Paired Period Calculator

Time one cycle or many. Keep the raw readings, compare periods in the same units, and see what the spread can — and cannot — tell you about your measurements.

Synthetic readings constructed by adding 0.3 seconds per cycle to readings centered on 2 seconds. Both methods agree around 2.3 seconds; the data alone cannot identify that shared offset.

Different intervals. Comparable periods.

Can both methods agree and both be wrong?

One cycle: 2.28 s ÷ 1 cycle= 2.28 s/cycle
Twenty cycles: 45.6 s ÷ 20 cycles= 2.28 s/cycle

One matched pair: Pair 1. Divide elapsed time by complete cycles before comparing methods. The raw totals remain in the table.

Complete pairs: 5
First only: 0
Second only: 0
Missing both: 0

A · One cycle

B · Twenty cycles

Same complete pairs, same vertical scale. Sloping lines show within-pair changes; vertical clustering shows each method’s spread. This is not a time-series plot or a test of accuracy.

The sample spreads are equal within numerical precision.

Spread describes the readings, not their accuracy. The same 5 complete pairs are used in every summary below; unmatched observations are excluded from these comparisons. All summary values are in seconds per cycle.

Complete-pair summaries, all in seconds per cycle. Sample SD is not standard error.
Method	Mean	Sample SD	Conditional SEM
One cycle	2.3	0.0158114	Independence not assumed
Twenty cycles	2.3	0.0158114	Independence not assumed
Paired difference (second − first)	8.88178e-17	1.98603e-16	Independence not assumed

SEM is not calculated: independence has not been assumed. It is not total measurement uncertainty. Shared bias and instrument effects are not estimated; even zero SEM does not establish zero uncertainty.

Raw timings and normalized periods · all 5 rows

Raw totals and cycle counts are retained. Calculated values are displayed to six significant digits; small nonzero values use scientific notation. All periods and paired differences are in seconds per cycle. Missing observations are not zeros.

Every submitted pair, including incomplete observations. Normalized periods and differences are in seconds per cycle.
Pair ID	One cycle: raw ÷ cycles	A period	Twenty cycles: raw ÷ cycles	B period	B − A
Pair 1	2.28 s ÷ 1 cycle	2.28	45.6 s ÷ 20 cycles	2.28	4.44089e-16
Pair 2	2.29 s ÷ 1 cycle	2.29	45.8 s ÷ 20 cycles	2.29	0
Pair 3	2.3 s ÷ 1 cycle	2.3	46 s ÷ 20 cycles	2.3	0
Pair 4	2.31 s ÷ 1 cycle	2.31	46.2 s ÷ 20 cycles	2.31	0
Pair 5	2.32 s ÷ 1 cycle	2.32	46.4 s ÷ 20 cycles	2.32	0

Use your paired readings

Results above use the last calculated data. Apply edits to recalculate; retain each method’s raw elapsed totals, not periods you have already divided.

Comparison title

Method A

First method

First elapsed-time unit

Cycles timed by the first method

Method B

Second method

Second elapsed-time unit

Cycles timed by the second method

Paired elapsed times

Paste three tab-separated columns, without a header: anonymous pair ID, first elapsed time, second elapsed time. Leave a duration blank for a missing observation. Each row must be a genuine matched pair; rows are not matched by sorted value. Up to 600 rows. No comma-separated format is inferred.

Statistical assumptions

Choose the assumption only for independent repeated pairs measuring the same stable quantity under unchanged conditions within each method. Different pendulums or repeated trials from one person are not automatically independent repeats. Dependence within a matched pair is allowed.

Use anonymous IDs, not student names. This form sends the table for calculation without putting raw readings in the address bar. It works without JavaScript or an account.

A narrower spread is not proof of a better answer

Period = elapsed time ÷ complete cycles. Millisecond totals are first converted to seconds (1,000 ms = 1 s). Sample SD describes spread among readings and uses n − 1 in the variance denominator. Conditional SEM = sample SD ÷ √n. For the change between methods, it uses the SD of the paired differences, not an assumption that the methods are independent.

Timing more cycles can reduce the effect of a fixed total-timing uncertainty after division, but counting errors, changing motion and shared bias can remain. This calculator does not simulate a pendulum, acquire stopwatch readings or calculate total measurement uncertainty. Its educational input limits are 1 microsecond to 1 day and 1–10,000 complete cycles.

Read NIST’s Type A uncertainty guidance, NIST’s paired-data explanation, and JILA’s SPRUCE objectives. These are teaching references, not endorsements or a claim of validated learning gains.

Explore the physics classroom library. This calculator does not yet create or save a live paired-measurement lesson.