“Ask the right questions, and nature will open the doors to her secrets.”

— C V Raman.

Today I woke up early as my phone started vibrating. When I looked at my phone I found that I had got a call from my friend Narugopal. Narugopal is one of my oldest friends. I was certain that he had got another topic to prove. I can clearly remember what he had done for the first time. It happened some years back. He called me exactly the same way. When I picked up he said, "Come home. It's urgent." and hang up the call. I became really worried and jumped off my bed and set out for his home.

When I reached I found that he had gathered 5 dogs including Bruno. Bruno is his 8 years old Dalmatian.

I asked, "What happened?"
He said, "My Bruno is not short and I can prove it mathematically."
I said, "What? You called me so early just for this?"
"Sahi pakde hai. (You are right)", he said and handed over me a book.

I stared reading the page that was open.

Variance:
Variance is the average of the squared differences from the Mean.

Standard deviation:
Square root of variance is standard deviation. It is denoted by σ (the greek letter sigma).

I showed him the book and said, "What do you mean?"
"Oh yes! I missed the mean. Mean is sum of observation divided by the total number of observation", he said.

Then I became angry and shouted, "Will you clearly tell me why you called me here? I didn't came here for these non-sense things, exams are still two months ahead."

He said, "Okay! Come with me and help me to measure the height of the dogs." and handed me over a tape measures.

I didn't have any other choice as he helps me so much. We started measuring the height of dogs. (The second one from the left is Bruno.)

After measuring the dogs in centimeters here we got the result

He said, "Let's find the mean now."
I had completely forgotten the formula as I study just some days before exams. So I said, "You find and show me."

          50 + 90 + 25 + 45 + 65        275
mean = ————————————————————————————  = —————— =  55
                      5                  5

He said, "Now draw a line at the mean."
I did as he said.

Then he asked me to subtract the height from mean and I did because I know that's easy.

My next instruction was to find squared mean difference (Same process like mean but in this case you have to square the numbers before adding) of the differences.

                            (5)2 + (-35)2 + (30)2 + (10)2 + (-10)2
squared mean difference = ——————————————————————————————————————————
                                            5
            
                            25 + 1225 + 900 + 100 + 100
                        = —————————————————————————————————
                                        5

                            2350
                        = —————————
                              5

                        = 470

"Can you guess what you just did? You just found the Variance. Check the definition.", he said and showed me the same page again.
I read the definition again and said in my mind, "Wow! This was so easy. Why didn't I get it earlier?"

He said, "Also read the definition of standard deviation and try to find it."
I read the definition and trust me finding the standard deviation was never been so easier as I just had to find the square root of the variance.

standard deviation (σ) = square root of variance
                       = √470
                       = 21.6794...
                       = 22 (approx.)

Then he said, "Subtract this number from the mean and also add this number to the mean and draw the lines and mark the area." I did what he said.

Then he said, "Awesome! Can you see the red rectangle? All the dogs between this range are normal, below the range is short and above the range is tall. So the moral of the story is, my Bruno is not short."

Then I said, "That's fine but don't you think we should have measured the height from shoulders ?"
He became silent. This is one of the rare moments I witnessed ever. Making him quiet is an impossible job.

It was the story of some years back. I had no idea about what he got to prove today. I picked up the call and he said, "Come home. It's urgent." and hang up the call.