Richard Sprague

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Gideon's Wisdom of Crowds Experiment

Created: 2019-11-23 ; Updated: 2019-11-23

My grad school friend conducted a fun experiment back in 2012 which I’m embarrassed to say I just learned about. He asked his zillions of Google+ followers to guess the number of cheerios in a jar, hoping to test the idea of the “wisdom of crowds”. He has released the raw data, so I used it as an excuse for another R-tude.

After placing his original spreadsheet into an R dataframe called gideon_woc, I generated this quick overall summary:

gideon_woc %>% group_by(type) %>% summarise(mean = mean(value), median = median(value)) %>% knitr::kable(digits = 0, caption = "Final summary of all data collected.")
Table 1: Final summary of all data collected.
type mean median
Combined: All Groups 514 402
GR Group 1 Guesses 502 400
GR Group 2 Guesses 520 420
GR Group 3 Guesses 525 423
GR Group 4 Guesses 528 424
GR Group 5 Guesses 493 400
Shared Group 1 Guesses 458 333
Shared Group 2 Guesses 485 386
Shared Group 3 Guesses 514 348
Shared Group 4 Guesses 497 418
Shared Group 5 Guesses 221 175

Gideon received 2,238 valid guesses made in multiple rounds, here organized by type. Some of the people guesses had access to the other people’s guesses (“Shared”) while others were blind to the other guesses (“GR”). Since the actual number of cheerios in the jar is 467, you can see that blinding appears to have made a significant difference in the final guesses.

The good news is that so far our math agrees.

Here’s what we get when we graph each version of the guess. The red dots are outliers, i.e. guesses that fit outside the middle 75% of all guesses.

gideon_woc %>% dplyr::filter(type != "Combined: All Groups") %>% ggplot(aes(x=type,y=value)) + geom_boxplot(outlier.color = "red") + theme(axis.text.x = element_text(angle=90))

Outliers

One thing that astounds me is the total number of such outliers, as you can see in this table. A lot of people thought the jar contained many multiples more objects than it actually did.

gideon_woc %>%
group_by(type) %>%
summarise(min = min(value), max = max(value), mean = mean(value), median = median(value)) %>% knitr::kable(caption = "Summary after removing outliers")
Table 2: Summary after removing outliers
type min max mean median
Combined: All Groups 42 3600 513.5567 402.00
GR Group 1 Guesses 96 2400 502.1863 400.00
GR Group 2 Guesses 98 2584 520.3391 420.00
GR Group 3 Guesses 42 2700 525.4684 423.00
GR Group 4 Guesses 42 2073 528.1578 424.50
GR Group 5 Guesses 98 2880 493.3125 399.75
Shared Group 1 Guesses 50 2012 457.8618 333.00
Shared Group 2 Guesses 120 1598 485.4426 386.00
Shared Group 3 Guesses 58 3600 514.0122 347.50
Shared Group 4 Guesses 120 1080 497.4000 418.50
Shared Group 5 Guesses 85 380 220.6667 175.00

Now let’s remove those outliers and see what we get.

outlier <- function(x) {
b <- boxplot.stats(x)
b\$out}
outliers <- gideon_woc %>% dplyr::filter(type != "Combined: All Groups") %>% group_by(type) %>% pull(value) %>% outlier()
gideon_woc %>% dplyr::filter(!(value %in% outliers)) %>%
group_by(type) %>%
summarise(min = min(value), max = max(value), mean = mean(value), median = median(value)) %>% knitr::kable()
type min max mean median
Combined: All Groups 42 1178 447.7969 390.0
GR Group 1 Guesses 96 1178 444.6712 396.0
GR Group 2 Guesses 98 1162 463.8041 400.0
GR Group 3 Guesses 42 1150 448.2045 400.0
GR Group 4 Guesses 42 1178 461.5106 401.0
GR Group 5 Guesses 98 1080 420.8006 374.0
Shared Group 1 Guesses 50 1099 396.1875 326.5
Shared Group 2 Guesses 120 1050 433.1552 384.0
Shared Group 3 Guesses 58 1100 422.3553 325.0
Shared Group 4 Guesses 120 1080 497.4000 418.5
Shared Group 5 Guesses 85 380 220.6667 175.0
gideon_woc %>% dplyr::filter(!(value %in% outliers)) %>%
ggplot(aes(x=type,y=value)) + geom_boxplot(outlier.color = "red") + theme(axis.text.x = element_text(angle=90))

Gideon’s post asks a bunch of questions for statistics experts, hoping to understand just how significant the results were. Unfortunately that’s all the time I have for today. Hopefully I can revisit this post to learn some other interesting insights.