You have 3 types of storybooks — Book A, Book B, and Book C — of varying quantities. You need to find the average number of words in the storybooks. <ul><li>Each Book A has an average of 100 words/book (quantity = 5, total words = 500) </li><li>Each Book B has an average of 200 words/book (quantity = 10, total words = 2000) </li><li>Each Book C has an average of 500 words/book (quantity = 15, total words = 7500) </li><li>The total number of words is 500 + 2000 + 7500 = 10000 </li><li>The total number of books is 5 + 10 + 15 = 30</li></ul>

Now, imagine this weird, complicated scenario:

If you’re a chemist, you may feel underwhelmed or disappointed by this idea. “Isn’t this just a failed experiment then?”. If you took some time to explore the controversy online, however, you may discover a treasure trove of interesting theories proposed by science fanatics that tries to prove this phenomenon right.

The Takeaway

<ul><li>This is the classic ‘weighted average’ approach and is likely what most people use to calculate averages. </li><li>This kind of average takes into consideration the weight (or quantity) of each item group.</li></ul>

The Reason Method 2 total words / total number of books

Both averages are perfectly valid and one may be more suitable than the other depending on the context. Surprised? Well… the learning point here isn’t about math.It’s about realizing how we could adopt a shift in perspective and still derive the right answer.

Both Approaches Are Valid

Indeed, it is the quality of our thoughts that counts — not the quantity.

<ul><li>In 1963 when Erasto was a student at Magamba Secondary School, he wanted to sell ice cream to earn some pocket money. </li><li>In the urgency to prepare the ice cream quickly, he skipped the cooling phase of the preparation and placed the hot milk mixture directly into the freezer. </li><li>To his surprise, he observed that his hot milk mixture froze faster than his classmates’ cooled milk mixture. </li><li>The observation ultimately spun into a research paper and was termed the Mpemba effect.</li></ul>

The Story

<ul><li>Logically speaking, it makes sense that cold water freezes faster than hot water since it can skip the cooling process that the hot water needs to undergo. </li><li>With this logic in mind, the Mpemba effect seems highly unintuitive. </li><li>Indeed, this phenomenon has caused significant controversy in the scientific community. </li><li>Although some very reputable scientific bodies have debunked this theory, the validity of this effect is still largely disputed, with some attributing the strange phenomenon being observed due to impurities in the mixture.</li></ul>

The Controversy

Are you more likely to receive help in a room with 3 people, or a room with 30 people? The answer may be surprising to some.

Idea #3: The Bystander Effect (Psychology)

A study showed that the average person has more than 6000 thoughts every day. That is approximately 4 thoughts every minute, or 1 thought every 15 seconds. Despite having thousands of thoughts every day, we hardly could claim ourselves to be deeply creative and innovative. In this stash, I will share 3 unintuitive ideas to challenge your thinking and encourage you to think outside the box.

When intuition fails, how would you react?

<ul><li>The probability that a person in distress will be helped by a bystander is inversely proportional to the number of bystanders present. </li><li>To put it simply, this means the more bystanders are present, the less likely someone in distress will receive help.</li></ul>

According to the Bystander Effec

Diffusion of Responsibility<ul><li>“Someone else will do it” (i.e. someone else will help the person in distress) </li><li>In this case, the presence of others makes individuals feel less responsible to take action. </li><li>Here, the responsibility is thought to be shared among those who are present. </li></ul>Social Influence<ul><li>“I will monitor how others act before I do anything” </li><li>As humans, we tend to subconsciously adjust our behaviors to fit our current environment. </li><li>In this case, when those around us fail to react to the person in distress, we often take that as a signal that nothing is wrong.</li></ul>

Two Major Factors causing the Bystander Effect

Both diffusion of responsibility and social influence makes it less likely for a person to receive help in a room with 3 people than in a room with 30 people. The Bystander Effect, a counter-intuitive theory that the more people are present, the less likely you’ll receive help, is yet another idea that challenges conventional logic.

Strange Paradox of an Idea

Here are your TWO options: <ul><li>Get the average of the averages (i.e. average of the sum of averages in Book A, B, and C). (100 + 200 + 500) / 3 = ~267 words per book </li><li>Get the average by dividing the total number of words by the total number of books. 10,000 / 30 = ~333 words per book </li></ul>Even if you’re not a mathematician, you may be inclined to pick the 2nd option. Perhaps surprisingly to some, both approaches are actually correct.

How would you find the average num of words in the storybooks?

<ul><li>The term ‘average of the averages’ is often frowned upon by mathematicians. </li><li>However, in this case, the term ‘average of the averages’ is merely a disguise. The calculation of the ‘average of the averages’ actually leads to the calculation of Simple Average, a perfectly valid way of calculating the average. </li><li>Not to get too math-y here, simple average merely means that you’re taking the average of a particular dataset without considering the weight (or in this case, the quantity) of each item group.</li></ul>

The Reason  Method 1 — ‘average of the averages’

<ul><li>This strange paradox of human behavior was first formulated by psychologists John Darley and Bibb Latané. </li><li>This social psychological theory was first prompted by a dark incident where it was wrongly reported that 38 bystanders watched passively during an attack on a victim. </li><li>The theory is then demonstrated and popularized in 1968 and many social scientists soon launch a series of experiments to test the theory. </li><li>Today, this theory is widely established and taught in many Psychology classes.</li></ul>

Background of The Bystander Effect

Known as the Mpemba effect, the phenomenon that hot water freezes faster than cold water was named after Erasto Mpemba.

Idea #1: Hot water turns to ice faster than cold water (Science)

What is the average age of 3 boys aged 10, 15, and 20? Well, you can simply add all their ages up 10 + 15 + 20 = 45 and divide it by 3 — giving you the answer of 15.

Idea #2: More than one average (Math)

That’s it! Do you feel weird, surprised, ambivalent, or even confused? If you did, then I’m glad that I have sparked some thoughts in your mind. Recap of Our 3 Counter-Intuitive Ideas <ul><li>Idea #1 — Hot water turns to ice faster than cold water (Science) </li><li>Idea #2 — More than one average (Math) </li><li>Idea #3 — The Bystander Effect (Psychology) </li></ul>I hope you have experienced some level of paradigm shift through learning about these thought-provoking ideas from numerous disciplines. Keep learning. Keep thinking. Keep improving. I’ll catch you in the next stash. Cheers!

Summary

A study showed that the average person has more than 6000 thoughts every day. That is approximately 4 thoughts every minute, or 1 thought every 15 seconds. Despite having thousands of thoughts every…

Idea #2: More than one average (Math)

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