Absolute, Mean Absolute, Relative and Percentage Errors

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. I take one ordinary scale, with a least count of zero point one centimetre. With it, I measure a length of twenty five centimetres, and then a tiny length of two point five centimetres. So in both readings, the error is the same: zero point one centimetre, one least count. Now tell me, are both measurements equally good? Same error, so same quality, right? Oh, not so fast! By the end of this video, you will see that one of them is ten times worse than the other. And look, every error question in JEE starts from the few lines we learn today. So, let's begin.

0:43Many readings: the best value is the mean

So, suppose we measure the same quantity again and again, n times. We call the readings a one, a two, and so on, up to a n. Those three dots on screen just mean, and so on, okay? Now, the readings come out a little different from each other. So which one value should you report? The first one? The biggest one? Yes, the average! The best value is the mean. Add all the readings, and divide by n, the number of readings. We write it as a, with a small bar on top. We call it a bar. Clear?

1:18Absolute error and mean absolute error

Next question: how far is each reading from this best value? That distance is the absolute error of that reading. Delta a sub i, the little triangle you see on screen, equals the modulus of a sub i minus a bar. Those two straight bars are the modulus, okay? Now, one thing to think about. Suppose a reading is smaller than the mean. Then a sub i minus a bar is negative. So, is its error negative? No, no! The modulus makes it positive. The error only tells you how far the reading is from the mean, not on which side. Got it? Now take the average of all these absolute errors. Add them, and divide by n. This is the mean absolute error: delta a, with a long bar on top. We read it delta a bar. The big sigma sign simply means, add them all. And now we can write the result properly: a equals a bar, plus or minus delta a bar. Best value, plus or minus its mean absolute error. Nice and clean. One small rule that many students forget. If the mean absolute error comes out smaller than the least count of your instrument, do not quote it. Quote the least count instead, okay?

2:36Relative and percentage error

Now back to our puzzle. The error alone does not tell you how good a measurement is. You have to compare it with the value itself. So, divide the mean absolute error by the mean. This is the relative error, also called the fractional error. Multiply it by a hundred, and you get the percentage error. Same idea, just written in percent. Right? So, now you tell me. Twenty five centimetres, and two point five centimetres, both with an error of zero point one centimetre. Which one is measured better? Let's calculate and see. For twenty five centimetres: zero point one, divided by twenty five point zero, times a hundred. Zero point four percent. Very small. For two point five centimetres: zero point one, divided by two point five, times a hundred. Four percent! See, the same error, but ten times the percentage error. Puzzle solved! So here is the lesson toppers remember. Short lengths need a finer instrument. Think about it: when the length is small, the same least count becomes a big percentage. And one quick trick: it works backwards too. Suppose a resistance of one hundred and fifty ohms is known to two percent. The omega sign on screen is ohm. What is its absolute error? Just two percent of one fifty: zero point zero two times one fifty, that is three ohms. So the resistance is one fifty, plus or minus three ohms. With me?

4:09Example 1: pendulum periods, mean and percentage error

Now, let's use all of this. Example one. The period of a pendulum in five trials is two point six three, two point five six, two point four two, two point seven one, and two point eight zero seconds. Find the mean period, the mean absolute error, the relative error and the percentage error. Pause the video, and try it first. Done? Okay, let's check. First, the mean. Add the five readings: thirteen point one two seconds. Divide by five: two point six two four. We round it to two point six two seconds, okay? Now the distance of each reading from two point six two. Two point six three gives zero point zero one. Two point five six gives zero point zero six. Two point four two gives zero point two zero, two point seven one gives zero point zero nine, and two point eight zero gives zero point one eight. All positive. Remember the modulus, right? Add them: zero point five four. Divide by five: zero point one zero eight, which we round to zero point one one seconds. That is the mean absolute error. So the period is two point six two, plus or minus zero point one one seconds. Clear? Last, the relative error: zero point one one, divided by two point six two, about zero point zero four. Times a hundred: about four percent. And that's it, all four answers done.

5:44Example 2: six weighings, mass with percentage error

Now a slightly harder one. Six weighings of a body give twenty point four, twenty point two, twenty point six, twenty point five, twenty point three, and twenty point four grams. Report the mass, and its percentage error. Pause the video, and try it. Done? Okay, let's check. The six readings add up to one hundred and twenty two point four grams. Divide by six: twenty point four grams, exactly. Now the deviations from twenty point four. Zero, zero point two, zero point two, zero point one, zero point one, and zero again. Their sum is zero point six. Now careful. Two of these deviations are zero. So do we divide by four, only the readings that have some error? That gives zero point one five, right? No, no, this is the trap! The formula says divide by n, and n is the number of readings. Six readings, so divide by six, the zero deviations included. Zero point six by six: zero point one gram. Got it? So the mass is twenty point four, plus or minus zero point one grams. And always write the unit, okay? And the percentage error: zero point one, divided by twenty point four, times a hundred. About zero point five percent. And we are done!

7:12Quick revision: mean, mean absolute and percentage error

So, quick revision. The best value is the mean of the readings. The mean absolute error is the average distance of the readings from the mean, and the result is the mean, plus or minus this error. And the percentage error compares the error with the value itself, so the same error hurts a small length much more. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!