Rightmost Digits Research Articles

Digits of powers of 2 in ternary numeral system

We study the digits of the powers of 2 in the ternary number system. We propose an algorithm for doubling numbers in ternary numeral system. Using this algorithm, we explain the appearance of “stairs” formed by 0s and 2s when the numbers $2^n (n=0,1,2, \ldots)$ are written vertically so that for example the last digits are forming one column, the second last digits are forming another column, and so forth. We use the patterns formed by the leftmost digits, and the patterns formed by the rightmost digits to prove that the sizes of these blocks of 0s and 2s are unbounded. We also study how this regularity changes when the digits are taken between the left end and the right end of the numbers.

On digits of Mersenne numbers

Motivated by recently developed interest to the distribution of q -ary digits of Mersenne numbers M_p = 2^p-1 , where p is prime, we estimate rational exponential sums with M_p , p \le X , modulo a large power of a fixed odd prime q . In turn this immediately implies the normality of strings of q -ary digits amongst about (\log X)^{3/2+o(1)} rightmost digits of M_p , p \le X . Previous results imply this only for about (\log X)^{1+o(1)} rightmost digits.

Cause-related marketing and the effect of 99-ending pricing

PurposeCause-related marketing (CRM) campaigns have become common features of the marketplace. CRM often involves a for-profit business agreeing to contribute a specified amount to a cause when the business’s customers engage in revenue-generating exchanges. Despite the central role that price is likely to play in a consumer’s decision to purchase or not to purchase an offer associated with a CRM campaign, to the best of the authors’ knowledge, very few have examined price framing effects in a CRM context. This paper aims to explore the effect of rightmost digits manipulation in prices on participation intentions for CRM campaigns.Design/methodology/approachIn Study 1, 241 college students participated in an online experiment for class credit. The experiment used a 3 (price level: low, medium and high) × 2 (price ending: 99 ending and no ending) between-subjects design. The dependent variable was participation intention, and several moderators and mediators were considered. PROCESS was used to test the moderated mediation. In Study 2, 351 subjects participated in an online experiment with a design similar to the earlier study. In Study 2, however, new mediators were added and the moderated mediation was tested using SPSS PROCESS macro.FindingsThis research shows that price ending impacts the effectiveness of CRM as a tactic on consumers’ purchase intentions. Consistent with the authors’ prediction, this study shows that consumers exposed to a 99-ending CRM offer are more likely to participate in the offer compared to consumers exposed to a no-ending priced offer. Offer attractiveness, elaboration and corporate social responsibility were also shown to have a strong effect on participation intentions.Practical implicationsThis research indicates that for moderately priced products, 99-ending prices led to an increased influence on consumer purchase intentions; on the other hand, no-ending/even-ending prices were more effective for high-priced products. Thus, the use of the right digit effect by managers in a CRM context as way of increasing consumers’ participation likelihood is likely to be more successful for moderately priced offers.Originality/valueThis research extends previous work on CRM and right digit effect in pricing. This study’s findings, in both Studies 1 and 2, demonstrate that the effectiveness of CRM campaigns on consumer choice is dependent on the offer price ending. Consumers exposed to the no-ending priced CRM offers tend to be affected less by CRM campaigns compared to consumers exposed to 99-ending offers, who perceive the offer as more attractive.

Currency Substitution and Price Endings: Right Digit Effect

ABSTRACTThe strategic manipulation of prices. rightmost digits has been a tactic used by retailers in the western world for decades. By studying the internationalization of pricing tactics in a global economy, our research adds a much needed contribution to the literature of price endings and pricing tactics in global markets. We find that at lower price levels, consumers exposed to a 99 ending price in a currency substitution market are more likely to purchase the product compared to consumers in the US market. At higher price levels, on the other hand, consumers in either market situation exhibit no change in purchase intentions. Thus, the 99 ending tactic has no effect on consumers when the product is expensive. The use of the right digit effect by managers in a currency substitution/ dollarized economy as a way of persuading consumers to buy is still likely to be more successful compared to the USA market. As such, firms in a dollarized economy should structure their pricing strategies while taking into consideration the type of product they are offering and the consumer market they are dealing with.

Statistical analysis of numerical preclinical radiobiological data

Abstract Background Scientific fraud is an increasingly vexing problem. Many current programs for fraud detection focus on image manipulation, while techniques for detection based on anomalous patterns that may be discoverable in the underlying numerical data get much less attention, even though these techniques are often easy to apply. Methods We applied statistical techniques in considering and and comparing data sets from 10 researchers in one laboratory and three outside investigators to determine whether anomalous patterns in data from a research teaching specialist (RTS) were likely to have occurred by chance. Rightmost digits of values in RTS data sets were not, as expected, uniform. Equal pairs of terminal digits occurred at higher than expected frequency (>10%) and an unexpectedly large number of data triples commonly produced in such research included values near their means as an element. We applied standard statistical tests (chi-square goodness of fit, binomial probabilities) to determine the likelihood of the first two anomalous patterns and developed a new statistical model to test the third. Results Application of the three tests to various data sets reported by RTS resulted in repeated rejection of the hypotheses (often at p-levels well below 0.001) that anomalous patterns in those data may have occurred by chance. Similar application to data sets from other investigators was entirely consistent with chance occurrence. Conclusions This analysis emphasizes the importance of access to raw data that form the bases of publications, reports, and grant applications in order to evaluate the correctness of the conclusions and the importance of applying statistical methods to detect anomalous, especially potentially fabricated, numerical results.

Odd pricing effects: an examination using adaptation‐level theory

PurposeUsing adaptation‐level theory as a conceptual framework, the purpose of this research is to determine what effect, if any, marketplace conditioning has on consumer price estimates and product evaluations.Design/methodology/approachA total of 475 subjects participated in two experiments that required them to read a scenario, evaluate a series of advertised products, and perform an aided price recall task.FindingsThe results suggest consumers are more likely to recall the correct price when more of the rightmost digits end in 0 or 9. Moreover, when prices are incorrectly recalled, consumers are likely to inadvertently assume prices end in commonly used rightmost digits (i.e. 5 and 9). Combined, the results demonstrate odd pricing effects are likely a result of marketplace price conditioning rather than truncation of rightmost digits as suggested by the analog model of numerical cognition.Practical implicationsFindings suggest that use of atypical rightmost digits in odd prices fails as a method to differentiate products in the mind of the consumer. This would explain the use of larger right ending digits by retailers in an effort to maximize profit without impacting consumer perceptions of quality, value, and purchase likelihood. In the absence of strong quality image effects, retailers are encouraged to continue the practice of setting prices with digits ending in 9.Originality/valueA key theoretical implication of this study is that the underestimation heuristic based on leftmost digit processing fails to explain the results of the incorrectly recalled price estimates. As a result, adaptation‐level theory may provide a more robust explanation for odd pricing effects.

Hospitality Managers’ Price-Ending Beliefs

The use of “just-below” pricing (such as pricing an item at $6.99 or $6.95, rather than $7.00) has been common in some segments of the hospitality industry (such as quick-service and mid-scale dining). The results of a detailed survey of the price-ending beliefs of hospitality managers show that many believe that just-below prices connote good value and round-number prices connote high quality. Furthermore, the majority of these managers believe that consumers tend to drop off or otherwise give insufficient consideration to a price’s rightmost digits. Although one might expect this drop-off belief among those managers who use just-below price endings, it is found also in a majority of those managers who do not use just-below price endings. The survey results suggest that managers who decline the benefits of a consumer drop-off tendency do so because they also believe in one or more possible negative consequences of using just-below price endings. These include the beliefs that just-below endings (1) impair perceptions of high quality, (2) work against an upscale image, (3) give an impression of not being fully honest or straightforward, and (4) involve inconvenience in calculating or communicating the price or in making change.

The politics of digits: evidence of odd taxation

From the concept of odd pricing, i.e., setting rightmost price digits below a whole number, this paper advances the political counterpart of odd taxation using a panel of Danish municipal taxes. First, the distribution of tax decimals is non-uniform and resembles the distribution of price-endings data. Second, nine-ending and other higher-end decimals are found to be over-represented which echoes odd pricing research. It suggests that incumbents take voters’ biases into account and apply odd taxes to minimize the political costs of taxation while maximizing revenue. Attention should be given to how policy digits are arranged to exploit voters’ cognitive biases.

99 Cent: Price Points in E-Commerce

This paper attempts to shed light on the pricing behaviour of firms or sellers in market places with price competition. In particular, it aims at testing the mechanism that lies at the heart of one of the most famous models in economics, namely the Bertrand model of competition. This model suggests that firms' profits will be zero as soon as there are at least two sellers in the market. The model is built on the idea that consumers will go to the cheapest shop, even if the differences in prices are infinitesimal. As a result, firms are tempted to undercut their opponents' price by a small amount and they do so over and over until they end up charging the cost at which they produce their products. However, the underlying assumption is at odds with the observation that consumers typically do not care very much about a single cent and the predicted outcome clearly goes against the fact that many prices end in 99c. If sellers really charged the price at which they buy or produce a good or a service, we should observe all possible price endings. Basu (2006) argues that the prevalence of 99 cent prices in shops can be explained by consumers who disregard the rightmost digits of the price. This bounded rational behaviour leads to an Bertrand equilibrium with positive mark-ups allowing firms to make at least small profits. We use data from an Austrian price comparison site, which bears the advantage that in such an environment other factors than price both can be observed online and play a minor role. Price comparison sites are therefore highly suited for our test. We find results highly compatible with Basu's theory. We can show that price points - in particular prices ending in 9 - are more frequent than other endings and have significant impact on consumer demand. This shows that the mechanism that was postulated by Bertrand is not in place. Consumers do not necessarily choose the cheapest price and they even less do so when the Euro digits of the price are the same (i.e. when the difference is small). Moreover, the nine-ending prices are sticky: neither the price-setter itself wants to change them nor the rivals do underbid these prices, if they represent the cheapest price on the market. This might either show that shops seem to believe that it is no use to only slightly undercut a rival's price if this doesn't trigger a change in the Euro, or it is an indication that there might be some sort of silent agreement to not undercut a price ending in 9. Our findings shed new light on the behaviour of consumers and firms, and propose that theoretical investigations on competitive behaviour should consider discrete state spaces, i.e. markets where the smallest plausible price change is by a Euro or at least by a full Cent.

Is the visual analyzer orthographic-specific? Reading words and numbers in letter position dyslexia

Letter position dyslexia (LPD) is a deficit in the encoding of letter position within words. It is characterized by errors of letter migration within words, such as reading trail as trial and form as from. In order to examine whether LPD is domain-specific, and to assess the domain-specificity of the visual analysis system, this study explored whether LPD extends to number reading, by testing whether individuals who have letter migrations in word reading also show migrations while reading numbers. The reading of words and numbers of 12 Hebrew-speaking individuals with developmental LPD was assessed. Experiment 1 tested reading aloud of words and numbers, and Experiment 2 tested same–different decisions in words and numbers. The findings indicated that whereas the participants with developmental LPD showed a large number of migration errors in reading words, 10 of them read numbers well, without migration errors, and not differently from the control participants. A closer inspection of the pattern of errors in words and numbers of two individuals who had migrations in both numbers and words showed qualitative differences in the characteristics of migration errors in the two types of stimuli. In word reading, migration errors appeared predominantly in middle letters, whereas the errors in numbers occurred mainly in final (rightmost) digits. Migrations in numbers occurred almost exclusively in adjacent digits, but in words migrations occurred both in adjacent and in nonadjacent letters. The results thus indicate that words can be selectively impaired, without a parallel impairment in numbers, and that even when numbers are also impaired they show different error pattern. Thus, the visual analyzer is actually an orthographic visual analyzer, a module that is domain-specific for the analysis of words.

Patterns of price endings used in US and Japanese price advertising

PurposeThe purpose of this paper is to guide managers' choices of rightmost digits in retail prices by acquiring a better understanding of the psychological mechanisms by which price endings can influence sales.Design/methodology/approachThe paper observes and compares the price endings used in large matched samples of advertised prices in two countries with considerable cultural differences, the USA and Japan.FindingsAlthough the digit 9 predominates among the rightmost digits of advertised prices in the USA the digit 8 predominates in Japan. In contrast to this difference, the US and Japanese prices are similar in that both show greater use of 9 or 8 endings when this choice lowers the price's leftmost digit and when the advertised price is claimed to be a discount price.Research limitations/implicationsFuture research should include a wider range of price advertising media and should examine usage patterns of less frequently occurring digits.Practical implicationsSetting a price that falls just below a round number can be helpful in creating a low‐price image. Setting this just‐below price with a 9‐ending would be appropriate in the USA and European countries, but in Japan and other Asian countries, it would be more appropriate to set this just‐below price with an 8‐ending.Originality/valueThese results provide guidance to the retail price setter and illustrate to the basic researcher how universal psychological processes and specific cultural meanings can interact to determine consumer perceptions of marketing stimuli.

The 99 price ending as a signal of a low-price appeal

There is evidence that the rightmost digits, or endings, of retail prices can communicate meanings to consumers. To better understand how such meanings are formed, this paper addresses the question of how the 99 price ending can have a low-price meaning even though 99-ending prices tend to be higher rather than lower competitive prices. Analysis of two large samples of newspaper price advertising indicates that there is a strong and robust correlation between the use of the 99 price ending and the presence of a low-price appeal such as a claimed discount. It is suggested that the salience of price advertising leads it to dominate other sources of information in the consumer's learning of price-ending meanings.

Nines in the Endings of Stock Prices

In this paper, we ask whether investors' responses to nines in the rightmost digits of stock prices are similar to consumers' responses to nines in the rightmost digits of retail prices. In contrast to the increase in demand that is often observed for 9-ending retail prices, excessive overnight selling tends to follow stocks that close on 9-ending prices. Additional analysis indicates this may be due to a resistance level created by round-dollar stock prices. This finding sheds light on alterative explanations offered for the clustering of ending digits in prices and may provide guidance on how to expand or adjust the maximizing explanations for the phenomenon.

Consumer Cognition and Pricing in the 9's in Oligopolistic Markets

The paper fully characterizes the Bertrand equilibria of oligopolistic markets where consumers may ignore the last (i. e. the right-most) digits of prices. Consumers, in this model, do not do this reflexively or out of irrationality, but only when they expect the time cost of acquiring full cognizance of the exact price to exceed the expected loss caused by the slightly erroneous amounts that is likely to be purchased or the slightly higher price that may be paid by virtue of ignoring the information concerning the last digits of prices. It is shown that in this setting there will always exist firms that set prices that end in nine though there may also be some (non-strict) equilibria where a non-nine price ending occurs. It is shown that all firms earn positive profits even in Bertrand equilibria. The model helps us understand in what kinds of markets we are most likely to encounter pricing in the 9’s.

Image Communicated by the Use of 99 Endings in Advertised Prices

In a controlled experiment, use of the 99 rather than the 00 price ending (e.g., using $49.99 rather than $50.00) affects the impression created by a price advertisement. The 99 ending increases the likelihood that viewers judge an advertised price as relatively low and as representing a discount. However, in addition to these price-image effects, the 99 ending has negative effects on quality image in the ads sponsored by higher quality retailers. These results suggest that the choice of rightmost digits is an important executional variable in price advertising.

Price clustering in foreign exchange spot markets

This paper documents the existence of price clustering in the foreign exchange spot market for the German mark, the Japanese yen, the United Kingdom pound, the French franc, the Italian lira, and the Swedish krona. The U.S. dollar exchange rate indicative quotes for these currencies tend to exhibit clustering around right-most digits that end in either a “zero” or a “five.” The tendency for exchange rates to cluster has increased with increases in trading volume and volatility. Moreover, the tendency for exchange rates to cluster differs across currencies.

Patterns of Rightmost Digits Used in Advertised Prices: Implications for Nine‐Ending Effects

Analysis of the rightmost digits of selling prices in a sample of retail price advertisements confirmed past findings indicating the overrepresentation of the digits 0, 5, and 9. The high cognitive accessibility of round numbers can account for the overrepresentation of 0- and 5-ending prices and suggests the existence of two effects that could account for the overrepresentation of 9-ending prices: (1) a tendency of consumers to perceive a 9-ending price as a round-number price with a small amount given back and (2) a tendency of consumers to underestimate a 9-ending price by encoding it as the first round number evoked during incomplete left-to-right processing. Analysis of the patterns of rightmost digits observed in the sample provides supportive evidence particularly for the second of these two 9-ending effects.

Increased consumer sales response though use of 99-ending prices

Through the cooperation of a direct-mail women's clothing retailer, we were able to conduct a well-controlled experiment testing the sales effect of using retail prices that end in the digits 99 rather than 00 (e.g., $29.99 rather than $30.00). The results indicated that the use of 99 endings led to increased consumer purchasing. This finding demonstrates the importance of the manager's decision concerning a price 's rightmost digits.

Data fabrication: Can people generate random digits?

Many people have difficulty in generating random numbers. This difficulty suggests that potentially fabricated numbers encountered in investigations of scientific misconduct be examined for nonrandom behavior. The present paper shows that even with a conscious effort to construct random digits, many subjects are unable to produce digits with a uniform distribution. For this study, subjects were directed to try to produce random digits in three places in order to fabricate a series of “pick 3”; lottery numbers. Subjects were most successful at producing a random (uniform) distribution of digits for the leftmost place; however, success at one place was not associated with success at another. In addition, subjects did not select all digits with equal frequency. Of 8,280 digits chosen in this study, the order from most to least chosen was 1, 2, 3, 6, 4, 9, 7, 0, 8, 5. Finally, no strong correlations among subjects’ digit choices were found. The conscious effort by these subjects to produce random digits stands in contrast with the usual case of data fabrication in which the fabricator must devote a conscious effort to choose leftmost digits so the number has the magnitude desired and pays little or no attention to the fact that the rightmost digits should be random. The results of the present paper indicate that even if a data‐fabricator were aware that error digits would be examined for uniformity, success in constructing uniform error distributions is not guaranteed. The difficulty that people have in creating random error digits supports the utility of examining such digits in investigations of scientific misconduct.