How is Amazon Sales Rank determined?


ATTAIN YOUR IDEAL WEIGHT generally updates the sales rank for each of the several million books listed every hour.What does Amazon sales rank mean? How does Amazon sales rank work?How is Amazon sales rank computed? Understanding Amazon sales rank is easy when you examine the hour-by-hour variation of sales rank for several books over an extended period of time. It provides great insight into the method Amazon probably uses to assign sales rank for the vast majority of the books listed.It enables authors to understand variations in their sales rank and how to tell exactly when anybody buys their book.


This Amazon sales rank analysis began in mid February 2009 with the publication of RIVER CITY a nurse's year in Vietnam and concluded after eleven months in mid January 2010. At the end of this analysis there is a link to an Addendum that uses data from 2012 and the first seven months of 2013 for some comments on Amazon ebook sales rank.


Figure 1 shows the hourly Amazon sales rank variation for three books: (1) the just published RIVER CITY a nurse's year in Vietnam, (2) a four month old book (4MOB) published in November 2008, and (3) a twenty year old book (20YOB) which was offered on Amazon only by outside sellers.The three books of very different age were used to see if Amazon treated them differently.The dots in Figure 1 indicate hourly sales ranks for each of the books over an eight day interval beginning with the first sale of RIVER CITY on Amazon.How did Amazon determine that RIVER CITY had a sales rank of 100,044 on 12 February 2009, just after it had sold its very first book on Amazon?And how did Amazon determine its sales rank was 114,294 one hour later when it wasnít going to sell its second book on Amazon for more than a week?




†† Figure 1.


Figure 2 shows the time zone distribution of the U.S. population.The dashed vertical line indicates that the centroid of the U.S. population for the 48 contiguous states would be local time in the central time zone and that is what is used in this analysis.Since sales ranks for a given hour are based on what happened in the previous hour, one could equally think of the times as indicating when sales actually took place in the eastern time zone.


Aaron Shepard provides a convenient way to check Amazon sales ranks with which allows you to store the query information for two books.It was used in the initial phase of data collection for this analysis.The day after the first sale of RIVER CITYon Amazon it was decided to accumulate the sales rank data hourly.Amazon sales ranks generally update at about five or ten minutes before the hour.If you wake up at quarter to the hour you can get the previous hourís rank before the update, get the next hourís rank after the update, then go back to bed for an hour and forty-five minutes.After six days, exhaustion set in and the quest for hourly data was abandoned.


In May, was discovered and acquisition of hourly sales rank data on RIVER CITY was resumed.For a $6/month membership fee will collect all 24 Amazon sales ranks each day for you to export.The sales ranks are tagged year round in daylight savings time in the U.S. eastern time zone.In summer, hour 0 is midnight and hour 23 is 11 PM east coast time.In winter, hour 0 is 11 PM the previous day and hour 23 is 10 PM in east coast time.The sales ranks accumulate hourly each day and can be exported at any time until the array is reset at about 10 minutes before hour also archives daily averages of,, and sales ranks for an unlimited number of months on the number of books designated by your account level.


The overall sales rank question will be approached by addressing two component questions.How does not selling a book affect your Amazon sales rank?How does selling a book affect your Amazon sales rank?



Figure 2.



How does not selling a book affect your Amazon sales rank?


Neither RIVER CITY nor 20YOB sold a book during the eight day interval shown in Figure 1 and both their sales ranks increased monotonically over that interval.Although the sales rank of RIVER CITY was about 300,000 lower that 20YOB initially, Amazon afforded the newcomer less respect and its status decayed more rapidly, resulting in a greater sales rank than 20YOB by the end of the eight days.


It appears that Amazon assigns an average status decay rate to books based on both their present sales rank and their long term history.Figure 3 shows a progression of decay curves for RIVER CITY. The green curve is identical to the one in Figure 1 after the first Amazon sale on 12 February.The blue curve shows a slower decay after the second Amazon sale on 21 February, although it still reached almost 1,700,000 before the next sale.The red curve shows the sales rank actually leveling off at about two weeks after a sale in mid-October before continuing the status decline.There also seems to be an inflection point two weeks after the sale on 21 February.All three curves have about the same decay rate for sales ranks less than 200,000.††††



Figure 3.


Figure 4 shows data from three different days spanning more than a month during the summer.The circles indicate the variation of sales ranks over portions of each day when no copy of RIVER CITY sold on Amazon.Although Amazon nominally updates sales ranks hourly, the values will frequently be identical for two or three hours in a row.The curves in Figure 4 were generated by applying a cubic spline interpolation to the sales ranks that were not frozen.The three curves, which are almost identical in shape, make it easy to visualize what the frozen sales rank values should have been.The frozen values occur randomly but when they do, the sales ranks for all books fail to update.It is as if some glitch interrupted the computation for a given hour but did not disturb the values at later times.


The most gentle slope for each curve in Figure 4 is around 5 a.m. and the steepest slope is around noon.This diurnal oscillation is also apparent in Figures 1 and 3.Figure 5 makes that characteristic in the Figure 1 data more apparent by plotting the change in the sales rank for each adjacent pair of points divided but the difference in the observation times.The result is the change in sales rank per hour, plotted midway between the two observations.That allows consistent changes to be computed even across large data gaps, although those points are certainly less accurate than when the observations are an hour apart.


What is apparent in Figure 5 is a pronounced diurnal oscillation about the mean trend in sales rank when no sale occurs.The magnitude of the oscillation is similar for all three books.The sales rank increases most rapidly in the early afternoon and it increases most slowly around 5 a.m.That seems reasonable.If you havenít sold a book, but it is 5 a.m. and very few books are selling, you shouldnít be penalized.But if you havenít sold a book at noon when lots of books are selling, youíre going to lose status fast.



Figure 4.


It is interesting that most of the diurnal variations are very similar.There is a peak in the early afternoon followed by a sharp drop, then a sort of shoulder in the evening.The points for 4MOB are more scattered because it was selling books during this period.That will be addressed in the next section.



Figure 5.


One might intuitively expect that the status of a book would decay in any hour in which no sale occurred, which was true for both RIVER CITY and 20YOB in Figures 1 and 5.But the situation is more complex and the sales rank of 4MOB actually decreased in Figure 5 for several hours after hour 120, evidenced by the negative changes in sales rank, even though no sales occurred.


The algorithm Amazon uses to determine sales rank in the absence of a sale seems to have two parts.There appears to be a mean decay rate, based on sales history and present sales rank, and an oscillation about the mean rate whose amplitude is probably based on the total sales of all books during that hour.The amplitude of the oscillation does not seem to be affected by the mean decay rate.So if the mean decay rate is slow, the oscillation can actually cause the sales rank to decrease without a sale in the early morning hours.This happened often in the case of the 19 October curve in Figure 3.


Figure 6 shows a scatter plot of all the changes in sales rank for two books during hours when they sold no copy on Amazon.The books were RIVER CITY and a three month old book (3MOB) published in October 2009.The criteria for each sales rank change used in the scatter plot was that the previous and following sales rank values had to be different from the present value, and the three values had to be separated by one hour.This simple procedure ensured that no frozen values (Figure 4) contaminated the computation and that the accuracy of the change in sales rank was not degraded by time gaps in the data.A great deal of data were eliminated by the criteria, but enough remained to provide a quantitative picture of Amazon sales rank decay rates.


Most of the data points in Figure 6 are for sales ranks less than 100,000.The circles indicate the average values of all the changes in sales rank in 25,000 interval bins (0-25,000, 25,000-50,000, 50,000-75,000, Ö).Changes in sales rank larger than 18,000 were considered outliers and excluded from the averages.†††††



Figure 6.


The diurnal oscillation in sales rank decay rates spreads out the sales rank changes vertically, sometimes causing them to go negative and actually decrease the sales rank, but the circles provide a good overall view of the average sales rank decay rate as a function of sales rank.Sales rank status decays most rapidly for sales ranks in the 100,000-150,000 range.For higher sales ranks the decay rate decreases approximately linearly with increasing sales rank to a small value at about 800,000.


An interesting feature is indicated by the dashed line in Figure 6 which has a slope of 0.18 and, for the most part, provides an upper limit on the changes in sales rank.What it indicates is that if you have a high status (low sales rank), that status will decay slowly if you donít sell a book in a given hour.But as your status does decay, the decay rate will accelerate, shoving you down faster and faster until your sales rank exceeds 150,000.Once your status is that low, Amazon starts easing up on you and your status diminishes at a slower and slower rate.


Figure 7 shows idealized Amazon sales rank decay rate plots generated for use in a simulation.The first six circles are identical to the ones in Figure 6.The circles for sales ranks larger than 150,000 are a linear approximation of the mean trend in Figure 6, with the change in sales rank arbitrarily fixed for sales ranks greater than 800,000.Amazon might apply fixed sales rank decay rates for sales ranks below 150,000 (first six circles in Figure 7) and then vary the decay rates for larger sales ranks depending on your recent and cumulative sales (dashed lines in Figure 7).The steep dashed curve would slow your status decline and the shallow dashed curve would move you more rapidly toward higher sales ranks.



Figure 7.


The distribution of sales rank increases shown in Figure 7 helps better selling books maintain their status.The lower your sales rank, the slower the initial increase in sales rank if you go several hours without a sale.Figure 8 shows the effect the three curves in Figure 7 would have on two books which achieved sales ranks of 1,000 (red) and 3,000 (blue) and then stopped selling.The thick curves are actually composed of a sequence of dots indicating the hourly sales rank variation over the three week period corresponding to the circles in Figure 7.The upper and lower dashed curves in Figure 8 correspond to the lower and upper dashed curves in Figure 7.


The Amazon respect for status is relatively short lived.What appears to be a plateau during the first day (blue) or two (red) in Figure 8 is actually a significant decay in status apparent in the enlargement shown in Figure 9.For the book with an initial sales rank of 3,000, the sales rank triples in about 19 hours and has increased by a factor of 10 after a day and a half without a sale.The sales rank for the book starting at 1,000 triples in just over 20 hours.After that it follows the same sales rank increase trajectory as the blue book, but with a 20-hour lag.



Figure 8.



Figure 9.


The diurnal oscillation was not factored into the sales rank decays shown in Figures 8 and 9 but will be incorporated in the overall sales rank simulation developed in the next section.Figure 10 shows the diurnal variation used, which seemed to well approximate the variation for sales ranks less than 100,000.To obtain the sales rank decay for any given hour without a sale, the mean decay rate (Figure 7) corresponding to the sales rank of the previous hour would be multiplied by the diurnal oscillation factor (Figure 10) corresponding to the present hour of the day.The resulting increase in sales rank would vary by almost an order of magnitude depending on the time of day.Another important aspect of Figure 10 is that one might be able to monitor the sales rank decay rate and use it as a surrogate for how total book sales on Amazon varied throughout any given day.Such an investigation would probably show significant differences between the weekday and the weekend distributions.



Figure 10.



How does selling a book affect your Amazon sales rank?


Morris Rosenthal has provided an excellent analysis of the significance of Amazon sales rank and its relationship to the average number of books sold.His data will be used to put the present analysis in perspective.It would be possible for Amazon to make a straight forward sales rank calculation for the top several hundred books.The book that sold the most copies in the previous hour would be number 1. The book that sold the second most copies would be number 2, and so on.But that procedure would already break down by sales rank 900.Rosenthal indicates that books with sales ranks between 800 and 1,000 are selling about 30 books per day.With 200 books each selling about one or two an hour, Amazon could not order them into unique sales ranks on that basis.Even considering the total number of books sold in the previous 24 hours and updating that number each hour would not be able to differentiate between the books.


For Amazon sales ranks between 3,000 and 140,000, Figure 11 indicates the number of books sold per day, extracted from Morris Rosenthalís graphs at the sale of its first book on Amazon, RIVER CITY jumped past millions of other books to a sales rank of 100,044.That status was not actually unreasonable since the first sale guaranteed that RIVER CITY was selling at least one book per day.When no book sold in the next hour its status began a rapid decay.Figure 11 indicates that a book selling one copy a day would have a sales rank of about 140,000.It was not possible to determine the average RIVER CITY sales rank over the first 24-hour interval because of the missing data, but it was probably about 200,000.



Figure 11.


The sales rank variation of 4MOB in Figure 1 exhibits what some refer to as erratic behavior, but the blowup shown in Figure 12 indicates the kind of information that can be gleaned.Vertical lines have been drawn in Figure 12 to indicate the fifteen times during the eight day period that contained sales of 4MOB.On one of the days (Sunday, hours 72 to 96), sales were made during four different hours (with the final sale just before midnight).One day had sales in each of three different hours, two days had sales in each of two hours, and four days had sales only during one hour.


With so few hours when books were sold, it is not unreasonable to assume that only one book sold during any particular hour.An interesting insight from this assumption is that the larger the sales rank, the larger the decrease in it from the sale of a book.The lower the starting position in Figure 12, the greater the vertical jump.As a reminder, the circle at 100,044 near the left edge indicates the jump from infinity that RIVER CITY made with the sale of its first book on Amazon.For sales ranks less than 20,000, the decrease with the sale of a single book is relatively small.Of course, the jump at about hour 30 in Figure 12 is an estimate assuming that the sales rank just decayed over the missing data interval.


The important point is that once one recognizes that the diurnal oscillations in the no-sale decay rate can slightly reduce sales rank without a sale (during the early morning hours of Tuesday and Thursday in Figure 12, for example), the actual hours when sales occur become quite obvious.The same general characteristics are on display in Figure 13 for the sales rank variation of 3MOB published in October 2009.Because the average sales of 3MOB during the interval shown in Figure 13 were higher than those of 4MOB in February 2009, it can be used to provide additional insights.



Figure 12.


The top panel of Figure 13 shows the sales rank variation of 3MOB during the first 23 days of December.Figure 4 showed typical occurrences of frozen sales ranks, but during the first three days of December there were both 18-hour and 11-hour intervals when the Amazon sales ranks did not update.During the first three days there were also three 3-hour intervals and five 2-hours intervals when the sales ranks did not update.Sales ranks frozen for long intervals are uncommon, but sales ranks not updating for a two or three hour interval once or twice a day are not unusual.



Figure 13.


Since the sales rank data acquired for 3MOB (and RIVER CITY after late May) were hourly without gaps, thanks to, the same strict editing criteria used for Figure 6 were applied in quantifying changes in sales rank resulting from a sale.The previous and following sales rank values had to be different from the present value, and the three values had to be separated by one hour.So no frozen values (Figure 4) contaminated the computations and the accuracy of the change in sales rank was not degraded by time gaps in the data.


The bottom panel of Figure 13 shows all the changes in sales rank, both positive and negative, that satisfied the editing criteria.The dashed lines in Figure 13 are located at 5 AM each day.The diurnal oscillation in sales rank decay during hours without a sale is readily apparent during two intervals, December 3-12 and 16-23.There appears to have been a promotional push on December 12 that pumped up sales of 3MOB for a few days.Figure 11 indicates that a sales rank of 3,000 corresponds to selling about 17 books per day, averaging more than one per hour during the late morning - early afternoon peak selling interval (Figure 10), suppressing the diurnal oscillation in sales rank decay.


Figure 11 indicates that books with sales ranks of 10,000 or larger are selling about 8 copies a day or less.That seems like a reasonable threshold for assuming that only one book will sell in any given hour.By looking at the negative excursions in sales rank in the bottom panel of Figure 13 when the sales rank is 10,000 or larger in the top panel, you can get the same general impression of the effect of a sale as was apparent in Figure 12.The lower the starting position (higher sales rank) in the top panel, the greater the jump.While the sales rank is small during December 13-15 the reductions in sales rank are quite modest, even if more than one book sold in an hour, as might have occurred.



Figure 14.


The dots in Figure 14 show the relationship between the sales ranks of 3MOB and RIVER CITY before and after each hour in which a sale occurred.The dashed curves show the positions within the scatter plot for which the sales of a single book would cause the sales rank to be reduced by 85%, or 80%, or 75%, ÖWhen the sales rank is about 750,000, the sale of a single book will generally bring it down to around 100,000, a reduction of about 85%.When the starting sales rank is lower, the percent reduction is also lower.


There is a significant amount of scatter in Figure 14, which is reasonable considering the time interval covered is May 2009 through January 2010 for RIVER CITY and October 2009 through January 2010 for 3MOB.Morris Rosenthal indicates that the data of Figure 11 represent an average behavior and that there are weekly to seasonal variations in sales as well as trends over longer time periods.The solid curve in Figure 14 was used as a piecewise linear approximation of the average variation of the change in sales rank resulting from the sale of a single book.


Figure 15 shows an enlargement of Figure 14 for sales ranks less than 100,000.The minimum reduction in sales rank for the piecewise linear approximation (solid curve) is fixed at 10.7% for sales ranks of 15,000 or less.As was indicated earlier, the reductions in sales rank used in this analysis were assumed to have resulted from the sale of a single book.That assumption would tend to break down for sales ranks lower than 10,000.The % reduction in sales rank per copy sold for books in low sales ranks which sell many an hour would have to be much less than 10.7%.For example, Morris Rosenthal indicates books with sales ranks of about 900 sell roughly 30 a day on average.Amazon would probably increase the sales rank for such a book if it only sold one copy in an hour during the middle of the day.The solid curve in Figure 15 indicates that the sales rank would decrease by 96.So this analysis would not be valid for books with sales ranks much less than 10,000.But sales ranks of 10,000 and greater encompass 99.8% of all the books for sale at



Figure 15.


The final step taken to assess the reasonableness of this analysis was to run an overall simulation of the variation of sales rank assuming that a book started selling 1, 3, 5, or 7 copies a day on Amazon.Figure 16 shows the sales rank variation over a two week period for a book with an initial sales rank of 800,000 that started selling one book a day.The blue dots indicate the hourly sales rank values if the book sales always occurred at 5 AM.The red dots indicate the hourly sales rank values if the book sales always occurred at 1 PM.


If no sale occurred in a given hour, the average sales rank decay rate (circles in Figure 7) corresponding to the sales rank of the previous hour was multiplied by the diurnal coefficient for that hour of the day (Figure 10) to get the actual increase in sales rank.For each hour in which a sale occurred, the new reduced sales rank was determined from the piecewise linear approximation (solid curve) of the average reduction in sales rank indicated in the scatter plot shown in Figures 14 and 15.†††



Figure 16.


The daily variation in sales rank shown in Figure 16 stabilized after about four days and there was not much difference in the mean values of the red and blue distributions.The bottom panel of Figure 17 shows an enlarged portion of Figure 16 to compare with similar enlargements for sales of 3, 5, or 7 books per day shown in the upper panels.The multiple sales in a day were assumed to occur one in each sequential hour centered on either 5 AM (blue) or 1 PM (red).The initial sale in all cases reduced the sales rank to about 120,000.



Figure 17.


When there is only one sale per day, the sales rank immediately begins to increase.When there are multiple sales in a day, the sales rank continues to decrease during the interval of sales, but by a smaller amount each time the sales rank is reduced (Figures 14 and 15).Once the sales conclude for the day the sales rank increases hourly until sales resume the following day.In this algorithm the time of day when a sale occurs is important.In the top panel of Figure 17 the seven reductions in sales rank which were centered on 1 PM (red dots) took the place of the seven largest increases in sales rank that would have occurred had there been no sale (Figure 10).In contrast, the blue dots showing the seven reductions in sales rank which were centered on 5 AM replaced the seven smallest increases in sales rank that would have occurred had there been no sale.With no sales in the late morning and early afternoon, the sales rank of the blue dots increased relatively quickly because of the diurnal variation (Figure 10), resulting in an average sales rank for the blue dots that is significantly larger than the average sales rank of the red dots, even though the same number of books were sold.


Figure 18 repeats the Morris Rosenthal curve from Figure 11 to which the average sales ranks resulting from the simulation (Figure 17) have been added.The red dots for the sales centered on 1 PM are remarkably close to the Rosenthal curve while the blue dots for the sales centered on 5 AM tend to have significantly larger sales ranks for the same number of books sold.It is reasonable that the red dots should be closer to the curve since midday is the most likely time for people to buy books and that is why the diurnal variation curve (Figure 10) has the shape it does.



Figure 18.


Although the algorithm developed here was based on statistics assuming that only one book sold in any given hour, it can be applied to books selling multiple copies an hour as long as the sales rank of the book is not too low.If a book happened to sell five copies in an hour, you would sequence through the sales rank reduction process five times in a row, just as you would if the sales occurred in sequential hours.The difference is that if all the sales occurred in a single hour, the sales rank will immediately begin to grow in subsequent hours and you will end up with a higher sales rank than if the sales had been spread out over time.There are three green circles on Figure 18, although only two of them are visible.They correspond to books which sold 3, 5, or 7 copies a day, but with all sales occurring exactly at 1 PM.For the book selling 3 copies a day at 1 PM, the sales rank is only a little lower than the book whose sales were spread out around 5 AM.For the book selling 5 copies a day at 1 PM, the green circle is directly behind the blue circle.For the book selling 7 copies a day at 1 PM, the sales rank is higher than the book which sold its 7 copies spread out around 5 AM.Your sales rank will be lowest if your books sell midday and spread out over time.


As a general conclusion of this analysis, it seems likely that, for books whose sales ranks are about 10,000 or greater, Amazon uses a mathematical algorithm to determine the sales rank of each book that does not depend on what is happening to the sales rank of any other book in that range.One might naively think that in every hour Amazon places several million books into a unique sales rank order, but that is probably not the case.In any given hour, who would know if two books had the sales rank 253,761 and no book had the sales rank 253,760?If any book with a sales rank greater than 10,000 doesnít sell during a given hour, Amazon seems to just increase its sales rank by the amount indicated for its present sales rank and sales history (Figure 7), modified by a function of the total sales during that hour (Figure 10).If a book does sell, they seem to just decrease its sales rank by some amount that is determined by its present sales rank (Figures 14 and 15).


If Amazon were doing something more complex, the diurnal variation seen in Figure 10 probably would not exist in the changes they make to sales ranks.In fact, if a book sold 7 copies a day, day after day, it really should have the same sales rank whether those copies all sold in one hour, or the sales were spread out over a number of hours in any part of the day.But then Amazon would have to consider an entire dayís sales and could not update sales rank on an hourly basis.Amazon may well monitor how many books are in various sales rank ranges (50,000-100,000, 100,000-150,000, 150,000-200,000, Ö) to see if the coefficients they are using are producing reasonable results statistically.If there are about 50,000 books in each bin, the various coefficients would be doing a decent job of showing authors what their status is.If books were starting to bunch up in certain bins, the coefficients would need adjustment.




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