration bound now follows by counting block partitions. Choosing the gaps between consecutive nonempty blocks givespossibilities. By the strict-decrease argument in part (b), no such partition can recur before algorithm termination. Counting a step as an assignment, any empty-cluster repairs, and a centre update, with the stability check performed after that update, givesFor , one update to the overall arithmetic mean suffices. Arbitrary changes between tied centres would invalidate the strict-decrease argument, which is why the stable tie convention in part (b) matters.
For , choose the uniform partition of an interval into pieces and its midpoint tagsTheir arithmetic mean is exactly . Apply the given Jensen inequality with equal weights :Multiplying by gives a midpoint Riemann sum on the left. Its mesh of a partition is , so the Riemann sum convergence theorem givesFor the equality is immediate. Together with part i, this is the Hermite-Hadamard inequality with the directions appropriate to a concave function.
Past exam of the optimisation for data science course of the University of Oxford 2023 2 f Solution 2026-10-25
The printed distance needs a square. The native PDF, paper page 3, prints in the final bound, but the general conclusion from (5) contains . We first prove the corrected statement and then show that the printed version can fail.
Take the expectation in (5), use the tower property, and sum over . The squared Euclidean distances form a telescoping sum:Discard the final nonnegative squared Euclidean distance. The smallest of numbers is at most their arithmetic mean, soThis is an bound above the stochastic gradient noise floor for the best expected objective function value. By Jensen's inequality, it also bounds for the averaged iterate , because is a convex function.
To test the printed version, take one component function, , with , and . This convex quadratic function has , hence it has a -Lipschitz gradient. The stochastic gradient descent update is deterministic, its conditional variance is zero, and the assumed positive bound is valid. At , the new iterate is , soThus the printed inequality is false under its stated assumptions. The squared Euclidean distance in the corrected bound resolves the defect.
Maintain the latest five temperature readings in a circular buffer, together with their running sum. The following C programming language code spells out the message passing interface rather than assuming a particular library:
send(dst,&m) is a blocking synchronous message passing send; receive(src,&m) receives from the specified sender or ANY and returns the sender's identity; sendrec(dst,&m) atomically sends the request and waits for that sender's reply in the same message object. Successful, reliable operations are assumed. Message contents are copied by the interface.enum { ANY = -1, TIMER = 1, TEMP = 2 };
enum { PULSE, PING, REQUEST, REPLY };
typedef struct {
int type;
int period_ms;
double value;
} Message;
extern void send(int dst, const Message *m);
extern int receive(int src, Message *m);
extern void sendrec(int dst, Message *m);
void smooth(void) {
double history[5] = {0};
double sum = 0;
int count = 0, next = 0;
Message m = { .type = PULSE, .period_ms = 1000 };
send(TIMER, &m); /* subscribe once */
for (;;) {
int sender = receive(count < 5 ? TIMER : ANY, &m);
if (sender == TIMER && m.type == PING) {
Message t = { .type = REQUEST };
sendrec(TEMP, &t); /* matching REPLY contains t.value */
if (count == 5)
sum -= history[next];
else
++count;
history[next] = t.value;
sum += t.value;
next = (next + 1) % 5;
} else if (count == 5 && m.type == REQUEST) {
Message answer = { .type = REPLY, .value = sum / 5.0 };
send(sender, &answer);
}
}
}Before the fifth reading, selective reception from
TIMER prevents premature client requests from being accepted. Such clients wait at their synchronous sends; they are not given a partial arithmetic mean. Thereafter the operating-system process receives both timer messages and client requests. Each accepted PING initiates one measurement; each accepted client request returns the arithmetic mean of the five most recently completed measurements, without initiating another measurement or changing the circular buffer.The loop invariant is that
count is the number of valid readings, capped at five, sum is their sum, and next is the next unused slot during startup or the oldest slot afterwards. Removing the overwritten reading and adding the new one preserves this loop invariant. Each update and reply uses time complexity and storage, since the window length is fixed. Under timely sensor replies, timer delivery and process scheduling, measurements are initiated once per second. The next parts explain failures of these timing assumptions; the code does not promise one-second sampling when the sensor itself cannot sustain it.The arithmetic mean is , the total measured concentration divided by the number of observations. The median is the central value of the sorted observations, or an interpolation between the two central values under the chosen even-sample convention. It is the 50th percentile.
The table's “75th quartile” means the 75th percentile, or upper quartile , with approximately three quarters of observations at or below it. Its “25th quartile” means the 25th percentile, or lower quartile . There are four quarters of a distribution, rather than 75 different quartiles. Quantile interpolation rules can differ for a finite sample; their intended population interpretation is unchanged.
TODO: Attachment C is absent. Check its contents before treating this as a source-complete answer; the computation below uses the printed table, independent observations, a common within-season variance model and an explicitly chosen significance level.
For a one-factor analysis of variance, let be the four seasonal sizes, means and standard deviations. The weighted grand arithmetic mean isThis agrees with the rounded annual mean . The between-season and within-season sums of squares areThere are between-season statistical degrees of freedom, because the four fitted seasonal means are compared subject to their weighted grand-mean constraint. There are within-season statistical degrees of freedom, because one mean was estimated in each group. The total has , equal to their sum. HenceUnder , compare with the F-distribution on statistical degrees of freedom. The critical value is about , with p-value approximately from the rounded summaries. Reject equality of all four means very strongly under the stated model. The omnibus hypothesis test establishes that at least one mean differs; it does not identify every pairwise difference.
The summer variance is substantially lower than the other three, so the exact common-variance ANOVA assumptions deserve checking, as does temporal independence. Raw observations would permit diagnostics and a heteroscedastic or time-dependent analysis. As a cross-check on arithmetic, implies annual sample standard deviation , matching the table's . The numerical significance must not be mistaken for verification of the unobserved Attachment C or of the sampling model.
The null space consists of concentration patterns on which transport produces no change. Here it is the familyEvery box has the same concentration. Ocean circulation may continue, but inflow and outflow of the chemical tracer balance in each box. With these are exactly the steady states; physically admissible members have . With nonzero , adding to any particular steady state leaves its transport balance unchanged, but a null space vector alone need not balance the sources and sinks.
To identify the tracer inventory, apply conserved inventory from a left null vector. Directly summing each column givesConsequently, for the supplied ordinary differential equations,For , is a conserved quantity. The connected box model equalizes concentrations, approaching the unique member of the null space with the initial inventory:The arithmetic mean is appropriate here because the supplied matrix has zero column sums. In a general box model the conserved quantity is , where is a positive left null space vector; for physical concentrations expressed per unit volume, these weights represent the box volumes up to a common factor. One must establish the weights from the equations, rather than infer equal volumes from a schematic drawing.
Nonnegative initial concentrations remain nonnegative under transport alone. Indeed is a nonnegative matrix, sohas nonnegative entries for : it is a positive matrix semigroup. Its column sums are one by . To see why equalization occurs, set for ; then andThe right side vanishes only for constant , which in the zero-sum vector subspace means . Its strict negative bound on the unit sphere in that vector subspace gives decay to zero. Thus the matrix exponential redistributes inventory without creating negative concentrations; arbitrary prescribed sinks can still exhaust the tracer if they are allowed to act indefinitely.
Pauling bond-energy excess 2026-10-14
A Pauling bond-energy excess subtracts a chosen like-atom baseline from the unlike-atom bond-dissociation energy. For an unequal-input arithmetic example, stipulate , and , all with one dissociation convention. The arithmetic mean baseline and excess areThe arithmetic-baseline Pauling scale model therefore gives . The bond-dissociation energy is not itself the excess: subtracting the baseline is essential. All three numbers are stipulated, not experimental values for named chemical elements.
Constructing a Pauling bond-energy excess from stipulated bond-dissociation energies and converting it to a dimensionless electronegativity difference
. Details. Changing the baseline changes the model. For the same stipulated inputs, a geometric mean baseline would be , yielding an excess of and a difference of approximately . It is a different construction, even though both start from the same bond-dissociation energies. State which baseline is used before comparing electronegativity differences.
The arithmetic-baseline Pauling scale also has a useful inverse: a difference of corresponds to an excess of , and a difference of corresponds to . Doubling the electronegativity difference quadruples the modeled excess. Interchanging the endpoint labels changes neither quantity; the square root alone cannot choose the direction of bond polarization.
In The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms (printed pp. 3576–3578, Table II and equation (1)), Linus Pauling organized excess energies for chemical bonds into an approximately additive map of their square roots. He also identified a limitation: his H–F map separation of predicted an excess of , exceeding his table's value by . Those are historical values and a historical model discrepancy, not modern recommended inputs.[ref]
The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms by Linus Pauling. printed pp. 3576–3578, Table II, equation (1), and paragraph beginning “These coördinates”
Pauling electronegativity Created 2026-10-13 Updated 2026-10-20
Pauling electronegativity is a relative, dimensionless scale motivated by bond-dissociation energies. The interesting observation is that a chemical bond between unlike chemical atoms can be stronger than a simple baseline inferred from the corresponding like-atom chemical bonds. An empirical excess energy can then be associated with an electronegativity difference. The 1932 paper The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms built such a map; its original coordinates used hydrogen as zero, so those coordinates should not be mistaken for the modern tabulated Pauling electronegativities.[ref]
One arithmetic-baseline version of the Pauling scale relation isHere , and are consistently defined bond-dissociation energies per molecule, and the arithmetic mean is the chosen baseline. With molar energies instead, write , approximately . The denominator makes the result dimensionless; the factor is not a unit-independent magic constant. This is an empirical relation with a specified baseline, not a definition valid for every chemical bond.[ref]
Energy Electronegativity and Chemical Bonding by Stepan S. Batsanov. §1 equation (1), coefficients for eV, kcal/mol and kJ/mol; §2.2, dependence on other bond properties
For an explicitly synthetic example, suppose and with the same temperature and dissociation convention. Then and the model gives an electronegativity difference of . Replacing by gives zero; a negative excess would make this real square-root model inapplicable, not produce a negative electronegativity difference. These invented inputs illustrate the formula and are not measurements of named chemical substances.
Only differences appear, so adding the same constant to every Pauling electronegativity leaves this relation unchanged. A reference value must fix the origin. The bond-dissociation energies alone also do not tell which endpoint attracts electron density more strongly, because interchanging and leaves the formula unchanged. Chemical comparisons fix that direction. Thus Pauling electronegativity is useful as a compact ordering, while the underlying measured energies and approximations remain essential.
Consider the discrete linear system withThis is a finite periodic analogue of the Poisson equation, using the negative second finite difference with unit grid spacing. It is also a Graph Laplacian equation on a cycle graph; no continuum approximation is being claimed here.
Summing the four equations gives zero on the left, since each value occurs twice positively and twice negatively. A necessary condition is therefore . The Fourier diagonalization of a circulant matrix shows that this is also sufficient: the only zero eigenvalue of is the constant mode , and .
For the displayed forcing, the discrete Fourier transform givesDivide only at the three nonzero eigenvalues, and choose to impose zero arithmetic mean:Direct substitution gives . All other vectors satisfying the linear system are , because the kernel is the space of constant vectors. Thus balanced forcing determines the differences between values, while a reference level is an extra choice. This small model explains why a periodic Poisson equation needs both a compatibility condition and a normalization.
Periodic sequence Created 2026-10-14 Updated 2026-10-20
A sequence indexed from zero with an integer period satisfying the displayed identity for all . Repeating a finite cycle is an example. The Cesàro mean article explains why its cumulative averages approach the one-cycle arithmetic mean.
Phase folding of a light curve 2026-10-14
Phase folding of a light curve maps timestamps onto a trial cycle of duration . Given a reference time , define the phase in cycles bywhere is the fractional part, using the floor function also for negative . Plot brightness against rather than ; measurements separated by an integer multiple of occupy the same phase. For a periodic function , this reveals the shared shape of many cycles. Folding changes coordinates; it neither averages samples nor chooses the correct by itself. Welsh et al.'s folded Kepler plot separates the exoplanet transit and orbital brightness modulation. [ref]
https://www.ucolick.org/~jfortney/papers/Welsh10.pdf. Welsh et al. (2010), figure 1
.The second figure uses a completely synthetic trapezoidal dip: days, day, and the same depth and four-hour duration as the first figure. Different colours identify four successive cycles. Correct folding superimposes the dips; a wrong trial orbital period spreads them out. If , the centre of the th repeated event shifts by approximately cycles when folded on , for . With days, the fourth plotted event is displaced by cycles. Transit-timing variations can also smear a constant-period fold, so retain the original timestamps and inspect individual events.
Toy phase folding: repeated synthetic dips align at the correct period; a flux dip becomes a positive magnitude change
. Details. One may combine nearby phases with a weighted mean, but the uncertainty reduction requires assumptions. For independent measurements with common standard deviation and effectively the same expected brightness within the bin, their arithmetic mean has standard deviation . Thus sixteen independent -precision samples give precision on that phase-bin mean. Shared calibration errors, changing starspots, and bins wide enough to mix ingress with out-of-transit brightness violate these assumptions. A folded plot can conceal changes between cycles. Gaussian process models can represent correlated variability and interpolate irregular samples, but their predictions and uncertainties depend on their assumed covariance function.
The bottom panels show the same model as relative brightness and as magnitude difference in one fixed photometric passband with a fixed magnitude zero point:A dip to gives magnitudes: fainter means a larger apparent magnitude. The figure places larger magnitude differences lower on the page so the two dip shapes are easy to compare; reversing that plotting convention does not change the formula. For and small , expansion of the logarithm gives . The exact transform is nonlinear: transforming an exposure's mean brightness is generally different from averaging its instantaneous apparent magnitudes. The ordinary logarithm cannot be applied to a nonpositive background-subtracted measurement; keep such measurements and their uncertainties in brightness units for an appropriate statistical model.
Pollen tube germination assay Created 2026-10-14 Updated 2026-11-05
A pollen tube germination assay makes tube emergence visible outside a flower. The published Practical Biology experiment, Observing the growth of pollen tubes (Practical Biology), Technical note 3, lists florist lilies among successful materials without specifying a species. Record the actual identification; success with one sample cannot establish a universal growth rate.[ref]
Observing the growth of pollen tubes (Practical Biology). Technical note 3 and Teaching notes: material dependence and contorted versus straight tubes
Pearson's Core practical 4: Investigate the effect of sucrose concentration on pollen tube growth by Pearson Education Ltd, Teacher sheet, pp. 1–2, describes an inspectable experiment. Mix equal volumes of sucrose solution and mineral-salt medium on clean cavity slides. Dust in pollen from mature anthers with a mounted needle, record the addition time, and incubate without coverslips in covered dishes containing wet filter paper. The moist chamber reduces drying; the uncovered drops avoid oxygen limitation. Brief observations with an optical microscope at , using an eyepiece graticule calibrated against a stage micrometer, reveal tube emergence and length. Return slides to the chambers and switch off the lamp between observations to reduce heating and drying. For comparisons of sucrose concentration, use the same flower across treatments to control maturity, then repeat with other flowers.[ref]
An additional proposed experimental design check is a medium-only slide: it tests whether debris is being mistaken for pollen tubes. Compare matched pollen samples at different final sucrose concentrations, keeping temperature, elapsed time and observation rules fixed. A sample known to produce pollen tubes in a previously successful medium would test whether an apparent failure comes from handling or the tested condition. These controls are questions to test, not guaranteed outcomes. A negative result cannot distinguish dead pollen from an unsuitable medium without further comparisons.
Two original calculations clarify the readout. First, take of sucrose solution and of sucrose-free medium. Assuming additive volumes, conservation of the amount of substance givesThis dilution also halves the medium's original component concentrations. Label the final molar concentration, not the starting bottle.
Second, stipulate a toy field containing pollen grains, of which have emerged pollen tubes. Its emergence fraction is . If those twelve pollen tubes have total measured length , their arithmetic mean length is . Assigning length zero to the other grains instead gives per grain. These invented numbers show why emergence fraction and mean length among emerged tubes answer different questions. Specify the denominator and the emergence criterion before counting, and avoid selecting only conspicuous long pollen tubes.
Tube emergence in a drop demonstrates growth under that medium's conditions. It observes no stigma interaction, passage through a style, delivery into an ovule, gamete fusion, or seed development. Thus a positive pollen tube germination assay does not demonstrate successful fertilization in a living flower. Its beauty is making one otherwise hidden part of sexual reproduction directly observable.[ref]
Observing the growth of pollen tubes (Practical Biology). Teaching notes: steps between pollen germination and seed formation; prepared style slides as a separate observation
Population variance Created 2026-10-20 Updated 2026-11-05
For a finite population with arithmetic mean , its variance is . It averages squared deviations over the whole population.
Positioning repeatability 2026-11-05
Positioning repeatability describes the scatter of positions reached when a motion system repeatedly returns to the same commanded position under specified conditions. It is an application of measurement repeatability, a form of measurement precision. A system may return to almost the same wrong position each time, so small scatter does not establish measurement accuracy.[ref]
International vocabulary of metrology by JCGM. VIM3 entries2.15,2.20 and2.21: precision and repeatability conditions
The approach direction belongs in the test description. Unidirectional positioning repeatability keeps it fixed; bidirectional positioning repeatability includes returns from both directions. To measure the assembled stage, independently observe its final position rather than only the motor's rotary encoder count. Physik Instrumente describes a static test that lets the platform stop before reading it, using an interferometer for displacement and an autocollimator for angular error, with two reflecting mirrors on the stage.[ref]
How to Consistently Verify the Performance of Nanopositioning Motion Control Equipment by Physik Instrumente. §§3.1–3.2, static readout, direction convention and paired-mirror apparatus
For an original numerical example, repeatedly command . Suppose an independent position readout with known measurement calibration gives the errors , in micrometres,Each direction has the same sample standard deviation,Yet their arithmetic means are 0 and . The magnitude of their mean difference, a positioning reversal error at this target, is 2.0 micrometres, far larger than the within-direction scatter. Pooling all eight values gives a sample standard deviation of ; that statistic mixes the directional shift with the smaller return-to-return variation.
Two tightly repeated return positions separated by approach direction
. Original chosen errors at one target. Four trials per direction; small bars show mean plus or minus one sample standard deviation. The figure is not measured product performance. Details.The calculation uses a declared sample standard deviation, not an unspecified manufacturer's tolerance or a universal repeatability convention. A positioning specification must state its approach directions, target positions, settling/readout procedure, load, environmental conditions, statistic and measurement uncertainty. A single target does not establish performance over the travel range. The manufacturer's method explicitly separates directional means from scatter and combines information from multiple targets when forming axis-level specifications.[ref]
Backlash is one possible cause of direction-dependent behavior, while thermal expansion and instrumental drift can move the apparent target over time. A repeatability experiment must distinguish those effects from random scatter. Measuring small returns to one target also answers a different question from minimum incremental motion, which asks whether a small commanded change can actually be executed.
Prefix sum Created 2026-09-24 Updated 2026-10-14
A prefix sum stores the accumulated sum through a specified position in a finite sequence or array. For inputs , include the empty sum and setFor the original input , the prefix array including is . A negative entry makes the accumulated sum decrease; a prefix sum does not have to be monotone. Conversely, the original entry is recovered by subtraction: .
The useful cancellation is that any contiguous block has sumBoth prefixes include the entries before ; subtracting cancels those entries. In the example, the sum from positions two through four is , recovered as . The arithmetic mean of that block is , obtained by dividing its sum by its length. Prefix sums therefore make many repeated questions about an unchanged array cheap: reuse accumulated totals instead of resumming every block.
The sequential recurrence builds all prefixes in time complexity and storage; each block query needs two reads and one subtraction, hence time complexity. These counts assume constant-cost arithmetic and access to array entries. If one input changes, all later stored prefixes change, so this simple representation is most useful when queries outnumber updates. For integer inputs, sufficient numeric capacity must be retained; with floating-point arithmetic, subtracting large nearby prefixes can lose relative accuracy for a small block total. The exact mathematical identity does not eliminate representation error.
The apparently sequential recurrence also motivates a prefix scan: associativity permits a tree of partial calculations that yields all accumulated results in parallel. Guy E. Blelloch's 1990 technical report systematically develops this operation and its applications; its authored chapter version explains the two tree sweeps used to obtain all prefixes. [ref]
https://www.cs.cmu.edu/afs/cs.cmu.edu/project/scandal/public/papers/CMU-CS-90-190.html. Blelloch, Prefix Sums and Their Applications, CMU-CS-90-190, November 1990
. [ref]https://www.cs.cmu.edu/~guyb/papers/Ble93.pdf. Blelloch, authored chapter version, §§1.1–1.2: sequential scans and parallel tree sweeps
. Process scheduling 2026-10-13
Process scheduling chooses which runnable operating-system process or software thread uses a CPU next, and when it gives up that CPU. Runnable means able to execute now; work waiting for an input-output operation is not eligible until that wait ends. A ready queue records eligible work. The policy chooses a recipient; a context switch supplies the mechanism for transferring execution. Sharing a single CPU lets a long calculation and an interactive program both make progress, although only one execution stream runs on that CPU at an instant.
Consider three single-software thread operating-system processes on one CPU, with abstract time units, zero context switch or scheduler overhead, and no input-output blocking. Their arrival times and CPU bursts are , and . Each CPU burst is all the execution that its task needs. With FCFS, the current task runs to completion: A occupies , B and C . With round-robin scheduling and scheduling quantum , the intervals are A , B , C , A , B and A . At a boundary, arrivals join the ready queue before the unfinished task whose scheduling quantum has expired; this fixes the otherwise ambiguous ordering at time 2.
The following triples always list A, B, C. Completion is an absolute time; scheduling turnaround time, scheduling waiting time and scheduling response time are durations measured from the relevant events.
For example, B first runs at time 2 under round-robin scheduling, so its scheduling response time is . It finishes at time 8, giving scheduling turnaround time and scheduling waiting time . Under FCFS, C waits before its sole execution interval. Under round-robin scheduling, A waits during and , totaling . Both timelines do exactly units of useful CPU work; changing the order changes who benefits first.
The appeal is that shared progress can improve responsiveness: B and C get their first CPU service sooner under round-robin scheduling, while A completes later. The arithmetic mean of the scheduling response times falls from to ; the arithmetic mean of the scheduling turnaround times is for both policies in this particular example. These are different goals, and this workload establishes no universal ranking. Giving each continuously runnable task repeated turns is a practical form of fairness. A smaller scheduling quantum can shorten waits between turns, but makes context switches more frequent when competitors remain runnable. If each full slice performs units of useful work and incurs a fixed switching cost , the overhead fraction in that simplified repeated cycle is . Real costs and workloads determine the choice. [ref]