Abstract Consumers often encounter product-related numerical information, such as attribute ratings and version numbers. This research demonstrates that a smaller (compared to a larger) numerical difference can increase perceived improvement and enhance product appeal. We find that when a product’s version number or rating changes from a decimal number to an integer (e.g., 2.4 to 3), product appeal is enhanced compared to when the change is between two integers (e.g., 2 to 3), even though the latter difference is mathematically larger. This effect occurs when the meaning of the numerical information is unclear, leading consumers to try to infer what it represents. We suggest that a decimal number is inferred to be part of a fine-grained scale, in which decimals are the intermediate values and integers are endpoints or category boundaries. The switch from a decimal to an integer is therefore perceived as skipping over intermediate values and crossing a category boundary. This suggests that the product has made a substantive improvement, making it more appealing. A consecutive integer-to-integer change does not provide a cue to support such inferences. In five studies, we demonstrate the decimal-to-integer effect, its underlying process, and its boundary conditions. numerical information, categorical perception, product versions, product attributes, product ratings Seven highly polished, brand-new handles and seven sets of fine gold lettering spelling the words Nimbus Two Thousand and One gleamed under the Gryffindors’ noses in the early morning sun. “Very latest model. Only came out last month,” said Flint carelessly, flicking a speck of dust from the end of his own. “I believe it outstrips the old Two Thousand series by a considerable amount.” (Harry Potter and the Chamber of Secrets, Rowling 1998) Consumers often encounter numerical information regarding products they might consider buying. This numerical information can include model and version numbers, ratings, or information regarding various product attributes. Yet, unlike J. K. Rowling’s young wizards, consumers are not always easily able to assess the extent to which a product has improved over time: numerical information can be unclear, and the meaning of the differences difficult to understand (Gunasti and Ross 2010; Hsee et al. 2009). Surprisingly, we suggest and demonstrate that a smaller numerical difference can sometimes be perceived as greater, which in turn can affect product judgments. In this research, we focus on numerical differences involving decimals and integers, and the perceived differences between two such numbers. We suggest that, in some cases, a smaller numerical difference can be perceptually larger, subsequently increasing product attractiveness. More specifically, we show that when a product-related number or rating increases from a decimal number to an integer (e.g., from 3.4 to 4), consumers will find the product more appealing than when the increase is from one integer to the next (e.g., from 3 to 4), even though the difference is objectively greater in the latter case. This effect should occur when consumers encounter unclear numerical information, and make inferences about the meaning of the differences between numbers based on cues such as precision. Specifically, we argue that a decimal number can trigger an inference that it is drawn from a more precise numerical scale. Distances between units on more precise scales can be perceptually greater, and may signal that intermediate values exist (Burson, Larrick, and Lynch 2009; Pandelaere, Briers, and Lembregts 2011). Moreover, when decimal numbers are perceived as intermediate values, integers may be seen as endpoints or category boundaries. Crossing a category boundary can enhance the perceived magnitude of a change (Isaac and Schindler 2013). We therefore suggest that the switch from a decimal to the next integer implies that several intermediate values were skipped and a category boundary has been crossed. When the difference is between two integers, however, consumers have no reason to assume that intermediate values exist, and the greater integer is merely the next consecutive number rather than a category boundary. Consequently, the switch from a decimal number to an integer should be perceptually larger than an integer-to-integer change, increasing product appeal. In other words, the perception of an integer as a category boundary is contingent on the existence of intermediate values; if enough intermediate values are skipped and a category boundary is crossed, the perceived magnitude of the change is enhanced. This in turn drives the positive effect of a decimal-to-integer change. We base this proposition on several streams of research. First, numerical precision can affect evaluations and estimates (Janiszewski and Uy 2008; Thomas and Park 2014; Xie and Kronrod 2012), and precise numbers draw more interest and attention than round numbers (Santos, Leve, and Pratkanis 1994). The precision of a numerical scale can also affect consumer perceptions (Pandelaere et al. 2011). Second, research has shown that differences between categories, and the perception that a category boundary has been crossed, can affect consumers’ perceptions and behavior (Irmak, Naylor, and Bearden 2011; Isaac and Schindler 2013). We link and extend these two streams of research to demonstrate that a decimal-to-integer difference can also be an indicator that a product has crossed a threshold into a new category. In addition, while past studies have found that similar differences can seem more substantial when crossing a category boundary, we show that smaller differences may seem larger in the case of a decimal-to-integer change. Third, we broaden the understanding of how consumers make inferences based on numerical information, focusing on precise numerical differences. Numerical information related to product attributes or contained in alphanumerical brand names can lead to consumer inferences (Yan and Duclos 2013), such as a “bigger is better” approach (Gunasti and Ross 2010). We demonstrate how differences between numbers can influence assessments of product improvement, and that smaller differences may sometimes be more effective in communicating product improvement. The remainder of this article is organized as follows. First, we review past literature on how numerical information, precision, and perceptions of category change affect consumers’ evaluations and decision making. We build upon these streams of research in formulating our hypotheses about the positive effect of smaller differences involving decimal-to-integer upgrades. We then report five studies that provide support for this effect and for its proposed underlying process and moderators. Finally, we discuss the theoretical and practical implications of our findings. THEORETICAL BACKGROUND Inferences Based on Numerical Information and Precision Numerical information has been studied in various contexts, including advertising (Schindler and Yalch 2006; Xie and Kronrod 2012), pricing (Thomas, Simon, and Kadiyali 2010; Wadhwa and Zhang 2015), confidence (Jerez-Fernandez, Angulo, and Oppenheimer 2014), and negotiations (Mason et al. 2013). Product-related numerical information may appear in a brand name, model, or version number (such as Apple iOS 9.3); as a rating of the product or its attributes (such as a CNET rating for tablet design); or in information regarding various product attributes (such as energy efficiency values). Past research has shown that consumers often draw inferences from product-related numerical information, including the sequence of numbers. For example, alphanumeric brand names (brand names that include a model number) can increase perceived technological improvement, product differentiation, and willingness to pay compared to brand names that include only verbal information, especially for radical innovations (Auh and Shih 2009). Yan and Duclos (2013) have suggested that consumers may use alphanumeric brand names as anchors when making inferences regarding unknown product attributes. In addition, consumers are often guided by a “bigger is better” approach to sequential numbering (Gunasti and Ross 2010; Pavia and Costa 1993). The relative precision of numbers may also affect consumer perceptions and behaviors. A number is considered more precise if it ends in fewer zeroes, or with more digits after the decimal point (Janiszewski and Uy 2008; Thomas et al. 2010). Research has demonstrated different effects for round as opposed to precise numbers. Round numbers are perceived as delivering more stability and benefits than precise numbers: Pena-Marin and Bhargave (2016) found that consumers perceive energy drinks and pills that came in round-number doses as more effective than those with a more precise volume. Round numbers are also more likely to serve as reference points for goal-directed behavior. For example, SAT test takers were more likely to retake the test if their score fell just shy of a round number than if their score was a round one (Pope and Simonsohn 2011). This indicates that round numbers may be perceived as boundaries of a desired level of performance. Precise numbers generate different perceptions than round numbers. They are seen as more accurate, credible, factual, and scientific, while round numbers may seem estimated or arbitrary (Santos et al. 1994; Schindler and Yalch 2006; Xie and Kronrod 2012). Consumers often assume that more precise numbers are used for a reason and convey important information (Isaac, Brough, and Grayson 2016; Zhang and Schwarz 2012). More precise numerical scales have also been shown to affect consumers’ evaluations. For example, the same relative numerical difference can be perceptually larger when expressed along a scale that is expanded (or finer-grained), which enhances product judgments (Pandelaere et al. 2011; Tao, Wyer, and Zheng 2017; Zhang and Schwarz 2012). We go further and argue that even a smaller numerical difference can sometimes be perceptually larger. The appearance of a precise number—in our case, a decimal number—in a product description may cue consumers that a relatively precise scale or measurement is in use. We expect consumers to further infer that decimal numbers are intermediate values, while integers are endpoints or category boundaries. If a numerical upgrade leaps over several intermediate values and crosses this perceptual boundary, it should affect consumers’ perceptions and preferences. In the next section, we discuss how the perception of crossing category boundaries can affect consumers’ evaluations. Categories and Boundaries People tend to organize information in categories. For instance, test scores that range between 0 and 100 can be categorized as letter grades, while hotels can be placed in top-10 lists based on guest ratings or certain features (e.g., “top 10 romantic hotels”). These categories can have a considerable effect on consumer evaluations and behavior. Research has shown that competitive behavior increases as a meaningful threshold is approached, such as a top 500 rating (Garcia, Tor, and Gonzalez 2006). Goal-directed behavior is also affected by proximity to numerical standards, such as a “.300 batting average” in baseball (Pope and Simonsohn 2011), or a time under four hours for marathon runners (Allen et al. 2016). While the specific number can shift due to contextual information or individual ability, being just below or just above numerical reference points may be perceived as the difference between a decent performance and a great one. Thus, numerical reference points may be seen as boundaries separating performance categories. Numerical boundaries are typically round numbers (e.g., top 10 or top 500; Garcia et al. 2006; Isaac and Schindler 2013), or other meaningful numbers, such as times conveyed in hour or half-hour increments (Allen et al. 2016). Because category boundaries are meaningful, crossing them can enhance the perceived magnitude of a change. For example, the difference in the evaluation of a student ranked 10th compared to a student ranked 11th is perceptually greater than the difference between the students ranked 11th and 12th, because it crosses a threshold into the top 10 (Isaac and Schindler 2013). Crossing category boundaries can also affect perceptions and behaviors in non-numerical contexts (Zhao, Lee, and Soman 2012). One example of this is the “out-of-region bias,” in which individuals estimate places to be closer when they are located in the same (geographical) category than in different categories (Irmak et al. 2011). Another is the “border bias,” which refers to greater estimations of risk when a disaster spreads from the same state, rather than from an equidistant location in a different state (Mishra and Mishra 2010). Tu and Soman (2014) reported a similar pattern of findings for perceptions of events that cross a time boundary, such as the end of a month. Taken together, there is a substantial body of research showing that crossing an explicit or perceptual category boundary can increase the perceived magnitude of a change, compared to a within-category change. We propose that a decimal number creates an inference of a more precise scale, in which decimals are the intermediate values and integers serve as category boundaries. If this is the case, moving up from a decimal number to the next integer will be seen as crossing a category boundary. This should make such a change perceptually greater compared to a change between two consecutive integers, in which there is no indication of intermediate values or the crossing of a category boundary. The Present Research In this research, we focus on perceptions of differences between decimal numbers and integers. Although such differences are often mathematically smaller than those involving two integers, they may nonetheless increase the perceived degree of product improvement and the appeal of the product. We therefore propose that when a product’s numerical information changes from a decimal number to the next integer (e.g., from 2.4 to 3), consumers will find the product more appealing than when the change is between two integers (e.g., from 2 to 3), even though the difference is larger in the latter case. We term this phenomenon the decimal-to-integer effect. This effect occurs when consumers try to infer the meaning of unclear numerical information. The decimal number suggests that additional intermediate values exist, and the integer is perceived as a category boundary between decimals. A change to the next integer suggests that several intermediate values were skipped and a category boundary has been crossed, making the decimal-to-integer difference more meaningful than an integer-to-integer difference. A scale comprised only of integers lacks sufficient information to allow for inferences about intermediate values or category changes. Thus, while the difference between two consecutive integers (e.g., between 2 and 3) is objectively larger, the difference between a decimal number and the next integer (e.g., between 2.4 and 3) can be perceptually larger, and this will increase product attractiveness. H1: In the context of product-related numerical information, a change from a decimal number to the next integer will enhance product appeal to a greater extent than a change between two consecutive integers (the decimal-to-integer effect). We proposed that the decimal-to-integer effect occurs because the decimals indicate that a scale or measure is more precise, with decimals as the intermediate values and integers as the endpoints or category boundaries. Skipping over intermediate decimal values to the next integer therefore suggests that the product has crossed a category boundary and has improved substantially, making it more appealing. If this proposed process is accurate, the decimal-to-integer effect should be affected by explicit cues about the precision of the scale, its intermediate values, and its boundaries. Explicit information about the precision of the scale and the presence of intermediate values should make integer-to-integer changes more meaningful. First, information about the precision of the scale can indicate that an integer-to-integer change has also skipped several intermediate values, making it perceptually greater than it would be without such information. This is in line with past research showing that the same relative numerical difference is perceived as greater when it is represented on a more precise scale (Burson et al. 2009; Pandelaere et al. 2011). Second, if a scale explicitly includes intermediate values, integers should be seen as category boundaries even when the change is from an integer to the next integer. Thus, providing an explicit indication that there are intermediate values means that both decimal-to-integer differences and integer-to-integer differences should provide a similar indication that intermediate values were skipped and a category boundary has been crossed. This should eliminate the decimal-to-integer effect. H2: The decimal-to-integer effect will not occur in the presence of explicit information about intermediate values. Perceptions of the meaning of the numerical change should also be affected by explicit cues as to whether a category boundary has been crossed. As indicated by past research (Irmak et al. 2011; Isaac and Schindler 2013), both numerical and non-numerical changes that cross a perceived category boundary can be experienced as greater than comparable changes that do not cross a boundary. We propose that the decimal-to-integer effect is caused by the inference that a category boundary has been crossed. If this suggested underlying mechanism is correct, we should be able to attenuate the decimal-to-integer effect by presenting explicit information about a category change. H3: The decimal-to-integer effect will be attenuated in the presence of explicit information about category change. While some types of numerical information are easy to understand and map onto existing knowledge, others are not. Past research has indicated that consumers rely on numerical information even when they do not fully understand what it represents (Gunasti and Ross 2010; Hsee et al. 2009). In this research, we consider several types of information that consumers might find unclear and therefore difficult to understand: product versions and model numbers (Gunasti and Ross 2010), ratings such as those provided on review websites, and unfamiliar attribute values such as camera specifications (Hsee et al. 2009). When the meaning of the numerical information is unclear and consumers lack the context needed to understand it, they are more likely to rely on the precision of the numbers as a cue, which should lead to the decimal-to-integer effect. When clearer numerical information is available, consumers can rely on existing knowledge and understanding (Gunasti and Ross 2010). This should allow them to assess the meaning of a decimal-to-integer change as well as that of an integer-to-integer change, so additional cues such as numerical precision are less likely to affect evaluations. Thus, clearer numerical information should lead to an attenuation of the decimal-to-integer effect. H4: The decimal-to-integer effect will occur when the meaning of the numerical information is unclear but not when its meaning is made clear. Overview of Studies In five studies, we demonstrate that the change from a decimal number to the next integer increases product appeal to a greater extent than a change from an integer to the next integer. Our suggested model is summarized in figure 1. Study 1 demonstrates the decimal-to-integer effect in the context of product version numbers (hypothesis 1). Study 2 replicates the effect and explores its limits. In study 3, we turn to product attributes, and examine how explicit information about the precision of the scale and its intermediate values moderates the effect (hypothesis 2). In addition, we show that the effect is mediated by the perception that a category was crossed. In study 4, we manipulate category boundaries directly in order to provide further support for our proposed underlying process (hypothesis 3). Finally, study 5 focuses on the clarity of the numerical information as a boundary condition of the effect (hypothesis 4) in the context of individual ratings. FIGURE 1 View largeDownload slide SUMMARY OF HYPOTHESES AND STUDIES FIGURE 1 View largeDownload slide SUMMARY OF HYPOTHESES AND STUDIES STUDY 1: THE DECIMAL-TO-INTEGER EFFECT The goal of study 1 was to establish the decimal-to-integer effect. Focusing on the effect in the context of product version numbers, we selected software that the participants use regularly as the target product, and varied the information regarding its existing version number. Thus, the study made use of a real-life context in which participants were familiar with the product, but not with the meaning of the numerical information. In line with hypothesis 1, we expect an upgrade from a decimal number to the next integer version number to be more appealing than an upgrade from an integer to the next integer version number, even though the latter is mathematically larger. Method Participants and Design Ninety-six undergraduate management students (60.4% female, Mage = 24.75, SD = 5.96) took part in an online study in exchange for credit or a chance to win a gift certificate to a local café. They were randomly assigned to one of two conditions: software whose existing version number was either 3 (integer condition) or 3.4 (decimal condition). The new version in both conditions was 4. Procedure Participants were told that they would be completing two questionnaires: one in which they would provide feedback about the faculty’s study sign-up system, and a second, unrelated study involving product reviews. The latter study was included to increase the credibility of the cover story. Following this introduction, participants read the manipulation for this study, in which they were told that the software that they had been using to sign up for studies and receive credit for studies was either version 3 or version 3.4, and that the university was considering an upgrade to the recently released and improved version 4 (see the web appendix for information about the stimuli used in all studies). Participants were then asked to rate their support for the upgrade using the following items: “It would be a good idea to upgrade the study management system,” “I would be happy with a decision to upgrade the study management system,” and “It is a good idea to upgrade when the possibility arises.” These three items had a Cronbach’s α = .89, and their mean score is used in the analysis. Participants also rated how satisfied they were with the current version of the system. Responses were given on a seven-point scale (1 = strongly disagree, 7 = strongly agree). In keeping with the cover story for the study, participants were asked to provide feedback about the current system, and to suggest features that they would like a new version to include. Participants then completed the unrelated study questionnaire before providing background information. Results and Discussion Participants in both conditions were equally satisfied with the current version of the study management system (Mdecimal = 4.83, SD = 1.37 vs. Minteger = 4.96, SD = 1.32; t < 1). However, we found the expected differences in participants’ preferences for an upgrade: participants who were told that the upgrade under consideration was from version 3.4 to version 4 were more favorable about an upgrade (M = 5.04, SD = 1.39) than those who were told that the upgrade being considered was from version 3 to version 4 (M = 4.49, SD = 1.13; t(94) = 2.16, p = .034). Study 1 provides support for hypothesis 1 by showing that a smaller numerical change in product-related information can increase product attractiveness compared with a larger numerical change. A new software version with an integer number was more appealing to participants who believed that it was preceded by a decimal version number than to those who were told that it had been preceded by another integer, even though the latter change was greater. While participants were familiar with the product itself, they were given no specific information about the new version or the benefits it offered, leaving them to rely on the version number as a cue. We did not explicitly state what versions had been previously introduced by the company, only that the faculty was using either version 3 or version 3.4. Participants in the decimal condition could have assumed that the company had released intermediate versions, such as 3.5–3.9, making the new version number a more meaningful category boundary. A follow-up study conducted among 60 undergraduate students (65% female; Mage = 23.79, SD = 2.26) using the same scenario provided evidence supporting this possibility. Participants agreed that “there were probably other versions of this software that were released between version 3 (or 3.4) and version 4” to a greater extent when the proposed upgrade was from version 3.4 to version 4 (M = 4.71, SD = 1.37) than when it was from version 3 to version 4 (M = 3.79, SD = 1.47; t(58) = 2.50, p = .015). STUDY 2: MULTIPLE COMPARISONS OF THE DECIMAL-TO-INTEGER EFFECT We proposed that a decimal number implies that a precise scale or measure is in use, with decimals as intermediate values and integers as category boundaries. If (1) enough intermediate values are skipped and (2) a category boundary is crossed, then the perceived magnitude of the change is enhanced and the decimal-to-integer effect occurs. In study 2, we test these limits by varying the version numbers involved in the comparison. Focusing on the first part of the process, we propose that when the difference between the decimal number and the next value is relatively small, the effect should diminish, as few intermediate values are skipped. This would be in line with Tao et al.’s (2017) finding that scale range effects occur at values that are moderately close to a scale endpoint but are reduced for values that are close to the endpoint. Turning to the second part of the proposed process, we do not expect the effect to occur when a comparable decimal change occurs within the same integer number (e.g., 2 vs. 2.6), since no perceptual category boundary is crossed. This prediction is also in line with Thomas and Morwitz’s (2005) research on price judgments, which showed that consumers tend to assign greater weight to the leftmost digits in a number. Thus, if the integer component of a number remains unchanged, consumers may focus on that rather than on changes in the digits after the decimal point. Method Participants and Design Three hundred eighty-seven participants (51% female, Mage = 33.51, SD = 9.22) from English-speaking countries were recruited online and were compensated for completing the study. Data collection was done in two stages, each on a different online survey platform and involving a different product: a camera on Amazon Mechanical Turk (n = 196), and photo-editing software on Prolific Academic (n = 191). We ran the same five conditions for each product. Both platforms and products yielded similar results, and we therefore report the results of the integrated data. This resulted in a 2 (product type: camera or software) × 5 (version change: existing version 2 vs. new version 3; existing version 2.4 vs. new version 3; existing version 2 vs. new version 2.6; existing version 2.8 vs. new version 3; existing version 2.7 vs. new version 3.3) between-subjects design, with participants randomly assigned to one of the version conditions. Procedure Participants read a description of one of two products: (a) a “social camera,” which was described as allowing users to produce and edit a variety of enhanced content for sharing on social media or other platforms, or (b) photo-editing software, which was described as good for photo editing and graphic design, offering a variety of features, and intuitive to learn and easy to use. These descriptions were based on existing products, but included several additional features in order to ensure that participants would not rely on preexisting knowledge in their evaluations. After reading about the product, participants were given information about the previous and the new model of the camera/version of the software: (a) existing version 2 with version 3 about to be launched; (b) existing version 2.4 versus new version 3; (c) existing version 2 versus new version 2.6; (d) existing version 2.8 versus new version 3; (e) existing version 2.7 versus new version 3.3. Participants were asked to rate their interest in obtaining the new version/model: “I would like to get the new version/model of the [product name].” Responses were given on a seven-point scale (1 = strongly disagree, 7 = strongly agree). Finally, participants answered a number of background questions. Results and Discussion A 2 × 5 ANOVA with product type and version change as the independent variables and interest in obtaining the new version of the product as the dependent variable revealed two significant main effects and no significant interaction (F(4, 377) < 1). Interest in obtaining the product (F(1, 377) = 45.43, p < .001) was greater for the software (M = 5.66, SD = 1.19) than for the camera (M = 4.64, SD = 1.69). Participants may have reasonably assumed that the software was free, which enhanced its appeal. Of greater importance, the difference in version numbers had a significant effect on participants’ interest in the product (F(4, 377) = 4.07, p = .003). We conducted a post hoc analysis using a Bonferroni correction for multiple comparisons. The expected decimal-to-integer effect was replicated, such that interest in the product was lower in the 2 versus 3 condition (M = 4.68, SD = 1.79) compared with the 2.4 versus 3 condition (M = 5.61, SD = 1.14, p < .001). Furthermore, there was a significant difference between the 2 versus 3 condition and the second condition in which a category boundary was crossed, the 2.7 to 3.3 change (M = 5.33, SD = 1.55; p = .021). Thus, the effect may be more accurately described as decimal-through-integer. However, there was no significant difference between the 2 versus 3 condition and the 2.8 versus 3 condition, which involved a small change of only one intermediate value (M = 5.14, SD = 1.55; p = .201). In addition, the 2 versus 3 condition did not differ significantly from the 2 versus 2.6 condition, in which the decimal change did not cross a category threshold (M = 5.00, SD = 1.64; p = .625). The results of study 2 provide additional evidence for a decimal-to-integer effect: smaller numerical differences between version/model numbers can increase product appeal, again supporting hypothesis 1. The effect was found for both tangible (camera) and nontangible (software) products. It occurs when the new version reaches a category boundary (version 2.4 vs. version 3) and when the category boundary is crossed, with the change reaching into the new category (version 2.7 vs. version 3.3). In both cases, the two parts of the proposed underlying process occur: multiple intermediate values are skipped and a category boundary is crossed. These results are in line with past research showing that changes that either reach or cross a category boundary can be perceptually larger than those that do not (Irmak et al. 2011; Isaac and Schindler 2013). In the next two studies, we provide support for the two parts of the proposed underlying process by examining how explicit information regarding the precision of the scale and category change can moderate the effect. STUDY 3: INTERMEDIATE VALUES AND THE DECIMAL-TO-INTEGER EFFECT In study 3, we explore the mechanism underlying the decimal-to-integer effect. We have suggested that the effect occurs when consumers encounter unclear numerical information, and seeing a decimal number leads them to infer that the scale is more precise. This precision suggests that the integers represent endpoints or category boundaries. The change from a decimal number to an integer signals that intermediate values have been skipped and a category boundary reached, making it perceptually greater than an integer-to-integer change. In hypothesis 2, we hypothesized that the effect would not occur in the presence of explicit information about intermediate values, as in this case both the decimal- and the integer-to-integer changes leap over intermediate values to cross a category boundary. In this study, we used a relatively unclear product attribute measure (a camera’s “color accuracy rating”) and provided participants with two illustrations (between-subjects) of possible scale values for this attribute. One scale contained only integers and the other scale contained integers and decimals (see figure 2). In the integers-only scale, there is no indication of intermediate values. This leads consumers to rely on the numerical information to assess the relative improvement and appeal of the product, resulting in a replication of the decimal-to-integer effect. When a decimal scale is provided, it is clear how many intermediate values there are between the old rating and the new one, and the integers represent category boundaries. Thus, several intermediate values are skipped and a category boundary is crossed in both the decimal-to-integer and the integer-to-integer conditions, which should eliminate the effect. FIGURE 2 View largeDownload slide SCALE ILLUSTRATIONS FIGURE 2 View largeDownload slide SCALE ILLUSTRATIONS In addition to manipulating intermediate values, we also sought to obtain preliminary evidence that the effect is the result of a perceived category change, and to explore whether this perception mediates the expected moderation effect. Method Participants and Design The sample included 190 English-speaking MTurkers (44.8% female; Mage = 32.36, SD = 7.45) who received 70 cents for their participation. Participants were randomly assigned to one of four conditions in a 2 (previous rating: integer or decimal) × 2 (scale illustration: round or precise) between-subjects design. Procedure We asked participants to imagine that they were considering buying a compact camera with a good zoom for an upcoming trip, and had found one that potentially fits their needs. They received a brief description of the camera and its features, which concluded with the following: “While the previous model of this camera had a color accuracy rating of 5 (integer condition)/5.4 (decimal condition), the new model has a rating of 6.” In the round-scale condition, participants were provided with an illustration of the color accuracy rating scale, which included integers between 2 and 7 and no intermediate values. In the precise-scale condition, they received an illustration that also included smaller marks denoting nine intermediate values between the integer points, similar to the marks on a ruler (see figure 2). After reading this description, participants were asked to evaluate the new model of the camera using two items: “I like the new model of the camera” (seven-point scale from 1 = strongly disagree to 7 = strongly agree) and “My evaluation of the new camera is __ (1 = not at all positive to 7 = very positive).” The two items had high reliability (Cronbach α = .86), and their mean score is used in subsequent analyses. As explained above, we also examined perceived category change as a mediator, as a precursor to manipulating it directly in the next study. Participants were therefore asked to rate their agreement with the following item on a seven-point scale from 1 (strongly disagree) to 7 (strongly agree): “The new model has crossed a significant threshold in terms of color accuracy.” Finally, participants answered a number of camera-related questions and provided background information. Results Camera Evaluation We ran a 2 × 2 ANOVA with previous accuracy rating and scale illustration as the independent variables and product evaluation as the dependent variable. This analysis revealed a significant interaction (F(1, 186) = 9.42, p < .001). In line with hypothesis 2, the decimal-to-integer effect was replicated in the round-scale-illustration condition. As figure 3 shows, the camera was evaluated more positively in this condition when its color accuracy rating had increased from 5.4 to 6 (M = 5.78, SD = .90) than when it had improved from 5 to 6 (M = 5.35, SD = .82; (F(1, 186) = 5.32, p = .022), replicating the decimal-to-integer effect (hypothesis 1). The effect was reversed among participants provided with a precise-scale illustration: the camera was evaluated more positively when its color accuracy rating had improved from 5 to 6 (M = 6.06, SD = .70) compared to an improvement from 5.4 to 6 (M = 5.59, SD = .97; F(1, 186) = 7.96, p = .005). The reversal likely occurred because the 5 to 6 increase in the precise-scale condition clearly contains more intermediate values than the 5.4 to 6 increase. FIGURE 3 View largeDownload slide CAMERA EVALUATIONS BASED ON RATING AND SCALE ILLUSTRATION FIGURE 3 View largeDownload slide CAMERA EVALUATIONS BASED ON RATING AND SCALE ILLUSTRATION Notably, the mean evaluation of the camera in the integer-to-integer conditions was significantly lower among participants presented with the round-scale illustration compared to those presented with a precise-scale illustration (F(1, 186) = 16.95, p < .001). However, evaluations in the decimal-to-integer conditions did not differ significantly based on the scale illustration (F < 1). This suggests that the scale illustration provided participants in the integer-to-integer condition with new information: whether the color accuracy rating includes intermediate values. The participants in the decimal-to-integer condition, however, were likely more aware that intermediate values were a possibility, regardless of the scale illustration. Perceived Category Change We ran an additional 2 × 2 ANOVA with previous accuracy rating and scale illustration as the independent variables and perception of crossing a meaningful category threshold as the dependent variable. This analysis revealed a significant interaction (F(1, 186) = 8.23, p = .005). As we expected, in the round-scale-illustration condition, the perception of category change was greater when the color accuracy rating had improved from 5.4 to 6 (M = 4.69, SD = 1.64) compared to an improvement from 5 to 6 (M = 4.10, SD = 1.45), an effect that was marginally significant (F(1, 186) = 3.35, p = .069). The effect was reversed among participants provided with a precise-scale illustration: here, the perception of category change was greater when the color accuracy rating had improved from 5 to 6 (M = 4.93, SD = 1.55) compared to an improvement from 5.4 to 6 (M = 4.27, SD = 1.34; F(1, 186) = 5.06, p = .026). We conducted a moderated mediation analysis, using bootstrapping mediation tests (Hayes 2013) with 5,000 replications. In Hayes model 7, scale illustration served as the moderator for the effect of previous accuracy rating condition on camera evaluation, and perception of category change served as the mediator. As expected, the effect of previous accuracy rating condition on the evaluation of the camera was mediated overall by the perception of category change (b = –.22, SE = .11; 95% CI: –.49 to –.06). Decomposing the mediation analysis into the different scale illustration conditions revealed that this mediation was positive in the round-scale-illustration condition (b = .1.0, SE = .07; 95% CI: .0009 to .27), and negative in the precise-scale-illustration condition (b = –.12, SE = .06; 95% CI: –.27 to –.017). Discussion The results of study 3 lend further support to our theorizing that the decimal-to-integer effect stems from the perception that such changes leap over intermediate values to cross a category threshold. In line with hypothesis 2, we found that the effect is replicated when the numbers provide the only cue as to whether the scale includes intermediate values. In this case, the change from a decimal number to an integer suggests that intermediate values have been skipped and a meaningful category boundary is crossed. When there is an explicit indication that intermediate values are possible—such as by means of a precise-scale illustration—participants no longer need to rely on the numerical information, and the effect does not occur. Interestingly, the effect was fully reversed in this study, possibly because the precise illustration caused participants to perceive the integer-to-integer change as leaping over a greater number of intermediate values. We examined this possibility in a follow-up test using the same manipulation on 173 different MTurkers (60.7% female; Mage = 32.57, SD = 8.16). Participants in the round-scale condition expressed greater agreement with the statement that “there are probably other possible color accuracy ratings between 5 (or 5.4) and 6” if the previous color accuracy rating was a 5.4 (M = 5.48, SD = 1.57) than when it was 5 (M = 4.07, SD = 1.59; F(1, 169) = 18.24, p < .001). Conversely, among participants in the precise-scale condition, there was no difference between the decimal condition (M = 5.28, SD = 1.52) and the integer condition (M = 4.93, SD = 1.36; F(1, 169) = 1.16, p = .283). Moreover, we found that when participants were asked to write how many intermediate values they thought there were between 5 (or 5.4) and 6, those in the round-scale condition indicated a greater number of intermediate values when the previous version was 5.4 (M = 3.40, SD = 2.24) than when it was 5 (M = 1.85, SD = 3.12; F(1, 169) = 5.14, p = .025). This pattern was reversed in the precise-scale condition, in which participants indicated a greater number of intermediate values when the previous version was 5 (M = 4.96, SD = 4.26) rather than 5.4 (M = 3.68, SD = 2.37; F(1, 169) = 3.66, p = .057). Both analyses yielded significant interactions: F(1, 169) = 5.36, p = .022, and F(1, 169) = 8.76, p = .004, respectively. Thus, participants exposed to a round-scale illustration were more likely to believe that there were intermediate values and to indicate a greater number of intermediate values when the previous version was 5.4 rather than 5. In the precise-scale condition, however, participants were equally likely to believe that intermediate values exist, but those in the integer (5) condition indicated a greater number of intermediate values, which is likely why the effect was reversed in this condition. Study 3 also provided an initial indication as to the role of perceived category change. The perception that the product had moved into a new category mediated the effect. In study 4, we provide stronger evidence for the impact of category change by manipulating this factor more directly. STUDY 4: MANIPULATING INFORMATION ABOUT CATEGORY CHANGE In study 4, we examine how information about category change moderates the decimal-to-integer effect. In hypothesis 3, we proposed that since the decimal-to-integer effect results from a perception of a category change, it should be attenuated if we use an explicit category change manipulation that overrides the relatively subtle decimal cue. In this study, we test this part of the proposed process by providing participants with an explicit indication that the numerical change has crossed a category boundary. When such an indication is given, we expect that the decimal-to-integer effect will be attenuated and product ratings will be similar in both the decimal-to-integer and the integer-to-integer conditions. Method Participants and Design We recruited 207 MTurkers (49.5% female, Mage = 36.96, SD = 11.43) to take part in an online study for which they received 50 cents. Participants were randomly assigned to one of four conditions in a 2 (previous rating: integer or decimal) × 2 (baseline or category change information) between-subjects design. To ensure the relevance of the product (a jacket), we did not include participants who indicated that winters where they lived were very mild (1 on a scale of 1–5; see details below). Procedure Participants were asked to imagine that they were seeking a new jacket for the coming winter, and that one of the leading brands had recently launched a new collection. They were told that this collection included a lightweight and stylish jacket whose warmth rating was 8, and that a similar jacket from the brand’s previous collection had a warmth rating of 7 (integer) or 7.3 (decimal). In the baseline condition, participants received no additional information. In the category change condition, participants were provided with a figure of the warmth rating scale that included seven levels, with blue representing the lowest rating and red representing the warmest (see figure 4; the full-color version appears in the web appendix). The warmth rating numbers were shown in colored font: the rating for the jacket from the previous collection was shown in light orange, and the rating for the new jacket was shown in dark orange. These levels were chosen to make the jacket appealing without placing it within the highest possible warmth category. FIGURE 4 View largeDownload slide ILLUSTRATION OF THE WARMTH RATING SCALE FIGURE 4 View largeDownload slide ILLUSTRATION OF THE WARMTH RATING SCALE After receiving this information, participants were asked to rate the attractiveness of the jacket from the new collection using two items: “I like the jacket from the new collection” and “I am interested in the new jacket.” Responses were given on a seven-point scale (1 = strongly disagree to 7 = strongly agree). The two items had high reliability (α = .91) and their mean is used in the analysis. Next, participants were asked about the weather where they lived: “How cold are winters where you live?” (1 = very mild to 5 = very cold); they then rated their interest in jackets in general and answered some background questions. Results We ran a 2 × 2 ANOVA with previous rating and category change information as the independent variables and jacket evaluation as the dependent variable. In support of hypothesis 3, we found a significant interaction (F(1, 203) = 4.46, p = .036). Planned comparisons revealed that the decimal-to-integer effect was replicated in the baseline condition, such that evaluations were significantly more positive (F(1, 203) = 4.15, p = .043) when the warmth rating had improved from 7.3 to 8 (M = 5.94, SD = 1.05) compared to when the warmth rating had improved from 7 to 8 (M = 5.54, SD = 1.16). Among participants who received an explicit cue about category change, the evaluation in the decimal condition (M = 5.62, SD = .98) did not significantly differ from that in the integer condition (M = 5.82, SD = .92; F < 1). Discussion The results of study 4 support our theorizing that the decimal-to-integer effect results from the perception of category change, and shows that this perception can be manipulated using explicit information to override inferences based on a subtler cue (the decimal number). More specifically, we replicated the effect when no information about category change was provided, and attenuated it when explicit information about whether a category boundary had been crossed was provided. The explicit information about category change mimicked the inference that occurred in the decimal-to-integer effect, thus attenuating the basic effect. In the last study, we explore a boundary condition of the decimal-to-integer effect. We suggest that the effect occurs when consumers encounter unclear numerical information, which may prompt reliance on the precision of the numbers as a cue regarding the magnitude of a change. Consequently, the effect should be attenuated when the numerical information is made clearer, allowing consumers to evaluate the meaning of the differences rather than inferring them. STUDY 5: INFORMATION CLARITY AND THE DECIMAL-TO-INTEGER EFFECT In study 5, we explore how the clarity of the numerical information moderates the decimal-to-integer effect, and extend the effect from product versions and product attributes to ratings of individuals. More specifically, this study used an unclear numerical measure (“reviewer expertise rating”), and provided information about the basis for this rating in order to attenuate the effect. We argue that when the numerical information is unclear, consumers are likely to rely on the decimal number as a cue regarding the magnitude of the change. If the numerical information is clearer, consumers can more easily evaluate the difference based on existing knowledge and understanding, so the decimal number is less likely to be used as a cue. Thus, the effect should be attenuated (hypothesis H4). Method Participants and Design The sample included 182 participants (50% female, Mage = 35.15, SD = 7.01), who received $1 for taking part in the study. Participants were approached via an Israeli commercial online panel that has over 30,000 registered users, representing a broad range of sociodemographic characteristics. They were randomly assigned to one of four conditions in a 2 (previous rating: integer or decimal) × 2 (numerical information clarity: baseline or greater clarity) between-subjects design. Procedure Participants were asked to imagine that they were planning to buy a new laptop computer, and had come across a review of the model they were interested in on one of the online retailing platforms. They were then told that the website displays “reviewer expertise ratings” next to the reviewer’s name. In the baseline condition, no additional information was given about this rating, keeping the meaning of the rating unclear. In the greater clarity condition, an additional sentence was included, explaining that the rating was based on the number of reviews written and how many helpful votes the reviewer had received. A pretest on 72 participants (51.4% female; Mage = 34.6, SD = 8.87) confirmed that participants in the greater clarity condition agreed with the statement “the reviewer expertise rating is a clear measure” (M = 4.84, SD = 1.35) more than those in the baseline condition (M = 3.76, SD = 1.84; t(70) = 2.80, p = .007). Participants were told that the reviewer’s expertise rating had recently improved from 8 to 9 (integer condition) or from 8.3 to 9 (decimal condition). They then responded to two items about the reviewer’s expertise: “The reviewer’s expertise rating has improved considerably” and “The reviewer’s expertise rating is significantly higher than it was before.” Responses were given on a seven-point scale (1 = strongly disagree to 7 = strongly agree). These two items had a Cronbach’s α of .79, and their mean score is used in the analysis. After responding to these items, participants responded to a number of questions involving their interest in laptops and their feelings about online reviews, and provided background information. Results A two-way ANOVA with previous rating and numerical information clarity as the independent variables and reviewer evaluation as the dependent variable revealed no main effects, but a significant interaction emerged (F(1, 178) = 4.32, p = .039). As expected based on hypothesis 4, the decimal-to-integer effect was replicated in the baseline (unclear) condition: participants who were not given information about the basis for the reviewer expertise rating rated the reviewer’s improvement as greater when the rating has changed from 8.3 to 9 (M = 4.90, SD = 1.35) than when the change was from 8 to 9 (M = 4.29, SD = 1.31; (F(1, 178) = 5.52, p = .020). However, the decimal-to-integer effect was attenuated in the greater clarity condition: when the rating was followed by an explanation, there was no difference between the decimal (M = 4.62, SD = 1.27) and integer conditions (M = 4.81, SD = 1.19; F < 1). Discussion Study 5 shows that the decimal-to-integer effect occurs when the numerical information lacks clarity, thus supporting hypothesis 4. Participants who encountered an unclear numerical measure (“reviewer expertise rating”) perceived a greater degree of improvement when there was a decimal-to-integer change than when there was an integer-to-integer change. This effect was attenuated when the meaning of the measure was made clearer. We propose that this is because consumers try to infer the meaning of unclear numerical information using whatever cues are available; a decimal number can indicate that the scale in question is more precise, making the change perceptually greater. When the numerical information and its basis is clearer, consumers are better able to evaluate changes using their own knowledge and understanding, so a decimal-to-integer change is less likely to bias evaluations. GENERAL DISCUSSION Can a smaller numerical difference between product ratings or between successive version numbers increase product attractiveness? In five studies, we show that this decimal-to-integer effect can indeed occur, and demonstrate how a smaller difference can become perceptually larger. More specifically, we argue that when presented with numerical information that is difficult to understand using existing knowledge, consumers may use additional cues to infer its meaning. When they encounter a decimal number, consumers infer that it is part of a more precise scale, which includes decimals as intermediate values and integers as endpoints or boundaries. As a result, the increase from a decimal number to an integer skips over intermediate values to cross what is perceived as a category boundary, which suggests greater improvement and enhances product appeal. This cue regarding the relative improvement of the product is not available when the change is from one integer to the next. Study 1 established the decimal-to-integer effect in a real-life context and found that when a version number increases from a decimal (in this case, version 3.4) to an integer (version 4), interest in upgrading is stronger than when the increase is from one integer to the next (version 3 to version 4). In study 2, we showed that the effect occurs only if enough intermediate values are skipped and a category boundary is crossed. In study 3, we turned to product attribute information, and provided further support for the underlying mechanism of the effect using explicit cues regarding scale precision. When a scale illustration included intermediate values, the effect was reversed, likely because the integer-to-integer change was seen as skipping more intermediate values. Study 3 also provided evidence for the importance of perceived category change as a driver of the effect using a moderated mediation analysis. This part of the process was further supported by study 4, in which we directly manipulated category change: a visual indication that a category boundary has been crossed attenuated the decimal-to-integer effect. Finally, study 5 showed that the clarity of the numerical information moderates the effect, which does not occur when consumers can evaluate the meaning of the information using existing knowledge. By demonstrating how and when smaller, more precise numerical differences can increase product attractiveness, we contribute to the literature on numerical precision and its effects. While past studies have demonstrated that the same relative differences can be perceptually greater depending on the numerical scale used (Burson et al. 2009; Pandelaere et al. 2011; Zhang and Schwarz 2012), we go further by showing that smaller numerical differences may also be perceived as representing a greater change, depending on the precision of the scale. We also contribute to the understanding of the effects of product-related numerical information. Previous research has established that numerical information can lead consumers to make inferences regarding product qualities (Pena-Marin and Bhargave 2016; Yan and Duclos 2013) and that alphanumeric brand names and product version numbers can be perceived as indicators of a product’s placement within a line of products (Auh and Shih 2009; Pavia and Costa 1993). We show that the nature of the numerical gaps between product ratings and version numbers can affect perceived product improvement, and that, in some cases, a smaller gap is more effective in communicating that a product has improved significantly. This finding diverges from past results that show that consumers tend to follow a “bigger is better” approach where sequential numbering is involved (Gunasti and Ross 2010). Furthermore, our results can be linked to past work on the impact of crossing category boundaries. Research has shown that consumer perceptions and behaviors are affected by geographic, temporal, and spatial boundaries (Irmak et al. 2011; Mishra and Mishra 2010; Tu and Soman 2014; Zhao et al. 2012), and by numerical changes that cross category boundaries in ranked lists (Isaac and Schindler 2013). Our findings suggest that the difference between a decimal and an integer in product-related numerical information (such as ratings, attribute values, and version numbers) may also be seen as an indication that a category boundary or threshold has been crossed. This can then be interpreted as a more substantial change, which in turn increases product attractiveness and purchase/upgrade intentions. The present research also offers some insights regarding new product adoption and upgrade. The adoption of new products has been extensively studied in the marketing and consumer behavior literatures. However, past studies have focused primarily on factors that increase or inhibit the adoption of newly released products, especially breakthrough and really new products (Gregan‐Paxton et al. 2002; Moreau, Markman, and Lehmann 2001; Mukherjee and Hoyer 2001; Zhao, Hoeffler, and Dahl 2009). Nonetheless, many products are multigenerational in nature, with ongoing updates involving various features and periodical new releases within the same product line (Auh and Shih 2009). It is therefore important to understand the cues consumers rely on when assessing the relative value of product upgrades and ongoing improvements, and how these affect their evaluation of the new version compared to existing ones. From a practical and managerial standpoint, our findings suggest that product-related numerical information should be used with care, so that it can constitute a reliable cue for consumers. Furthermore, marketers may benefit from communicating at least some minor updates and changes to consumers more prominently, in order to further boost the impact of the announcement of more substantive changes. Focusing specifically on version numbers, if the current version number is already close to a numerical threshold, it may be advisable to signal product improvement using other methods. The results of study 4 suggest that visual or verbal cues could provide a way to indicate category change when the difference between the numbers is small. This research has several additional limitations that should be noted. We focused on differences involving relatively small values of decimal numbers and integers. Whether the effect occurs when larger numerical values are involved, such as 52.4 versus 53, remains to be seen. It is possible that as decimal numbers become more distant from the left digit (Thomas and Morwitz 2005), consumers will pay less attention to them or view them as less important. Similarly, it is not yet clear what effect multiple decimal points (e.g., version 2.4.17 compared to 2.5) would have. Future research could shed more light on the conditions under which relatively precise gaps between numbers and ratings are beneficial. In addition to the possible impact of different numerical combinations and differences, individual differences might also impact the way consumers perceive numerical gaps and subsequent product attractiveness. For example, future research could consider need for cognition (Cacioppo et al. 1996; Cacioppo and Petty 1982), which reflects an individual’s tendency to engage in and enjoy effortful cognitive endeavors; or numeracy, which refers to the ability to understand and use numerical information (Peters 2012). Both are likely to influence the manner in which consumers utilize and interpret product-related numerical information when formulating their product evaluations. While larger numerical gaps might be expected to signal a greater difference between product versions or ratings, our results demonstrate that the opposite can also be true: smaller differences can sometimes be perceptually greater and boost product attractiveness. This effect occurs for small differences involving decimal and integer numbers, but not for other types of numerical changes. In other words, some small differences feel larger than others—and it is important to “mind the gap” in this regard. DATA COLLECTION INFORMATION The experimental concept was developed jointly by the authors. The first author developed the research design and analyzed the data from studies 1 and 4. The second author analyzed the data for study 2, and the third author analyzed the study 5 data. All three authors worked on the analysis of the data from study 3 and its follow-up test. The authors communicated frequently in order to discuss research development, design, and data analysis. The studies reported in this article were conducted online. Data for study 1 was collected in March and April 2016, via the behavioral lab at Tel Aviv University’s Coller School of Management; the follow-up test was conducted via the lab in November 2017. Data for study 2 was collected on Prolific Academic and Amazon’s MTurk in March and July 2017. We collected data for studies 3 and 4 on Amazon MTurk: study 3 in July 2017, its follow-up test in November 2017, and study 4 also in November 2017. Data collection for study 5 and its pretest was done in January 2018, using Israeli consumer panels. This research was supported by grants from the Israel Science Foundation (grant no. 1197/15), the Open University of Israel Research Fund, the Jeremy Coller Foundation, and the Henry Crown Institute of Business Research. The authors would like to thank the editor, associate editor, and three anonymous reviewers for their insightful comments and guidance throughout the review process. The authors are also grateful to Darren Dahl, Shai Danziger, Nira Munichor, Manoj Thomas, Kimberlee Weaver, and the participants of the HEC Paris Marketing Department seminar for their helpful feedback and suggestions regarding this research. Supplementary materials are included in the web appendix accompanying the online version of this article. References Allen Eric J. , Dechow Patricia M. , Pope Devin G. , Wu George ( 2016 ), “Reference-Dependent Preferences: Evidence from Marathon Runners,” Management Science , 63 ( 6 ), 1657 – 72 . 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Journal of Consumer Research – Oxford University Press
Published: Mar 22, 2018
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