Build a forecast from observed frequencies

Try

A fictional service desk records ten waiting times in minutes: 0, 0, 0, 2, 2, 2, 2, 5, 5 and 5. These are original synthetic observations for practicing a method, not measurements from a real organization. Make one mark for each observation at its numerical location. Which waiting time is most frequent? Does that location have to equal the mean?

Condense the marks into a count table without losing any observations. An empirical distribution places a fraction of the dataset at each observed value. The word empirical signals that the weights come from data, rather than a known coin or spinner law. It says nothing by itself about how representative the records are.

Predict

Predict the mean before calculating it. Three zero waits should pull the average below what a table of distinct values alone might suggest. A tempting but incorrect calculation is (0 + 2 + 5)/3. That treats the three values as equally frequent and discards the counts.

Waiting time in minutes	Count	Relative frequency	Value × count
0	3	3/10	0
2	4	4/10	8
5	3	3/10	15
Total	10	1	23

Predict what happens if every record is copied once. Then predict what happens if one genuinely new five-minute observation is added. Those actions have different meanings even when they both increase the number of rows in a file.

Experiment

The playground uses different synthetic waiting times: 1, 3 and 7 minutes. Set the count of each value, predict the resulting probabilities and mean, then reveal the graph and table. Notice that the numerical gaps on the horizontal axis are 2 and 4 minutes. Change only the count at 7 minutes and explain the direction of the mean’s movement.

Build a distribution from observed counts

Original synthetic observations take values 1, 3 or 7 minutes. Edit their counts. Predict how their relative frequencies and mean change; these observations do not establish the probabilities of a future population.

Count of 1-minute observations012345678910Count of 3-minute observations012345678910Count of 7-minute observations012345678910

Reveal calculationsReset explorer

Commit to a prediction, then reveal the table and fixed-scale plot. Change one control and explain the difference.

All settings and traces are temporary. This explorer does not save an answer or award mastery; keep your assumptions and reasoning in the written practice.

Next set all counts to zero. There is no empirical probability distribution and no mean to display. A zero-minute mean would incorrectly claim that observed waits were zero. Restore one nonzero count and explain why a one-value empirical distribution has variance zero without proving that the underlying service has no variability. Record one before-and-after count table in your written response.

Understand

For the fixed worked dataset, the mean is 23/10 = 2.3 minutes. It equals 0(3/10) + 2(4/10) + 5(3/10), the expected value under the empirical distribution. If one of the ten records is selected uniformly at random, this is exactly the expected recorded wait. Calling 2.3 minutes a forecast for a new customer requires an additional assumption that the records adequately represent the future process.

This distinction matters because the same arithmetic can answer different questions. A convenience sample collected during a quiet hour may summarize those records accurately but underrepresent busy-hour waits. A process change can make an older representative sample a poor guide to tomorrow. Missing waits must not silently become zero waits: zero is an actual measured value, while missing means the measurement is unavailable.

Copying the entire dataset doubles each count and the total. Every relative frequency stays unchanged because 2c/(2n) = c/n, and the mean stays 2.3. The copies do not provide independent new information. In contrast, adding one new five-minute wait changes the mean to (23 + 5)/11 = 28/11 minutes. Whether the new record improves representativeness depends on how it was selected, not just on the larger denominator.

For a nonempty dataset with values x and counts c, the mean is the sum of xc divided by the total count n. Empirical distribution variance similarly averages squared deviations using denominator n. An estimator of population variance sometimes uses n − 1 instead; that is a different purpose. The explorer labels its quantity as distribution variance so the denominator can be inspected rather than inferred from an abbreviation.

Apply

Prepare a short forecast note from the ten-record worked example. Include the empirical probability of waiting five minutes, the expected recorded wait, and the number of observations. State one condition under which using these frequencies for future customers would be reasonable, and one plausible change that would invalidate that use. Avoid wording such as “the chance is definitely 30 percent” when describing a future population from ten synthetic records.

Construct the PMF graph yourself with a common zero-to-one vertical scale and numerical horizontal positions 0, 2 and 5. Keep the gap between 2 and 5 larger than the gap between 0 and 2. A table supplies exact ratios; the graph reveals how mass is distributed. Both should express the same dataset. A bar of height zero at an allowed value means none was observed, not that the event is impossible in the population.

Continue

Expected values become decision inputs only after specifying what consequences the numbers represent. The next lesson compares two fictional strategies under an assumed probability model. Carry forward the distinction between calculating an expectation correctly and having trustworthy evidence for the probabilities used in it.