Introduction

According to existing research, men have a tendency to estimate their own IQs more highly than women, but there is no significant difference between men’s and women’s actual IQ scores (Furnham and Rawles, 1995). The same study found that when asked to rate the IQs of other people, such as their parents and grandparents, both men and women tended to rank other men as more intelligent than other women.

The reason why men’s assessment of their own intelligence tends to be higher than women’s is not entirely certain (Furnham and Robinson, 2023). In general, it is debated whether the difference is caused by gendered socialization or by biology. Furnham and Robinson also discuss emotional intelligence, for which the opposite trend is true: women tend to rank themselves more highly in emotional intelligence than men do. I noticed that there wasn’t much information in the studies I read that focused on the accuracy of the assessments, which is what I hope to do in my project. So, for my final project, I investigated the following question: do men and women demonstrate a difference in tendence to inaccurately estimate their own IQs?

My parameter for this study was the mean difference between men and women in mean IQ estimation gap within the town of Vardo. My initial conjecture, based on the real-life studies that I read, was that men would tend to overestimate their IQs while women would tend to underestimate. I suspected that this conjecture would be somewhat supported by Islander data.

Data Collection Methods

The observational units were each of the Islanders. The variable was measured by surveying 25 men and 25 women over the ages of 18 and asking them to estimate their own IQs, then having them take IQ tests and subtracting their actual IQ scores from their estimated IQ scores. I acquired a random sample within the population of Vardo, which has 570 houses. I had a random number generator pick 25 different numbers from 1 to 570. I then asked male residents of the houses with the numbers generated to participate in my study. If there were no men living in the house, or if the man declined consent, then I moved on to the next house down. On occasion, it took several tries to find a willing participant. I then generated 25 more numbers from 1 to 570 and repeated the same process to select female participants. I didn’t know that you could select people who declined by making repeat visits, so I did not make any repeat visits. This means that my sample was not entirely random. Other than that, the actual study went smoothly and there were not any problems. I accidentally made one participant take their IQ test before taking the survey, but it did not seem to affect their answer.

Descriptive Statistics

library(readr)
IQ_Data <- read_csv("~/Math 247 Spring 2024/Mini projects/mini project 3 - Sheet1.csv")
head(IQ_Data, n=2)
data(IQ_Data)
favstats(IQdiff ~ gender, data = IQ_Data)

Fig. 1. Side-by-side whisker boxplot comparing the IQ estimation gaps of male (top) and female participants.

bwplot(gender ~ IQdiff, horizontal = TRUE, data = IQ_Data)

Fig. 1. Side-by-side whisker boxplot comparing the IQ estimation gaps of male (top) and female participants.

histogram(~IQdiff | gender, data = IQ_Data, width = 1, layout = c(1, 2))

Fig. 2. Histograms of IQ difference distributions for women (left) and men.

My study compared the binary categorical explanatory variable of gender and the quantitative response variable of IQ estimate accuracy. For the purposes of this study, there were only two groups for gender: men and women. IQ estimate accuracy was measured by subtracting participants’ IQ scores from their reported estimates of their own IQs.

Just by looking at a side-by-side boxplot of the data (Fig. 1) and comparing summary statistics between men and women, there appears to be a significant difference between the groups that is in line with my initial conjecture.

The 25 male participants all overestimated their IQs by 1 to 16 points. The mean difference between estimated and actual IQ for men was 6.36 points. Conversely, the 25 female participants all underestimated their IQs by 4 to 16 points, and had a mean difference between estimated and actual IQ of -9.32 points. So, at first glance, there appeared to be a strong tendency for men to overestimate their own IQs and for women to underestimate.

My parameter was the difference between men and women in mean IQ estimation gap for the population of adults living in the town of Vardo. The null and alternative hypotheses were the following:

Null hypothesis: On average, men and women estimate their own IQs with equal accuracy.

\[H_{0}: \mu_{men} = \mu_{women}\]

Alternative hypothesis: On average, men overestimate their own IQs more than women do, and/or women underestimate their own IQs more than men do.

\[H_{0}: \mu_{men} > \mu_{women}\]

Analysis of Results

stat(t.test(IQdiff ~ gender, data = IQ_Data))
##         t 
## -15.02791
two.sided.p.value<-pval(t.test(IQdiff ~ gender, data = IQ_Data))
cat("the two-sided p-value is",two.sided.p.value)
## the two-sided p-value is 7.296303e-19
confint(t.test(IQdiff ~ gender, data = IQ_Data))

On average and in my sample, men overestimated their own IQs by 15.68 points more than women (\(\bar{x}\) = 15.68.) Because I chose participants randomly, my sample can be considered representative for the population of adults from Vardo.

A type I error is an incorrect rejection of the null hypothesis, and a type II error is an incorrect failure to reject the null hypothesis. So, in this setting, a type I error would mean that I concluded that there is a statistically significant difference between men and women in IQ estimate accuracy when there was actually not. A type II error would mean that I concluded that there was no statistically significant difference between men and women in IQ estimate accuracy when there actually was a difference.

To analyze my data, I used a two-sample t-test. There were more than 20 observations in each gender group, and neither of the sample distributions were strongly skewed (Fig. 2), so the data met the validity conditions for this test. I calculated a t-statistic of 15.02791, meaning that the observed difference in means between men and women for IQ accuracy was about 15 standard deviations away from the expected difference in means for the null hypothesis.

The two-sided p-value for my observed statistic was 7.296303 x 10^-19. This calculation means that there was a 7.296303 x 10^-17% probability of observing a difference in means of 15.68, assuming the null hypothesis is true. Because the p-value is so small, and the t-statistic is fairly far away from zero, I am choosing to reject the null hypothesis. Therefore, I am concluding that men tend to overestimate their own IQs more than women.

I calculated a 95% confidence interval of (17.7833, 13.5767). In other words, we can be 95% confident that women tend to underestimate their own IQs between 17.7833 and 13.5767 points more than men. This confidence interval does not contain zero, which aligns with my conclusion based on theory-based tests of significance and again allows us to rule out the null hypothesis.

Conclusion

In conclusion, within the population of adults living in Vardo, men tend to overestimate their IQs and women tend to underestimate. The data actually supported my conclusion to a point that seemed unrealistic: every single male participant overestimated their IQ, and every single female participant underestimated. I believe that the developers of the Islands program established this trend on purpose to reflect the results of real studies, but in real life, there would most likely be more variation within samples and not every participant’s data would follow the expected trend. As stated previously, the conclusion of my study can be generalized to adults from Vardo because I used a random sample.

My study could have been improved by surveying and testing children as well as adults. In addition to making the results more generalizable, it would be interesting to know if gender-based trends affect children as strongly as they do adults, since children have not been socialized as their assigned gender for as long as adults. I could have also broadened my focus to more cities so that my results could have been generalized to an entire island or even the entire Islander world. If I were to perform a follow-up study, I might focus on whether participants over- or underestimate the IQs of people other than themselves based on gender. Overall, studying gender psychology is a valuable way to understand gender-based prejudices, because often the conclusions can be applied to real-world scenarios.

Bibliography

Furnham, A., & Rawles, R. (1995). Sex differences in the estimation of intelligence. Journal of Social Behavior and Personality, 10(3), 741. Retrieved from http://ezproxy.whitman.ed u/login?url=https://www.proquest.com/scholarly- journals/sex-differences-estimationintelligence/docview/1292332901/se-2

Furnham, A., & Robinson, C. (2023). Sex difference in estimated intelligence and estimated emotional intelligence and IQ scores, The Journal of Genetic Psychology, 184:2, 133-144, DOI: 10.1080/00221325.2022.2140025 from https://doi.org/10.1080/00221325.2022.2140025