How many cycles should I time in a time study?

Enough that the average of each element is within the accuracy you need at the confidence you choose, usually within 5% of the true average at 95%. Time 5 or 10 cycles first, then n = (t × s ÷ (k × x̄))², where s is the standard deviation of those readings, x̄ their mean, k the accuracy as a fraction (0.05) and t the Student's t value for your confidence with one fewer degrees of freedom than readings. Ten readings averaging 11.3 seconds with s of 0.95 seconds need 15 cycles, so 5 more. A steady element needs few cycles; one that jumps around needs many.

Should I use t or z in the time study sample size formula?

Use t when the standard deviation comes from fewer than about 30 readings, which is almost every time study. Freivalds (Niebel's Methods, Standards, and Work Design) uses t with n − 1 degrees of freedom for this reason. With 10 readings t at 95% is 2.262 instead of z's 1.96, which asks for about a third more cycles. As the readings grow, t falls toward z, so recalculate after the extra cycles.

What is the ILO formula for the number of observations?

In Introduction to Work Study (ILO, 4th edition, 1992, chapter 21), the number of readings for a 95.45% confidence level and ±5% is n = (40 √(n′Σx² − (Σx)²) ÷ Σx)², where n′ is the number of preliminary readings and x each reading. The book's example: readings of 7, 6, 7, 7 and 6 hundredths of a minute give 8.81, so 9 readings, 4 more than were taken. The 40 is 2 ÷ 0.05: two standard errors (95.45%) over a 5% margin.

Why do the classic formulas say 95% but use 2 instead of 1.96?

Two standard errors either side of the mean cover 95.45% of a normal distribution, and the older work study texts round that to 95%. Using 2 with a 5% margin gives the 40 in the ILO and Barnes formulas. The difference from 1.96 is small: about 4% more cycles.

What should I do with an unusually long reading?

Find out why before you decide. If you saw a cause, such as a dropped part, a question from someone, a fumble or a timing error, mark the reading, write the cause on the study sheet and leave it out of the average. If there was no cause, it is part of the job's normal variation and stays in. The ILO calls such readings rogues and says exceptionally short ones need the same care. The calculator shows how much the reading furthest from the mean changes the answer.

Does the number of cycles make the standard time accurate?

Only the observed average. The sample size controls how close the average observed time is to the true average for that operator and method. The standard time also depends on the performance rating, which is a judgement, and on the allowances your site sets for rest, personal needs and contingencies. More cycles do not fix a wrong rating or a method that changed during the study.

Free tool

Time study sample size calculator

To know how many cycles to time, take a few preliminary readings of one element and use n = (t × s ÷ (k × x̄))²: s is the standard deviation of the readings, x̄ their mean, k the accuracy you want as a fraction of the mean and t the Student's t value for your confidence level. Ten readings averaging 11.3 seconds with s of 0.95 seconds need 15 cycles for ±5% at 95%, so 5 more. The calculator also gives the ILO formula and Barnes's range method, both fixed at about 95% and ±5%, and the General Electric rule of thumb by cycle length.

nequals(t × s divided by k × x̄)²

n (ILO)equals(40 √(n′Σx² − (Σx)²) divided by Σx)²

n (range)equals(40 × R divided by d2 × x̄)²

n is the number of cycles to time in all. s is the sample standard deviation of the preliminary readings, x̄ their mean, k the accuracy as a fraction of the mean (0.05 for ±5%) and t the two-sided Student's t for your confidence with n′ − 1 degrees of freedom, n′ being the number of preliminary readings (Freivalds, chapter 10). The ILO and range formulas are fixed at 95.45% and ±5%: 40 = 2 ÷ 0.05. R is the range of the preliminary readings and d2 the range-to-σ constant for their number (2.326 for 5, 3.078 for 10). Round n up.

Your preliminary readings

Paste the readingsType mean and s

Readings of one element

10 readings read. Separate them with spaces, commas or new lines, a point for decimals (up to 500). Leave out readings with a known interruption.

Time unit

SecondsMinutes

Confidence level

90%95%99%

Accuracy, ± % of the mean

5 is the usual figure

Whole work cycle (optional)

For the General Electric guide

Pre-filled with the worked example below: one element timed over 10 cycles. Run each element separately; the element that needs the most cycles sets the length of the study.

Cycles to time

For ±5% at 95% confidence. You have 10: time 5 more.

The working

Mean x̄ = 11.3 s, standard deviation s = 0.949 s from 10 readings.

t at 95% with 9 degrees of freedom = 2.262. k = 0.05.

n = (2.262 × 0.9487 ÷ (0.05 × 11.3))² = 14.43, rounded up to 15.

Accuracy you have now

±6%

Of the mean, at 95% confidence

With z instead of t

z = 1.960; right for 30 or more readings

The classic short methods, fixed at 95.45% (2 standard errors) and ±5%

ILO formula

n = 10.15; 11 in all: 1 more

Range method (Barnes)

R = 3, R/x̄ = 0.265

General Electric guide (whole cycle)

A rule of thumb by cycle length that ignores the spread

The reading furthest from the mean is 13 s (reading 7, 1.8 standard deviations from the mean). Without it the answer would be 11. Leave it out only if you saw a cause (a dropped part, a question, a fumble) and log it as an irregularity; if not, it is part of the job.

After the extra cycles, paste all the readings and recalculate: the mean and spread will change, and so may the answer (the ILO makes the same point).

Next step

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How to use it

From the first readings to the full study

1. Fix the elements first. Break the job into elements with an end point you can see or hear, and time a trained operator working the standard method. The sample size is for one element at one operator; another operator or method is another study.

2. Time 5 or 10 cycles. These are the preliminary readings (Barnes uses 5 or 10, the ILO example 5). Mark any reading with an interruption and leave it out.

3. Paste them and choose the confidence and accuracy. 95% and ±5% are the figures the ILO and Barnes use. Tighter accuracy costs a lot: halving the margin to ±2.5% needs four times the cycles.

4. Do it for every element. Each element gets its own answer. Since you time whole cycles, the element that needs the most sets the length of the study; the ILO says the same for cumulative timing.

5. Time the extra cycles and recalculate with all the readings. The mean and spread move as readings are added, and t gets smaller, so the answer can go up or down. Stop when the readings you have meet the answer.

Worked example

Illustrative numbers, not a benchmark

The element "fasten four screws" from the time study sheet, timed over 10 cycles in seconds: 11, 12, 10, 12, 11, 10, 13, 11, 12 and 11. The team wants the average within ±5% at 95% confidence. Illustrative numbers.

1Mean x̄ = 113 ÷ 10 = 11.3 s. Standard deviation s = √(8.1 ÷ 9) = 0.949 s. Range R = 13 − 10 = 3 s.
2t at 95% with 9 degrees of freedom = 2.262. k × x̄ = 0.05 × 11.3 = 0.565 s.
3n = (2.262 × 0.949 ÷ 0.565)² = 3.80² = 14.4, rounded up to 15 cycles: 5 more than the 10 already timed.
4The 10 readings already give ±2.262 × 0.949 ÷ (√10 × 11.3) = ±6.0%, which is why more are needed for ±5%.
5The classic short methods agree roughly: ILO (40 × √(10 × 1,285 − 113²) ÷ 113)² = (40 × 9 ÷ 113)² = 10.1, so 11; range method (40 × 3 ÷ (3.078 × 11.3))² = 11.9, so 12. They use 2 standard errors and the population-style spread, and ignore the extra uncertainty of only 10 readings, so they ask for fewer.

Time 5 more cycles, then recalculate with all 15. If the 13 s reading had a cause the observer saw, it is logged as an irregularity and left out, and the answer drops to 11. For a 45 s work cycle the General Electric guide says 40 cycles, a cautious rule that does not look at the spread.

What the number means

With n cycles, you can be 95% (or your chosen level) confident that the average you calculate is within ±k of the true average time of that element, for that operator, method and set-up. It says nothing about a single cycle: in the example below, single cycles still vary by about ±2 s (2 × s) around the 11.3 s average.

It does not cover differences between operators, shifts or materials. If those matter, study them separately, or sample across them on purpose and accept a larger spread.

When the formula breaks

Variation that is not random. The ILO says the method is valid only to the extent that the variation is due to chance and not made on purpose by the operator. A deliberate slowdown while being watched is not chance; nor is a steady drift as a new operator learns the job.
Two methods mixed. If the operator sometimes does the element a different way, the readings form two groups and the spread is meaningless. Split the element or fix the method first.
Too few readings. With 2 or 3 readings, s is a poor estimate. t allows for it, which is why the answer is so large. Time at least 5.
Skewed times. Manual elements often have a few long cycles and no very short ones. The formula assumes the average behaves normally, which holds better as n grows; with strongly skewed data, time more than the minimum.
Coarse readings. Whole seconds on a 3 s element hide the spread. If every reading is the same, read the watch more finely.

Abnormal readings and outliers

Ring any reading that is far higher or lower than the rest and write the reason next to it while you are still at the station. The ILO calls these rogues and says to examine them carefully: a long one may be a timing error, a fumble or a foreign element (something that is not part of the job), and a short one too may be the observer's error.

A cause you saw: leave it out of the average and log it as an irregularity.
No cause: keep it. Deleting real variation makes the standard too tight.
Occasional elements (changing a bin every 20 parts, say) are timed and counted separately, not mixed into the cycle readings.

The ILO does not drop the extra time of a rogue reading; it carries the time above the average into the contingency allowance.

From observed time to standard time

The sample size only makes the observed average reliable. The rest of the time study turns it into a standard:

Basic time = observed time × observed rating ÷ standard rating (the ILO uses 100 as the standard rating, chapter 22).
Standard time = basic time + allowances for relaxation, contingencies and any special allowances (ILO chapter 23).

Allowance percentages are set by each site or agreed with the workforce; this page does not give any. The time study sheet has the columns for rating, allowance and standard time.

The General Electric guide

Because working out each element is slow, the ILO notes that some authors and companies such as General Electric use a conventional guide by the length of the whole cycle (ILO Table 15; Freivalds Table 10.2):

Minutes per cycle	Cycles to time
Up to 0.10	200
Up to 0.25	100
Up to 0.50	60
Up to 0.75	40
Up to 1.00	30
Up to 2.00	20
Up to 5.00	15
Up to 10.00	10
Up to 20.00	8
Up to 40.00	5
Over 40.00	3

It is quick, but it ignores how much the element varies, so it can ask for far more or far fewer cycles than the formula. Use it to plan the study; use the formula to check it.

Sources and how this calculator was checked

Andris Freivalds, Niebel's Methods, Standards, and Work Design, 12th ed. (McGraw-Hill, 2009), chapter 10, Time Study: the t-based formula and Table 10.2, the General Electric guide.
International Labour Office, Introduction to Work Study, 4th rev. ed., ed. George Kanawaty (1992), chapter 21, section 8, Sample size, pp. 292 to 294, and Table 15; chapters 22 and 23 for rating, rogue readings and allowances.
Ralph M. Barnes, Motion and Time Study: Design and Measurement of Work, 7th ed. (Wiley, 1980), Table 13 (readings from the range of 5 or 10 readings, with d2 = 2.326 and 3.078).

Unit tests reproduce the ILO's worked example (8.81, so 9), the t values Freivalds prints (2.064 for 24 degrees of freedom, 2.145 for 14), Barnes's d2 constants and spot values of his table, the General Electric rows, and the example on this page. Edge cases tested: one reading, identical readings, 2 readings, accuracy 0.

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How LeanSuite helps

LeanSuite's AI time and motion study times every element from phone video and shows each element's time with its statistical variance, so timing 40 cycles means filming them rather than standing at the station with a stopwatch. Use this calculator to decide how many cycles the film needs to cover.

See Time and Motion Study

FAQ

Time study sample size calculator: common questions

Enough that the average of each element is within the accuracy you need at the confidence you choose, usually within 5% of the true average at 95%. Time 5 or 10 cycles first, then n = (t × s ÷ (k × x̄))², where s is the standard deviation of those readings, x̄ their mean, k the accuracy as a fraction (0.05) and t the Student's t value for your confidence with one fewer degrees of freedom than readings. Ten readings averaging 11.3 seconds with s of 0.95 seconds need 15 cycles, so 5 more. A steady element needs few cycles; one that jumps around needs many.
In Introduction to Work Study (ILO, 4th edition, 1992, chapter 21), the number of readings for a 95.45% confidence level and ±5% is n = (40 √(n′Σx² − (Σx)²) ÷ Σx)², where n′ is the number of preliminary readings and x each reading. The book's example: readings of 7, 6, 7, 7 and 6 hundredths of a minute give 8.81, so 9 readings, 4 more than were taken. The 40 is 2 ÷ 0.05: two standard errors (95.45%) over a 5% margin.
Find out why before you decide. If you saw a cause, such as a dropped part, a question from someone, a fumble or a timing error, mark the reading, write the cause on the study sheet and leave it out of the average. If there was no cause, it is part of the job's normal variation and stays in. The ILO calls such readings rogues and says exceptionally short ones need the same care. The calculator shows how much the reading furthest from the mean changes the answer.

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