School of Education, Culture and Communication
Division of Applied Mathematics
BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS
A model to examine how social influence on individual grades
may bias the aggregate grade in a sequential grading process
by
Anas Bairkdar
School of Education, Culture and Communication
Division of Applied Mathematics
Bachelor thesis in mathematics / applied mathematics
Date:
2018-08-26
Project name:
A model to examine how social influence on individual grades may bias the aggregate grade in a sequential grading process.
Author: Anas Bairkdar Supervisor: Fredrik Jansson Examiner: Kimmo Eriksson
Special thanks to:
Milica Rancic
Comprising:
Abstract
Businesses nowadays tend to ask their customers to grade a product or a service they have experienced, usually that grading is a number of stars on a scale of 5. When an individual grades a product, that grade goes into a system that calculates the average of all given grades and expose it to the next individual, and so on. Through primary data we estimated what could be a realistic distribution of uninfluenced grades as well as a realistic degree of social influence. This thesis aims to understand the social influence on individual grades and to what extent the order in which graders come may bias the aggregate grade. In addition, we aim to apply mathematical analysis and simulations to examine the social influence of a planted grade on aggregate grade, depending on the strength of social influence and the total number of graders.
It was concluded that ordering the same set of individuals in different ways will give different aggregate grades, mostly biased when ordering from largest to smallest and the other way around. In addition, we presented a perception on how large a planted grade can affect the aggregate grade, studying different main factors such as the degree of social influence and total number of graders.
Acknowledgment
Foremost, I would like to thank my supervisor Fredrik Jansson for the continuous support, my thanks also go to my reviewer Johan Richter for the insightful comments.
I would like to express my sincere gratitude and deepest appreciation to Milica Rancic, Head of Division of Applied Mathematics in Mälardalen University, for the unlimited dedication, support, inspiration and empathy.
Finally, I would love to thank my family and friends, this journey would not have been possible without you.
Contents
1. Introduction 1 2. Model 3 3. Experiments 5 3.1 Exp.1 6 3.2 Exp.2 7 3.3 Experiments analysis 8 4. Population-level Analysis 104.1 How much can social influence bias the aggregate grade? 10 4.2 How much can a planted grade affect the aggregate grade? 13 5. Discussion 20
1. Introduction
It is common nowadays that people are asked to grade a product or an experience they have been through, for example using a number of stars between 1 and 5. As a sign of trust and transparency, producers tend to expose the average of previously given grades on a product or a service, so that potential customers can have a grasp on how that experience was graded by others, before they decide to go through it or not.
When a new individual is stating a grade, this newly given grade will be added to the previously given grades, a new average will be calculated and exposed, and so on. We will refer to this grading process as a sequential grading process. In addition, and to make no confusion, we will be referring to products and experiences (services) as ‘products’, since experiences can be considered as intangible products.
In a generic case, when a certain individual is grading a product, the grade will be stated as how this particular individual -on his own- estimates the number of stars that this product deserves. However, in a sequential grading process, an average of previously given grades will be exposed to individuals while stating their grades. In this case, individuals can still state their own underlying grades, regardless of the exposed average of previous grades that this product earned by previous individuals. On the other hand, individuals might possibly be -to a certain extent- influenced by that exposed average, since it represents the opinion of previous individuals that has graded that product. That influence is generally defined by psychologists and
sociologists as ‘Social influence’. Sociologists defined social influence as a “change in an individual’s thoughts, feelings, attitudes, or behaviors that results from interaction with another individual or a group”[3].
The social influence has been studied by many sociologists and psychologists since the early years of the 20th century[1]. In 1898, Norman Triplett started the first experiment on the phenomenon of social facilitation. This theory implies that people tend to do well when they are watched by others in the things they are good at [2]. However, in 1959, one of the main theories of social influence was proposed by French and Raven, where they discussed the social power and provided formalization for the social influence concept[3].
For the particular case of a sequential grading process, social influence occurs when individuals shift their own underlying grade towards the exposed average of
Imagine the following situations, Situation one is that when a group of individuals are stating their underlying grades of a product, on a certain system that calculates the average of all stated grades and expose it in the end of the process. Situation two is that when the same group of individuals are sequentially stating their grades of the same product, on a system that calculates the average of given grades and expose it each time a new individual is stating a grade.
First situation allows no social influence of previously given grades on individuals’ decision, since the average of given grades is exposed right after the last individual has given his/her grade. The average of all given grades represents this group of individuals, since they graded it on their own, without any influence of others’ grades or recommendations. While second situation exposes each individual to the average grades given by previous individuals, through a certain system that adds each newly given grade to the previous ones, calculates a new average and expose it to the next individual, and so on until the last individual leaves his/her grade.
For a first glance, we might anticipate no clear difference in aggregate grades between both situations, since the same group of individuals are grading the same product. However, assuming that individual grades are influenced by the average grade of others, could this bias the aggregate grade to any serious extent?
When graders are influenced by previously given grades stated by previous graders, the order of the graders becomes important. Early graders will influence later graders. The aggregate grade of the same set of individual grades will therefore vary
depending on if the early grades happen to be the more positive ones or the more negative ones. In this thesis I aim to examine how large this bias can be using mathematical analysis and simulations.
Assuming that a certain party wants to influence the aggregate grade (say, in the positive direction), how planting an early grade that is extremely high might affect the aggregate grade?
Without social influence, the effect of such a planted grade on the aggregate grade would quickly be diluted when other graders submit their grades. However, in the presence of social influence, the effect of a planted grade will last longer. In this thesis I aim to examine how large this effect of a planted grade will be, depending on the strength of social influence and the total number of graders.
2. Model
The individuals’ decision problem, while stating a grade of a product, is simply to pick a number of stars in the finite set G = {1, 2, 3, 4, 5} to represent their rating of that product. With 5 stars as maximum satisfaction and 1 star as minimum. Individual graders are assumed to be numbered in the order in which they submit grades. We assume that an individual grader i has a true, or underlying, grade 𝑋" that is a real
number between 0.5 and 5.5. Before the grader submits a grade, this underlying grade is assumed to be processed in two steps.
In the first step, the underlying grade 𝑋" is transformed into a socially influenced
grade 𝑆", which in the next step is transformed into a submitted grade 𝑌" on a discrete
scale from 1 to 5 stars.
Grader number 1 cannot observe any previous grades and is therefore not subjected to social influence, so 𝑆% = 𝑋%. For all graders 𝑖 > 1, the effect of social influence is
assumed to be a move from the underlying grade towards the observed average of all previously stated grades 𝑍, with the parameter α determining the proportion of the full distance that the grade moves:
𝑆" = 𝑋"+ 𝛼 (𝑍"/% − 𝑋")
We further assume that social influence has the same strength for all graders, given by the parameter α.
Where 𝑍"/% denotes the average of the first i -1 grades which is observed by the 𝑖23
grader, that is,
𝑍"/% =(456476⋯64"/% 9:5) for any 𝑖 > 1
As grades can only be submitted as an integer number of stars between 1 and 5, the socially influenced grade must be rounded to the closest integer before submission: 𝑌" = [𝑆"],
where we use the notation [x] for the integer closest to a real number x. (In case of a tie, let us say that we round downwards.)
A maximum social influence is when we choose to move all the distance from X to Z, representing the highest possible influence of Z, or when a = 1.
𝑆" = 𝑍"
In parallel, a minimum social influence is when we decide to stick with our underlying grade and not consider Z at all, or when a= 0.
𝑆" = 𝑋"
Imagine that an individual’s underlying grade is 3, where the exposed grade is 5. The socially influenced grade will be 4 if that individual has a 0.5 degree of social
influence, or:
𝑆 = 3 + 0.5 (5 − 3) = 4
When there is no previous source of social influence, say when graders are not exposed to the average of previous grades, the stated grade will simply equal the underlying uninfluenced grades(after rounding).
3. Experiments
To estimate what could be a realistic distribution of uninfluenced grades as well as a realistic degree of social influence, two behavioral experiments are
conducted. Both experiments Exp.1 and Exp.2 are implemented in the center of Västerås city, Sweden, on Friday 18th of May 2018, when a crowded food festival was ongoing.
In the first experiment (Exp.1), we asked a group of passersby to smell a glass of coffee beans and grade it. More precisely, do they believe that these beans belong to a good source of coffee, on a scale of 1 to 5 stars. After a certain number of reviews, data was gathered. Then, we conducted a second experiment (Exp.2), where we asked a group of passersby to smell the same glass of coffee beans BUT with a piece of paper saying that the average grades that this kind of coffee beans has earned throughout previous experiments was respectively high (4.5 out of 5) and thus, new data from a new group of people was gathered. Furthermore, Both experiments were applied on random passersby in the center of Västerås city.
Determining whether a kind of coffee is good or not by only smelling a handful of coffee beans is a process that requires a high level of experience in coffee, this condition is hardly met among a random segment of people. Thus, this experiment will ask people to scale this coffee out of five stars.
In (Exp.1), random passersby are grading a product according to their own
underlying grades, without any interference of others’ previous grades on individuals’ decision making. That is the absence of social influence. While (Exp.2), and through exposing the average of previously given grades, opens the door to social influence, allowing an interaction between individual underlying grades and other individuals’ average grade.
Assuming the validity of the social influence model, a parameter for the strength of social influence can -in principle- be estimated from data.
3.1. Exp.1:
Asking the passersby to grade a handful of coffee beans on a scale of 1 to 5 stars by smelling it, we got 24 participants whose grades are shown as the following (Figure 3.1):
Figure 3.1
Remember that in Exp.1, participants have not been exposed to a social influence grade (previous recommendations) and that they are grading according to their own preferences only.
3.2 Exp.2:
In Exp.2, we asked the passersby to smell and grade the same handful of coffee beans, BUT with exposing a social influence grade (previous recommendations). These recommendations are shown as a ranking of (4.5) stars out of 5.
The participants have been informed that this grading is a result of previous
experiments in another city of Sweden, we got 24 participants whose grades are shown as the following (Figure 3.2)
The mean of these grades equals (3.5) and standard deviation equals (0.93).
3.3 Experiments analysis:
As mentioned before, assuming the validity of the social influence model, the parameter for the strength of social influence can -in principle- be estimated from data.
Note that the mean of stated grades we obtained in Exp.2 does not represent the aggregate grade of a sequential grading process in the presence of social influence, since the exposed grade in Exp.2 was not dependent on the stated grades, it was rather a constant value through the experiment.
There is a clear descriptive difference in the mean of Exp.1 and Exp.2 grades, which demonstrates a tangible influence of the exposed grade (the social influence) in our particular experiment. But more importantly, can we roughly argue that there is a significant difference in means of Exp.1 and Exp.2 grades? A significant difference will help us make inferences about the entire population, and whether the social influence on individual grades is a theme or a coincidence.
Our hypothesis is that the presence of information about previous grades indeed influences submitted grades. The null hypothesis is that that the presence of
information about previous grades does not influence submitted grades, and hence that any difference in the mean grade between the two experimental conditions is due to chance from the random sampling of participants. The statistical practice of null hypothesis testing involves calculating how unlikely it would be for chance alone to produce a difference in data from the two conditions that is at least as large as the difference observed in the experiment. This probability is conventionally denoted by p, and in case it is sufficiently small, typically if p < 0.05, the convention is to say that “the null hypothesis can be rejected” and that “the observed difference is statistically significant”.
Applying T-test statistic to compare the means of both groups, we find that (T=2.35) and (P=0.02). By conventional criteria, and since 𝑃 < 0.05 , the null hypothesis can be rejected and the observed difference is statistically significant.
We shall now fit the observed data to our model by estimating the value of the social influence parameter α. Note that we have only observed the submitted grades, the 𝑌" in our model. We have no direct access to the underlying grades 𝑋" or the socially
influenced grades 𝑆". However, submitted grades differ from socially influenced
grades only by rounding. We expect the average effect of rounding to be zero (rounding upwards should generally occur just as often as rounding downwards). Therefore, the mean of submitted grades should be an unbiased estimate of the mean of socially influenced grades in Exp 2 and of the mean of underlying grades in Exp 1.
According to the individual-level model, and since the values of X, Y, Z are known, we can calibrate the degree of social influence that resulted of the exposed grade along the experiment, considering it as it represents the group of individuals who participated in Exp.2.
Z = 4.5 X = 2.79 S = 3.5
And since S = X + α (Z - X), we find that the degree of social influence α is 0.40. We note for (α = 0.4), the mean has shifted from X = 2.79 to S = 3.5, towards the exposed social influence grade Z = 4.5.
4. Population-level Analysis
As assumed before, the aggregate grade of the same set of individual grades will vary depending on if the early graders happen to be the more positive or the more negative ones. How large can this bias be? In addition, How large will the effect of a planted grade be, depending on the strength of social influence and the total number of graders?
4.1 How much can social influence bias the aggregate grade?
As assumed before, the aggregate grade of the same set of individual grades will vary depending on if the early graders happen to be the more positive or the more negative ones. How large can this bias be.
In order to obtain what could be a realistic dataset of X-values, we apply the 24 Y-values stated in Exp.1, in the absence of social influence, as shown in Figure 3.1. Since we assumed that underlying grades are real numbers that are mapped to the finite set G = {1, 2, 3, 4, 5}, we add a random number between -0.5 and 0.5 to simulate reversing the rounding.
As calculated in Exp.1, the average of these grades is (2.79), that is the unbiased aggregate grade. However, we assumed that early graders will influence later
graders. Therefore, the order in which graders come and state their grades will play a major role in the bias of aggregate grades.
Ordering the X-values we obtained in Exp.1 from largest to smallest and applying the same degree of social influence (0.4) would (according to the model) result the most extremely biased aggregate grade, which equals (3.34). On the other hand, ordering the X-values from smallest to largest would result an aggregate grade equals (2.29).
Imagine that you are releasing a product to be graded by customers in two markets, A and B. And assuming that previous surveys shown that customers in market A are more likely to like the product than customers in market B. This
To illustrate the effect of the strength of social influence, we simulate the set of X-values ordered from largest to smallest, with different X-values of degree of social influence. In graph 4.1 we plot the points of coordinates (degree of social influence, aggregate grade).
graph 4.1
When the case is 𝛼 = 0, individual i will state a grade 𝑌" that equals the underlying
grade 𝑋"(after rounding), regardless of the exposed grade 𝑍"/%.
𝑌" = 𝑋"+ 0 . (𝑍"/%− 𝑋") ⇔ 𝑌" = 𝑋"
Since graders are not influenced by the exposed average of previous grades, there will be no social influence, and the aggregate grade equals the unbiased aggregate grade (2.79).
On the other hand, when the case is 𝛼 = 1, individual i will state a grade 𝑌" that
𝑌" = 𝑋"+ 1 . (𝑍"/%− 𝑋") ⇔ 𝑌" = 𝑍"/%
Therefore, the biased aggregate grade equals the largest X in the data set (when the case is ordering grades from largest to smallest), that is (5).
Observing the shape of the plot, the marginal effect of social influence on aggregate grade is increasing when the degree of social influence increases.
4.2 How much can a planted grade affect the aggregate grade?
As mentioned before, a grade might be planted by a certain party to influence the aggregate grade. How large will the effect of a planted grade be, depending on the strength of social influence and the total number of graders?
For simplifying reasons, we assume the planted grade P, to be the initial exposed grade 𝑍% = 𝑝. However, we assume that all other underlying grades are constant,
𝑋" = 𝑋, for 𝑖 > 1.
Lemma:
The aggregate grades satisfy the recursion 𝑍% = 𝑝 and, for 𝑖 > 1,
𝑍" =
(1 − 𝛼)𝑋 + (𝛼 + 𝑖 − 1)𝑍"/%
𝑖
Proof:
A recursive function as defined by Hazart R. ‘’is a function which calls itself in its definition’’[5].
As the model states:
𝑌" = 𝑋"+ 𝛼 (𝑍"/%− 𝑋) And since 𝑍" = (𝑖 − 1)𝑍"/%+ 𝑌" 𝑖 We can rewrite 𝑍" as 𝑍" = (𝑖 − 1)𝑍"/%+ 𝑋"+ 𝛼 (𝑍"/%− 𝑋) 𝑖 𝑖. 𝑍"/%− 𝑍"/%+ 𝑋"+ 𝛼 𝑍"/%− 𝛼 𝑋
Therefore
𝑍" =(1 − 𝛼)𝑋 + (𝛼 + 𝑖 − 1)𝑍"/% 𝑖
Theorem: the aggregate grades satisfy the explicit formula
𝑍D = 𝑝(1 + 𝛼)(2 + 𝛼) … (𝑛 − 1 + 𝛼)
𝑛! + 𝑋 I1 −
(1 + 𝛼)(2 + 𝛼) … (𝑛 − 1 + 𝛼)
𝑛! J
Proof:
Through an induction proof, as Gerardo Diaz defines:
“Mathematical Induction: Let A(n) be an assertion involving an integer n. If we can perform the following two steps:
(1) prove that A(1) is true,
(2) for a given arbitrary k, assume that A(k) is true, and prove that A(k + 1) is also true,
then we can conclude that the assertion A(n) is true for every positive integer n”[4].
When 𝑛 = 1, a grade is planted 𝑍% = 𝑝. However, when 𝑛 = 2:
𝑍K = 𝑝 (1 + 𝛼) 2! + 𝑋 I1 − (1 + 𝛼) 2! J = 𝑝 (1 + 𝛼) 2 + 𝑋 I1 − (1 + 𝛼) 2 J
𝑍K = 𝑌%+ 𝑌K 2 = 𝑝 + 𝑋 + 𝛼 ( 𝑝 − 𝑋 ) 2 = 𝑝 + 𝑋 + 𝛼 𝑝 − 𝛼 𝑋 2 = 𝑝 (1 + 𝛼) 2 + 𝑋 I1 − (1 + 𝛼) 2 J Therefore
We find that 𝑍K satisfies the explicit formula of 𝑍D and it is true for 𝑛 = 2.
Assuming that 𝑛 = 𝑘 satisfies the formula:
𝑍M = 𝑝(1 + 𝛼)(2 + 𝛼) … (𝑘 − 1 + 𝛼) 𝑘! + 𝑋 I1 −(1 + 𝛼)(2 + 𝛼) … (𝑘 − 1 + 𝛼) 𝑘! J Therefore, for 𝑛 = 𝑘 + 1 𝑍M6% = 𝑝(1 + 𝛼)(2 + 𝛼) … (𝑘 + 1 − 1 + 𝛼) (𝑘 + 1)! + 𝑋 I1 − (1 + 𝛼)(2 + 𝛼) … (𝑘 + 1 − 1 + 𝛼) (𝑘 + 1)! J 𝑍M6% = 𝑝(1 + 𝛼)(2 + 𝛼) … (𝑘 − 𝛼 + 1)(𝑘 + 𝛼) 𝑘! (𝑘 + 1) + 𝑋 I1 − (1 + 𝛼)(2 + 𝛼) … (𝑘 − 𝛼 + 1)(𝑘 + 𝛼) 𝑘! (𝑘 + 1) J Naming ∆ = (%6O)(K6O)…(M/%6O)M!
We can rewrite 𝑍M and 𝑍M6% as the following:
𝑍M6% = ∆𝑝 (𝑘 +𝛼)
(𝑘 + 1)+ 𝑋 I1 − ∆
(𝑘 +𝛼) (𝑘 + 1)J
Since we proved that 𝑍" satisfies:
𝑍" =
(1 − 𝛼)𝑋 + (𝛼 + 𝑖 − 1)𝑍"/%
𝑖
We assume that 𝑖 = 𝑘 + 1, therefore 𝑖 − 1 = 𝑘. Then
𝑍M6%= (1 − 𝛼)𝑋 + (𝛼 + 𝑘 + 1 − 1)𝑍M 𝑘 + 1 = (1 − 𝛼)𝑋 + (𝛼 + 𝑘)𝑍M 𝑘 + 1 = 1 − 𝛼 𝑘 + 1 𝑋 + 𝑘 + 𝛼 𝑘 + 1 𝑍M = 𝑋 1 − 𝛼 𝑘 + 1 +∆𝑝 𝑘 + 𝛼 𝑘 + 1 + 𝑋 𝑘 + 𝛼 𝑘 + 1−∆𝑋 𝑘 + 𝛼 𝑘 + 1 =∆𝑝 𝑘 + 𝛼 𝑘 + 1+ 𝑋 R 1 − 𝛼 𝑘 + 1+ 𝑘 + 𝛼 𝑘 + 1 −∆ 𝑘 + 𝛼 𝑘 + 1S =∆𝑝 𝑘 + 𝛼 𝑘 + 1+ 𝑋 R 1 − 𝛼 + 𝑘 + 𝛼 𝑘 + 1 −∆ 𝑘 + 𝛼 𝑘 + 1S Therefore,
The value of 𝑍M6% obtained from the recursion of 𝑍" equals the value of 𝑍M6%
obtained from the explicit formula of 𝑍D. Therefore, we can conclude that the
explicit formula of 𝑍D is true for every positive integer 𝑛 > 1.
Since we proved that 𝑍D is true, and in order to study the effect of a planted grade,
we plot 𝑍D (aggregate grade) for a range of values of 𝛼 (0, 0.1, 0.2,… ,0.9 ,1) for a
total number of grades n between (1) and (100).
In this case, we assumed that the underlying grade equals (3), and the planted grade equals (5).
figure 4.1
When 𝛼 = 1, the aggregate grade is a constant value (represented by a straight line as shown in figure 4.1) that equals the planted grade p, for any number of graders 𝑛 > 1, recall
𝑍D = 𝑝(1 + 𝛼)(2 + 𝛼) … (𝑛 − 1 + 𝛼)
𝑛! + 𝑋 I1 −
(1 + 𝛼)(2 + 𝛼) … (𝑛 − 1 + 𝛼)
𝑛! J
When 𝛼 = 1, for any 𝑛 > 1:
(1 + 1)(2 + 1) … (𝑛 − 1 + 1) = 𝑛! Therefore, (1 + 1)(2 + 1) … (𝑛 − 1 + 1) 𝑛! = 1 Then, 𝑍D = 𝑝(1) + 𝑋(1 − 1) = 𝑝 = 5
Which is a straight line equation that represents -theoretically- the highest social influence of a planted grade.
However, when 𝛼 = 0, the input (stated grades) is the true uninfluenced grades and the planted grade has no social influence on other grades. The aggregate grade is rapidly diminishing. 𝑍D = 𝑝(1 + 0)(2 + 0) … (𝑛 − 1 + 0) 𝑛! + 𝑋 I1 − (1 + 0)(2 + 0) … (𝑛 − 1 + 0) 𝑛! J 𝑍D = 𝑝 (1)(2) … (𝑛 − 1) 𝑛! + 𝑋 I1 − (1)(2) … (𝑛 − 1) 𝑛! J 𝑍D = 𝑝 1 𝑛+ 𝑋 R1 − 1 𝑛S
The planted grade has no significant influence on the flow of aggregate grades more than being calculated in the exposed average. Therefore, the curve of aggregate grades when (𝛼 = 0) is the most rapidly diminishing towards the true grade (X = 3) for any 𝑛 > 1.
Observing the shape of the intermediate curves of aggregate grades we obtained (Alpha between 0.1 and 0.9 ), there is a clear inverse relationship between total number of graders n and the influence of a planted grade on aggregate grades. A planted grade can demonstrate higher influence on aggregate grade when graders have higher degree of social influence and the drop in aggregate grade towards true grades is much quicker for lower degrees of social influence, even when carrying the same underlying grade. The marginal effect of a planted grade decreases quicker for lower degree of social influence
Higher alpha will help maintaining the effect of a planted grade carried to more graders, where lower alpha will not be able to pass that effect longer.
5. Discussion
Assuming individual grades are influenced by the average grade of others, this study found that this influence could possibly bias the aggregate grade to a certain extent in a sequential grading process.
We found that the aggregate grade of the same set of individuals can vary
depending on the order in which graders come. The bias of aggregate grades can be as large as possible when ordering the grades from largest to smallest, or the other way around. This bias of aggregate grades varies for varying factors like the degree of social influence and total number of graders. Moreover, the marginal effect of social influence on aggregate grade is increasing when the degree of social influence increases.
In addition, we found that planting a grade ,that is extremely high for instance, will influence the aggregate grade to different extents, depending on the total number of graders and the degree of social influence. Through mathematical analysis and simulations in R, we found that the marginal effect of a planted grade is decreasing when the total number of graders increases. That marginal effect decreases quicker for lower degree of social influence.. A planted grade can bias the aggregate grade to a certain extent, but that influence will quickly be diluted when more true underlying grades are stated. However, a planted grade has higher effect if planted for graders with higher degrees of social influence, even when carrying the same underlying grade.
Moreover, we focused on the general case of sequential grading, where average of previous given grades is calculated and presented to each individual. However, further work can study the bias of aggregate grade for other technical cases of sequential grading that have been observed on some online platforms, such as a sequential grading system that exposes the grade of the last individual only, instead of average given grades by all previous individuals.
In addition, further work can study the degree of social influence as a function of true and exposed grades, assuming that there is an inverse relationship between the degree of social influence and the distance between true and exposed grades, based on an assumption that graders are more likely to be influenced by exposed
Primary data is limited to the time, place and number of participants. It is
important to mention that the number of participants we got in our experiments was fairly small for inferring an accurate realistic degree of social influence. In addition, we studied the effect of a planted grade assuming that it is the initial exposed grade as well as the initial stated one. In a realistic scenario, a planted grade will be stated after a number of previous grades and will therefore make smaller difference in the average of all stated grades, when being exposed to the next grader.
6. Bibliography
[1] Wikipedia. Social psychology. Available at http://wikipedia.org/wiki/Social_Psychology;. Last Accessed: 22-May-2018.
[2] Triplett N. (1898 )The dynamogenic factors in pacemaking and competition. Am J Psychol ;9(4):507–533.
[3] Rashotte L. (2007)Social influence. In Blackwell Encyclopedia of Sociology. Blackwell Publishing, Malden, Massachusetts, pp. 4426–4429.
[4] Diaz G.(2005) Mathematical induction. Harvard University. Available at
http://www.math.harvard.edu/archive/23a_fall_05/Handouts/induction.pdf ;. Last Accessed: 24-Aug-2018.
[5] Hazrat R. (2010) Recursive functions. In: Mathematica®: A Problem-Centered Approach. Springer Undergraduate Mathematics Series, vol 53. Springer, London.
Appendix
Simulation code of 𝑍D in R: f<-Vectorize(function(a,n,p,x){#of course n>1 z=(p-x)*cumprod(seq(1,n-1)+a)/factorial(2:n)+x z=c(p,z) },vectorize.args="a") A<-as.data.frame(sapply(alpha<-seq(0,1,0.1),f,n=100,p=5,x=3)) names(A)<-as.character(alpha) A$N=1:dim(A)[1] library(tidyr)data_long <- gather(A, "Alpha", "Value",as.character(alpha), factor_key=TRUE) data_long$Alpha <- factor(data_long$Alpha, levels =
rev(levels(data_long$Alpha))) library(ggplot2)
ggplot(data_long,aes(x=N,y=Value,group=Alpha))+ geom_line(aes(color=Alpha))+
