Statistical Foundation of Work Sampling

The Fundamental Difference: Statistics vs Time Study

Work Sampling differs ontologically from traditional time tracking. Whereas a stopwatch measures durations, sampling is an inferential technique based on probability theory that estimates the frequency of occurrence of activities through instantaneous, random observations.

1. Binomial Distribution and Normal Approximation

Work sampling is grounded in the fact that when observing a system (worker or machine), the state at an instant is a dichotomous variable: the event either occurs or does not occur. This places sampling in the domain of the binomial distribution.

In practice, given that the number of observations is usually high (N > 30), the binomial distribution is approximated by the normal distribution (Gaussian curve). This approximation allows using standard formulas to determine:

• The required sample size
• Control limits
• Confidence intervals

2. Standard Error (σ_p)

The sampling error measures how much our estimate p̂ may vary relative to the real value p. It is defined as:

This formula shows that precision is inversely proportional to the square root of the sample size. Doubling the precision requires quadrupling the observations.

3. Sample Size Formula (N)

One of the most glaring shortcomings of generalist software is the absence of sample size calculators. The fundamental formula, derived from the normal approximation, is:

This equation incorporates three critical variables that define the rigor of the study:

3.1. Z Value (Confidence Level)

It represents statistical certainty. The most common values are:

3.2. Estimated Proportion (p)

It is the preliminary estimate of the occurrence of the key activity:

• If you have no prior data: Use p = 0.50 (maximum variance, maximum sample size)
• If you have data from previous studies: Use that percentage
• If you ran a pilot study: Use the preliminary result

Using p = 0.50 is the "worst case" because it maximizes the product p(1-p) = 0.25, guaranteeing a conservative sample size.

3.3. Absolute Error (e)

It is the acceptable tolerance (± X%). Examples:

• e = 0.10 (±10%): Exploratory studies → ~100 observations
• e = 0.05 (±5%): Standard studies → ~385 observations
• e = 0.03 (±3%): High precision → ~1,067 observations
• e = 0.01 (±1%): Maximum precision → ~9,604 observations

4. Absolute Precision vs Relative Precision

A technical nuance that distinguishes the expert is selecting the type of precision:

The formula for relative precision is:

Relative precision requires more observations when p is small, which is critical in studies of rare events (accidents, infrequent machine stops).

5. P Control Charts

To validate the consistency of the study during its execution, daily proportions (p_daily) should be plotted. Control limits are calculated as:

Points outside these limits indicate abnormal conditions on that day (e.g. breakdown, management visit, drill). The engineer must decide whether to include those data.

6. The Importance of Randomness

The validity of p̂ as an estimator of p depends strictly on randomness. If observations are not random, a systematic bias is introduced that sampling formulas cannot correct.

Practical Calculation Example

An engineer wants to study a production line with:

• Confidence level: 95% (Z = 1.96)
• Margin of error: ±5% (e = 0.05)
• Estimated proportion: unknown (p = 0.50)