Difference between revisions of "Estimating numbers"

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Revision as of 14:00, 2 March 2016

Estimating numbers in the field is one of the many challenges facing birders, and research has shown that most of us are not particularly good at it, and not just wrong but inconsistent as well. Fortunately, there are some simple tips and techniques that can help. Several studies suggest that we could be more accurate and more consistent in our estimates simply by referring to images with a known number of dots. This allows us to "recalibrate" our number sense, and make better judgements about how many birds we are seeing (lots more details below).

I've created a handy pdf page that you can print out and carry in the field. Refer to it regularly to recalibrate your sense of 100 or to compare to the flock of birds in front of you. You could also try using it while taking the quizzes. And please let me know if you think it helps.
Jump to the bottom to start a quiz.

Seeing clusters

The key to estimating large numbers is to break the flock down into smaller clusters. It's relatively easy to visualize two or three items. We look at them and know the number instantly, without consciously counting. Your goal is to develop the same intuitive sense of what "10" looks like. It won't be perfectly accurate, but once you develop a vision of "10", you can go through a flock counting by tens.


Four groups of ten arranged in 2s, 3s, 4s, and 5s. Any of these smaller numbers are easier to visualize than 10, and counting by 2s or 3s is often the best and quickest way to find 10.


Groups of ten dots in three different densities. Erwin (1982) tested birders and found no significant correlation between density and accuracy, although general psychology studies do find that estimates are higher with denser arrays.


100 dots arranged in groups of ten. Once you have a vision of what "10" looks like, you can simply scan across the flock counting by 10s.

Quiz 1cc IMG c197 edit w.jpg

A "flock" of 100 beans

Quiz 1cc IMG c197 counted w.jpg

Each oval holds ten beans, and it is easy to see that five groups of ten represent about half of the total, leading to an estimate of 100 beans.

When I see a flock of birds and want to get an accurate estimate I follow these steps:

  • (roughly) count the first ten by twos or threes to get that mental image
  • superimpose that group onto the adjacent part of the flock to find the next ten (making 20), and the next ten (30), and so on. Imagine drawing lines across the flock to separate each group of ten as you go.
  • (if the flock is 100 or more) count tens until you reach a larger round number, say 50, and then superimpose that to make 100, 150, etc.
  • (if the flock is several hundred or more) visualize a group of 100 and begin counting the rest of the flock by 100s.

The basic principle in any size flock is to get a sense of what some number looks like and then use that image to count the rest of the flock. Start with 10, then count by 10s to find 100, and use that to count by 100s up to 1000,

The next three images all show differently shaped groups, but all the same number – 100. To me the V-shaped flock looks like more, and the tightly-clustered flock looks like fewer, but your impression may differ.

Quiz 1cc IMG c2cc edit w.jpg

100 beans in a V formation.

Quiz 1cc IMG c221 edit w.jpg

100 beans in a loose group.

Quiz 1cc IMG c199 edit w.jpg

100 beans in a tight group.

And the image below shows 200 beans:

Quiz 2cc IMG c2c9 edit w.jpg

200 beans in a tight group.

Streaming flocks

The discussion above outlines the general principles of estimating, and it works well for discrete and well-defined flocks. A separate challenge is presented by birds streaming by in a continuous band. This is often seen in grackles flying to or from a roost, or shearwaters moving along a coastline. The birds form one unbroken stream, and there are no distinct flocks within the stream.

In this case the most common technique is to choose a fixed point and count every bird that passes that fixed point during some set period of time. It may be necessary to count by 10s or 100s if the flow of birds is that large. By "sampling" the stream at multiple times (e.g. counting for one minute every 3, 5 or 10 minutes) and recording the overall duration of the passage of birds, it is possible to calculate an estimate for the total number of birds. For example, if two one-minute counts show birds passing at 1000/minute, and the flight goes on for 12 minutes, the total estimate would be 12,000 birds.

There are many possible scenarios and complications that can make these estimates difficult and more speculative, but following a standard protocol and recording the details of your estimate will always be more useful and reliable than guessing or simply saying "thousands".

Mixed flocks

Another special challenge is presented by flocks including more than one species. The usual approach is to estimate the total of all species in the entire flock, and then estimate the percentage of each species and assign numbers accordingly. Obviously this introduces another source of potential error (estimating percentages).

Research on estimating

Number perception has been a focus of psychological research for decades. We have an innate ability to recognize small numbers (one, two, or three). We recognize these almost instantly and with complete accuracy. It takes much longer (over a second on average) to assess seven dots, and we are only about 75% accurate (Balakrishnan and Ashby 1992 ). These trends continue through higher numbers of dots.

There is a general and well-documented tendency to underestimate numbers higher than about ten (Krueger 1982 ). Estimates are also extremely variable, within and between observers, and the variability and degree of underestimation increases with larger numbers (Izard and Dehaene 2008 ).

The current psychological model for the process of estimating numbers involves two systems: an innate and vague "number sense" (an intuitive understanding of more or fewer) and the verbal number system (one, two, five, five hundred, etc.). We use our number sense to judge an array of objects and to recognize the difference between, say, 5 and 10, or between 500 and 1000, and then "map" that perception to the verbal number system to apply a more precise number concept (Vul et al 2013 ).

It is easy to use our general "number sense" and see that one array contains about twice as many objects as another, but not so easy to match that to a verbal description. Should we use the words 30 and 60? 50 and 100? 100 and 200? When testing rapid estimates of arrays of dots, Minturn and Reese 1951 found that initial estimates could be off by a factor of 4 (e.g., an array of 200 dots elicits estimates ranging from 50 to 700), but simply giving test subjects feedback on the accuracy of their estimates reduced the variability and increased the accuracy of answers, and these effects lasted for months.

Krueger 1984 and Izard and Dehaene 2008 both succeeded in increasing accuracy and reducing variablility in estimates simply by showing participants an image with a known number of dots. This allowed the participants to "calibrate" their number sense, and to produce very accurate estimates across the entire range of numbers.

Print out a handy pdf page linked here, carry it with you in the field, and refer to it regularly to recalibrate your sense of 100

Psychological effects

Since estimating numbers involves perception, other (unrelated) psychological factors can influence the results. Here are some examples:

  • A higher density of dots is perceived as a larger number, and this effect increases with increasing number (Class 1972 confirming earlier studies).
  • Overlapping dots (a very dense flock) is perceived as fewer
  • Arrays with dots regularly spaced are perceived as a larger number (Ginsburg 1978 ).
  • Estimates also tend to become more variable and less accurate (the tendency to underestimate gets worse) over the course of a session (Krueger 1982 , Vul et al 2013 ).
  • The size and shape of a frame surrounding the array of dots significantly alters estimates of the number of dots (Bevan and Turner 1964 ).
  • Increasing the monetary value of the objects shown leads to higher estimates of the number of objects (Ansbacher 1937 ).

Most of these effects are minor, but they do suggest some of the factors that might influence our estimates of bird numbers. Is a flock of Ross's Gulls more likely to be overestimated than a flock of European Starlings?

Studies involving bird flocks

Only a few studies have directly addressed the question of estimating bird flocks, and these match the more plentiful psychological research, showing large errors in estimating numbers, and large variation both within and between observers (Rappoldt et al 1985 ).

Some differences from the general psychology studies would be expected since birders generally have at least some experience estimating numbers, and may have developed different methods and biases than the average person, and this seems worthy of some research. Also, the psychological research focuses on numbers only up to a few hundred, but birders often deal with much larger numbers.

In Prater 1979 all observers slightly overestimated small flocks (hundreds) and greatly underestimated large flocks (thousands). The trend matches the psychological research, but the overestimation of smaller numbers is unusual. Erwin 1982 found that "inexperienced" observers matched the norm, tending to underestimate across all flock sizes, especially in large numbers (over 1000). But he found the opposite trend in "experienced" observers, who had a slight tendency to underestimate low numbers and to overestimate large numbers.

Other conclusions drawn from the studies on birders should be interpreted with caution due to small sample sizes. For example, testing observers' estimates of birds in one photograph Warren 2014 found a tendency for experienced observers to be more accurate than inexperienced, men more accurate than women, and older people (25 and older) more accurate than younger. Erwin 1982 and Frederick et al 2003 tested several variables (such as dot density or total number) and found no significant difference, but these trends have been confirmed by more controlled psychological studies with larger sample sizes. The value of feedback was confirmed by Erwin 1982

Further research could be useful to investigate how birders' experience affects their estimating abilities, and also whether the overall trend of underestimating larger numbers continues through the thousands and beyond, or whether some observers switch to overestimating very large numbers as found by Erwin 1982 .

The good news is that, even though our estimates are often very wrong, and variable, our sense of relative numbers (more or fewer) is quite good, and over multiple flocks our estimates average out to a fairly accurate number. Try to make multiple estimates of each flock, and confer with others about their estimates.

Elphick 2008


Below is an example of an estimation quiz. Estimate the number of birds or dots in the photo, type your answer in the box, and click "submit". Any answer within 20% of the actual count is scored as correct, and the answer will show green borders. Any answer more than 20% high or low will show red borders. You can click the link below to try another quiz.

1. How many beans?

Quiz 1cc IMG c197 edit w.jpg
Enter your answer here:
→ The actual number is 100.
Image details: - , . . . Image by David Sibley. link to image

Your score is 0 / 0



Ansbacher 1937 - H. Ansbacher . 1937. Perception of number as affected by monetary values. Archives of Psychology 215: .
Balakrishnan and Ashby 1992 - J. D. Balakrishnan F. Gregory Ashby. 1992. Subitizing: Magical numbers or mere superstition?. Psychological Research 54: 80-90. pdf
Bevan and Turner 1964 - W. Bevan F. D. Turner. 1964. Assimilation and contrast in the estimation of number. Journal of Experimental Psychology 67: 458-462.
Class 1972 - P. Class . 1972. Display density and judgments of number. Perceptual and Motor Skills 34: 531-534. pdf
Elphick 2008 - Chris S. Elphick . 2008. How you count counts: the importance of methods research in applied ecology. Journal of Applied Ecology 45: 1313-1320. article
Erwin 1982 - R. M. Erwin . 1982. Observer Variability in Estimating Numbers: an Experiment. Journal of Field Ornithology 53: 159-167. pdf
Frederick et al 2003 - P. C. Frederick B. Hylton, J. A. Heath, M. Ruane. 2003. Accuracy and variation in estimates of large numbers of birds by individual observers using an aerial survey simulator. Journal of Field Ornithology 74: 281-287. pdf
Ginsburg 1978 - N. Ginsburg . 1978. Perceived numerosity, item arrangement, and expectancy. American Journal of Psychology 91: 267–273.
Izard and Dehaene 2008 - V. Izard S. Dehaene. 2008. Calibrating the mental number line. Cognition 106: 1221-1247. pdf
Krueger 1982 - L. E. Krueger . 1982. Single judgment of numerosity. Perception and Psychophysics 51: 175-182.
Krueger 1984 - L. E. Krueger . 1984. Perceived numerosity: A comparison of magnitude production, magnitude estimation, and discrimination judgments. Perception and Psychophysics 35: 536-542.
Minturn and Reese 1951 - A. L. Minturn T. W. Reese. 1951. The effect of differential reinforcement on the discrimination of visual number. Journal of Psychology 31: 201-231. link
Prater 1979 - A. J. Prater . 1979. Trends in accuracy of counting birds. Bird Study 26: 198-200. pdf
Rappoldt et al 1985 - C. Rappoldt M. Kersten, C. Smit. 1985. Errors in large scale shorebird counts. Ardea 73: 13-24. pdf (free but registration required)
Rawinski 2015 - Alyssa H. Rawinski . 2015. Computer Modeling to Improve Flock-size Estimates. Colorado Birds 49: 239-246. pdf found that training with a computer simulation and feedback on accuracy significantly improved estimates; no link between experience, background, etc and estimate accuracy.
Vul et al 2013 - Edward Vul David Barner, Jess Sullivan. 2013. Errors in numerosity estimation arise from slow drive in magnitude-number mapping. Journal of Vision 13: 1044. pdf
Warren 2014 - Piers Warren . 2014. Human Factors Influencing Accuracy When Estimating Sample Sizes.  : . pdf

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