Rounding and heaping in non-software estimates

Round numbers are often preferred in software task estimation times, e.g., 1, 5, 7 (hours in one working day), and 14. This human preference for round numbers is not specific to software, or to estimating. Round numbers can act as goals, as clustering points, may be used more often as uncertainty increases, or be the result of satisficing, etc.

Rounding can occur in response to any question involving a numeric value, e.g., a government census or survey asking citizens about their financial situation or health. Rounding introduces error in the analysis of data. The Whipple index, described in 1919, was the first attempt to quantify the amount of error; calculated as: “per cent which the number reported as multiples of 5 forms of one-fifth of the total number between ages 23 to 62 years inclusive.” for errors of reported age. Other metrics for this error have been proposed, and packages to calculate them are available.

At some point (the evidence suggests a 1940 paper) a published paper introduced the term heaping effect. These days, heaping is more often used to name the process, compared to rounding, e.g., heaping of values; ‘heaping’ papers do use the term rounding, but I have not seen ’rounding’ papers use heaping.

The choice of rounding values depends on the unit of measurement. For instance, reported travel arrival/departure times are rounded to intervals of 5, 14, 30 and 60 minutes; based on reported/actual travel times it is possible to estimate the probability that particular rounding intervals have been used.

The Whipple index fails when all the values are large (e.g., multiple thousands), or take a small range of values (e.g., between one and twenty).

One technique for handling rounding of large values is to define roundedness in terms of the fraction of value digits that are trailing zeroes. The plot below shows the number of households having a given estimated balance on their first mortgage in the 2013 Survey of Consumer Finances (in red), and the distribution of actual balances reported by the New York Federal Reserve (in blue/green; data extracted from plot in a paper and scaled to equalize total mortgage values; code+data):

Households estimated outstanding mortgage (red) and actual outstanding mortgages in New York (blue).

The relatively high number of distinct round numbers swamps any underlying distribution of actual values. While some values having some degree of roundness occur more often than non-round values, they still appear less often than expected by the known distribution. It is possible that homeowners have mortgages at round values because they of banking limits, or reasons other than rounding when answering the survey.

The plot below shows the number of people reporting having a given number of friends, plus number of cigarettes smoked per day, from the 2015 survey of Objective and Subjective Quality of Life in Poland (code+data):

Number of people reporting having a given number of friends, plus number of cigarettes smoked per day.

The narrow range of a person’s number of friends prevents the Whipple index from effectively detecting rounding/heaping.

The dominance of round numbers in the cigarettes smoked per day may be caused by the number of cigarettes contained in a packet, i.e., people may be accurately reporting that they smoke the contents of a packet, rather than estimating a rounded number.

Simple techniques are available for correcting the mean/variance when values are always rounded to specified boundaries. When the probability of rounding is not 100%, the calculation is more complicated.

Rounded/Heaped data contains multiple distributions, i.e., the non-rounded values and the rounded values; various mixture models have been proposed to fit such data. Alternatively, the data can be ‘deheaped’, and various deheaping techniques have been proposed.

Given the prevalence of significant amounts of rounding/heaping, it’s surprising how few people know about it.

Surveys are fake research

For some time now, my default position has been that software engineering surveys, of the questionnaire kind, are fake research (surveys of a particular research field used to be worth reading, but not so often these days; that issues is for another post). Every now and again a non-fake survey paper pops up, but I don’t consider the cost of scanning all the fake stuff to be worth the benefit of finding the rare non-fake survey.

In theory, surveys could be interesting and worth reading about. Some of the things that often go wrong in practice include:

  • poorly thought out questions. Questions need to be specific and applicable to the target audience. General questions are good for starting a conversation, but analysis of the answers is a nightmare. Perhaps the questions are non-specific because the researcher is looking for direction: well please don’t inflict your search for direction on the rest of us (a pointless plea in the fling it at the wall to see if it sticks world of academic publishing).

    Questions that demonstrate how little the researcher knows about the topic serve no purpose. The purpose of a survey is to provide information of interest to those in the field, not as a means of educating a researcher about what they should already know,

  • little effort is invested in contacting a representative sample. Questionnaires tend to be sent to the people that the researcher has easy access to, i.e., a convenience sample. The quality of answers depends on the quality and quantity of those who replied. People who run surveys for a living put a lot of effort into targeting as many of the right people as possible,
  • sloppy and unimaginative analysis of the replies. I am so fed up with seeing an extensive analysis of the demographics of those who replied. Tables containing response break-down by age, sex, type of degree (who outside of academia cares about this) create a scientific veneer hiding the lack of any meaningful analysis of the issues that motivated the survey.

Although I have taken part in surveys in the past, these days I recommend that people ignore requests to take part in surveys. Your replies only encourage more fake research.

The aim of this post is to warn readers about the growing use of this form of fake research. I don’t expect anything I say to have any impact on the number of survey papers published.