Who Feels More Alone?

Are You Going To Meet a Friend Today?

In X Days You Will Meet a Friend

How Pissed Off Is Ben?

Are You Satisfied?

Conclusion

The “How Alone Do You Feel” Factor

Introverts Feel More Alone

Home / Essays / Who Feels More Alone?

Imagine you live in a world with only two types of people. Introverts (i’s) and extroverts (e’s).

In this world introverts are happy to be alone so they want to meet a friend only once every 14 days.
Extroverts on the other hand love to be around people, they want to hang out with a friend once every 2 days.

Let’s take a closer look at our protagonists:

I´s and e´s also have something in common. They have short term memory loss. Everyday at midnight their brain does a full reset. So they never remember what happened yesterday.

Now take a guess: Which personality type feels more alone? Introverts or extroverts?

What does this question even mean? Let's say you're an introvert. You wake up one morning and want to meet a friend. But when you call your friends, they don’t want to hang out with you. That day you will go to bed feeling alone.

Did you take a guess? Maybe you picked extroverts, because they need more human interaction in general. Or you picked introverts, because they shoot themselves in the foot by being such loners?

To answer this question I calculated the probability of meeting a friend today for each friend group mix.

What is a “friend group mix”?

If you have one introverted and one extroverted friend, your friend group mix is 1/1, one i and one e.
If you have 5 extroverted friends, your friend group mix is 0/5, zero i five e’s.

Each table cell is calculated like this: probability_self * sum_probability_friends.

If an introvert has one introvert friend its: 7.14% * (1-(1-7.14%)^1) = 0.51%.
If an introvert has two extrovert friends it’s: 7.14% * (1-(1-50%)^2) = 5.36%
If an introvert has two extrovert and one introvert friend it’s: 7.14% * (1-((1-50%)^2 * (1-7.14%)^1)) = 5.48%

x-axis = number of extroverted friends, y-axis = number of introverted friends

We can see that for extroverts the probability of meeting a friend today rises much faster than for introverts. Extroverts need less friends to have a high probability of meeting a friend today. But, if you are like me, your brain doesn't know how to work with percentages. So let's turn these % into days.

Now I calculated how many days i’s and e’s have to wait until they meet a friend.

i2 - Waiting days until meeting a friend for an introvert 🏴‍☠️

(i) / (e)	0	1	2	3	4	5	6	7	8	9	10
0	1095	28	18.67	16	14.93	14.45	14.22	14.11	14.05	14.03	14.01
1	196	26.13	18.23	15.84	14.86	14.42	14.21	14.1	14.05	14.03	14.01
2	102	24.61	17.85	15.69	14.8	14.39	14.19	14.09	14.05	14.02	14.01
3	71	23.35	17.5	15.56	14.74	14.36	14.18	14.09	14.04	14.02	14.01
4	55	22.28	17.2	15.43	14.68	14.33	14.16	14.08	14.04	14.02	14.01
5	46	21.38	16.92	15.32	14.63	14.31	14.15	14.08	14.04	14.02	14.01
6	40	20.6	16.67	15.22	14.58	14.29	14.14	14.07	14.04	14.02	14.01
7	35	19.93	16.45	15.13	14.54	14.27	14.13	14.07	14.03	14.02	14.01
8	32	19.35	16.24	15.04	14.5	14.25	14.12	14.06	14.03	14.02	14.01
9	29	18.83	16.06	14.96	14.46	14.23	14.11	14.06	14.03	14.01	14.01
10	27	18.38	15.89	14.89	14.43	14.21	14.11	14.05	14.03	14.01	14.01
11	26	17.98	15.74	14.82	14.4	14.2	14.1	14.05	14.02	14.01	14.01
12	24	17.62	15.6	14.76	14.37	14.18	14.09	14.05	14.02	14.01	14.01
13	23	17.3	15.48	14.7	14.34	14.17	14.08	14.04	14.02	14.01	14.01
14	22	17.01	15.36	14.65	14.32	14.16	14.08	14.04	14.02	14.01	14
15	21	16.76	15.25	14.6	14.29	14.15	14.07	14.04	14.02	14.01	14
16	21	16.52	15.16	14.56	14.27	14.13	14.07	14.03	14.02	14.01	14
17	20	16.31	15.07	14.51	14.25	14.13	14.06	14.03	14.02	14.01	14
18	20	16.12	14.99	14.48	14.23	14.12	14.06	14.03	14.01	14.01	14
19	19	15.95	14.91	14.44	14.22	14.11	14.05	14.03	14.01	14.01	14
20	19	15.79	14.84	14.41	14.2	14.1	14.05	14.02	14.01	14.01	14

x-axis = number of extroverted friends, y-axis = number of introverted friends

e2 - Waiting days until meeting a friend for an extrovert 🦅
e2 - Waiting days until meeting a friend for an extrovert 🦅
(i) / (e)	0	1	2	3	4	5	6	7	8	9	10
0	1095	4	2.67	2.29	2.13	2.06	2.03	2.02	2.01	2	2
1	28	3.73	2.6	2.26	2.12	2.06	2.03	2.01	2.01	2	2
2	14.52	3.52	2.55	2.24	2.11	2.06	2.03	2.01	2.01	2	2
3	10.03	3.34	2.5	2.22	2.11	2.05	2.03	2.01	2.01	2	2
4	7.8	3.18	2.46	2.2	2.1	2.05	2.02	2.01	2.01	2	2
5	6.46	3.05	2.42	2.19	2.09	2.04	2.02	2.01	2.01	2	2
6	5.57	2.94	2.38	2.17	2.08	2.04	2.02	2.01	2.01	2	2
7	4.94	2.85	2.35	2.16	2.08	2.04	2.02	2.01	2	2	2
8	4.47	2.76	2.32	2.15	2.07	2.04	2.02	2.01	2	2	2
9	4.11	2.69	2.29	2.14	2.07	2.03	2.02	2.01	2	2	2
10	3.82	2.63	2.27	2.13	2.06	2.03	2.02	2.01	2	2	2
11	3.59	2.57	2.25	2.12	2.06	2.03	2.01	2.01	2	2	2
12	3.4	2.52	2.23	2.11	2.05	2.03	2.01	2.01	2	2	2
13	3.23	2.47	2.21	2.1	2.05	2.02	2.01	2.01	2	2	2
14	3.1	2.43	2.19	2.09	2.05	2.02	2.01	2.01	2	2	2
15	2.98	2.39	2.18	2.09	2.04	2.02	2.01	2.01	2	2	2
16	2.88	2.36	2.17	2.08	2.04	2.02	2.01	2	2	2	2
17	2.79	2.33	2.15	2.07	2.04	2.02	2.01	2	2	2	2
18	2.72	2.3	2.14	2.07	2.03	2.02	2.01	2	2	2	2
19	2.65	2.28	2.13	2.06	2.03	2.02	2.01	2	2	2	2
20	2.59	2.26	2.12	2.06	2.03	2.01	2.01	2	2	2	2

x-axis = number of extroverted friends, y-axis = number of introverted friends

Think about how many introverted and extroverted friends you have. Then check the charts above: How many days do you (theoretically) have to wait until you meet a friend?

Did you also get 1095 days 😅? But hold on! Where do the 1095 days for 0 friends come from?

I chose to limit the number of waiting days to a maximum of three years. Otherwise the 0/0 cell would show the number infinity for i´s and e´s. This would have messed up the charts and following calculations.

Why did I choose three years (or 1095 days)? I guess in three years everybody should be able to make and meet at least one friend.

If you haven't met a friend in three years, let's hang out quickly so my charts are correct 😁.

Unfortunately the previous calculations still don't tell us who feels more alone. For that we need the relative difference between what a person wants and what a person receives.

An analogy:

Ben 👨 wants to earn 100€/h but gets only 50€/h. His salary is 50% below his expectations.
Lisa 👩 wants to earn 1000€/h but gets only 950€/h. Her salary is 5% below her expectations.

Who do you think is pissed off more? Ben 👨 or Lisa 👩? I think it's Ben. Because even though, just like Lisa, he earns 50€ less than what he wants. His relative difference (50%) is much higher than Lisas relative difference (5%).

Back to our own problem: Below you can see the same “relative difference calculation” for i’s and e’s and all friend group mixes.

i4 - Relative difference between actual waiting and the optimum of 14 days 🏴‍☠️

(i) / (e)	0	1	2	3	4	5	6	7	8	9	10
0	7721.43%	100.00%	33.36%	14.29%	6.64%	3.21%	1.57%	0.79%	0.36%	0.21%	0.07%
1	1300.00%	86.67%	30.23%	13.13%	6.16%	2.99%	1.47%	0.73%	0.36%	0.18%	0.09%
2	625.93%	75.79%	27.48%	12.08%	5.70%	2.77%	1.37%	0.68%	0.34%	0.17%	0.08%
3	401.65%	66.76%	25.03%	11.12%	5.27%	2.57%	1.27%	0.63%	0.31%	0.16%	0.08%
4	289.81%	59.17%	22.83%	10.25%	4.87%	2.38%	1.18%	0.58%	0.29%	0.15%	0.07%
5	222.96%	52.71%	20.86%	9.44%	4.51%	2.21%	1.09%	0.54%	0.27%	0.14%	0.07%
6	178.59%	47.17%	19.09%	8.71%	4.17%	2.04%	1.01%	0.50%	0.25%	0.13%	0.06%
7	147.07%	42.38%	17.48%	8.04%	3.86%	1.90%	0.94%	0.47%	0.23%	0.12%	0.06%
8	123.58%	38.19%	16.03%	7.42%	3.58%	1.76%	0.87%	0.43%	0.22%	0.11%	0.05%
9	105.45%	34.52%	14.72%	6.86%	3.31%	1.63%	0.81%	0.40%	0.20%	0.10%	0.05%
10	91.06%	31.29%	13.53%	6.34%	3.07%	1.51%	0.75%	0.37%	0.19%	0.09%	0.05%
11	79.39%	28.42%	12.44%	5.86%	2.85%	1.40%	0.70%	0.35%	0.17%	0.09%	0.04%
12	69.76%	25.86%	11.45%	5.42%	2.64%	1.30%	0.65%	0.32%	0.16%	0.08%	0.04%
13	61.71%	23.58%	10.55%	5.01%	2.44%	1.21%	0.60%	0.30%	0.15%	0.07%	0.04%
14	54.88%	21.53%	9.72%	4.63%	2.27%	1.12%	0.56%	0.28%	0.14%	0.07%	0.03%
15	49.04%	19.69%	8.96%	4.29%	2.10%	1.04%	0.52%	0.26%	0.13%	0.06%	0.03%
16	43.99%	18.03%	8.27%	3.97%	1.95%	0.96%	0.48%	0.24%	0.12%	0.06%	0.03%
17	39.61%	16.53%	7.63%	3.68%	1.81%	0.89%	0.45%	0.22%	0.11%	0.06%	0.03%
18	35.77%	15.17%	7.05%	3.41%	1.67%	0.83%	0.41%	0.21%	0.10%	0.05%	0.03%
19	32.38%	13.94%	6.51%	3.15%	1.55%	0.77%	0.38%	0.19%	0.10%	0.05%	0.02%
20	29.39%	12.81%	6.02%	2.92%	1.44%	0.71%	0.36%	0.18%	0.09%	0.04%	0.02%

x-axis = number of extroverted friends, y-axis = number of introverted friends

i4 - Relative difference between actual waiting and the optimum of 2 days 🦅

(i) / (e)	0	1	2	3	4	5	6	7	8	9	10
0	54650.00%	100.00%	33.34%	14.29%	6.67%	3.22%	1.58%	0.78%	0.39%	0.19%	0.10%
1	1300.00%	86.67%	30.24%	13.13%	6.16%	2.99%	1.47%	0.73%	0.36%	0.18%	0.09%
2	625.93%	75.79%	27.48%	12.08%	5.70%	2.77%	1.36%	0.68%	0.34%	0.17%	0.09%
3	401.65%	66.76%	25.03%	11.12%	5.27%	2.57%	1.27%	0.63%	0.31%	0.15%	0.08%
4	289.81%	59.17%	22.83%	10.25%	4.88%	2.38%	1.17%	0.58%	0.29%	0.14%	0.08%
5	222.96%	52.72%	20.86%	9.44%	4.51%	2.20%	1.09%	0.54%	0.27%	0.13%	0.07%
6	178.59%	47.17%	19.09%	8.71%	4.18%	2.05%	1.01%	0.51%	0.25%	0.12%	0.07%
7	147.08%	42.38%	17.49%	8.04%	3.87%	1.90%	0.94%	0.47%	0.24%	0.11%	0.06%
8	123.59%	38.19%	16.04%	7.42%	3.58%	1.76%	0.87%	0.44%	0.22%	0.11%	0.06%
9	105.45%	34.53%	14.72%	6.86%	3.32%	1.63%	0.81%	0.41%	0.20%	0.10%	0.05%
10	91.06%	31.29%	13.53%	6.34%	3.07%	1.51%	0.75%	0.37%	0.18%	0.10%	0.05%
11	79.39%	28.42%	12.44%	5.86%	2.85%	1.40%	0.70%	0.34%	0.17%	0.09%	0.05%
12	69.77%	25.86%	11.45%	5.41%	2.64%	1.30%	0.65%	0.32%	0.16%	0.08%	0.04%
13	61.71%	23.58%	10.55%	5.01%	2.45%	1.20%	0.60%	0.30%	0.15%	0.08%	0.04%
14	54.88%	21.53%	9.72%	4.63%	2.27%	1.12%	0.55%	0.28%	0.14%	0.07%	0.04%
15	49.04%	19.69%	8.97%	4.29%	2.10%	1.04%	0.51%	0.26%	0.13%	0.07%	0.03%
16	44.00%	18.03%	8.27%	3.97%	1.95%	0.96%	0.48%	0.24%	0.12%	0.06%	0.03%
17	39.61%	16.53%	7.63%	3.68%	1.80%	0.90%	0.45%	0.22%	0.11%	0.06%	0.03%
18	35.77%	15.17%	7.05%	3.40%	1.68%	0.83%	0.42%	0.21%	0.11%	0.05%	0.03%
19	32.39%	13.94%	6.52%	3.16%	1.56%	0.77%	0.38%	0.19%	0.10%	0.05%	0.03%
20	29.39%	12.81%	6.02%	2.92%	1.44%	0.71%	0.35%	0.18%	0.09%	0.05%	0.02%

x-axis = number of extroverted friends, y-axis = number of introverted friends

Wow, both tables are exactly* the same. This means that introverts and extroverts are both feeling alone the same.

*The only difference is the first cell. Why is that? Earlier in the essay I told you I set a maximum waiting time of 1095 days if somebody has no friends at all 😅.

Now because this “zero friends limit” is exactly the same for i´s and e´s the relative difference for extroverts is much higher.

This means if you lock an extrovert in a single prison cell for 1095 days he will feel much more alone than an introvert. That's a no brainer. You probably already knew that.

I know I`m not. And since I´m the mayor of this “thought experiment world”, I'm making up one more rule: Extroverts have more friends than introverts. Which might also be true in real life?

To simplify this last chapter I calculated how alone i’s and e’s feel depending on the number of friends they have. Here’s this calculation for all possible friend group mixes:

Mixed friend groups
Friends	🏴‍☠️ Introvert	🦅Extrovert
1	700.00%	700.00%
2	248.65%	248.64%
3	130.49%	130.49%
4	80.76%	80.77%
5	54.77%	54.77%
6	39.36%	39.36%
7	29.46%	29.46%
8	22.71%	22.71%
9	17.92%	17.92%
10	14.40%	14.41%

x-axis = introvert or extrovert, y-axis = number of friends, colored cells = “feeling alone factor”

Now the numbers look different from what we saw before. Why is that? If an introvert has 1 friend, this one friend can either be an introvert or an extrovert. So the average “feeling alone factor” for one friend is: (1300% + 100%) / 2 = 700%

As you can see, the feeling alone factor for an introvert with 1 friend is higher than for an extrovert with 2 friends. We made progress. Introverts feel more alone than extroverts 🥲.

But to be honest this feels too much like cheating. So I kept thinking: Isn't there something I might have forgotten? Yes, there is: We become friends with people who are like us. Which means extroverts have more extroverted friends & introverts have more introverted friends.

If i’s and e’s only have friends of the same kind, the new table looks like this:

Only friends of same kind
Friends	🏴‍☠️ Introvert	🦅Extrovert
1	1300.00%	100.00%
2	625.93%	33.34%
3	401.65%	14.29%
4	289.81%	6.67%
5	222.96%	3.22%
6	178.59%	1.58%
7	147.07%	0.78%
8	123.58%	0.39%
9	105.45%	0.19%
10	91.06%	0.10%

x-axis = introvert or extrovert, y-axis = number of friends, colored cells = “feeling alone factor”

As you can see, introverts feel shockingly more alone 😱. If both, i’s and e’s have only one friend, introverts feel 12 times more alone. If they both have 10 friends, introverts feel over 900 times more alone.

Wants to meet a friend once every

Probability he wants to meet a friend today

Introverts 🏴‍☠️ need some extroverts in their life to reduce feeling alone.
Extroverts 🦅 can benefit from an introvert friend if they have social stress.

Now there's only one question left to ask: Are you an extrovert and want to hang out in 14 days?

Back to all Essays

Thanks to Maria for reading drafts of this.