Types of Errors

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Suppose the end of your metre scale is worn off, by one millimetre. You don't notice, and you measure a pencil with it again and again, many times. Then you take the average. Now, averaging removes errors, everyone knows that. So your final answer is perfect, right? Hmm. No, no, not so fast! By the end of this class, you will know exactly which errors averaging can remove, and which errors it cannot touch at all. And look, JEE loves this: name the type of error, and pick the right way to remove it. So, let's begin.

0:41No measurement is perfect

So, let me start with a simple question. Take the best instrument you can imagine, and the most careful student in the class. Will the reading be perfectly exact? No! No measurement is perfect. There is always some difference between the value you measure and the true value. And that difference is what we call the error. Simple, okay? Now, errors are not all of one kind. Today we meet four kinds: systematic, random, least-count, and gross. For each one, we ask two things. Where does it come from? And how do we get rid of it?

1:17Systematic errors: always the same direction

First, systematic errors. Here the key word is direction. This kind of error pushes every reading in the same direction. If one reading comes out too big, every reading comes out too big. Okay? Where does it come from? Three places. One: a faulty instrument, with a zero error, or a wrong calibration. Two: an imperfect method. For example, you ignore air resistance, or you ignore heat loss. The method itself leaves something out, right? And three: a personal habit, like parallax, in the way you look at a scale. You see, a habit repeats every time, so its error also repeats, in the same direction. Now the big question. You take many readings and average them. Will a systematic error go away? No! Every reading is pushed the same way, so the average is pushed the same way too. Repeating the measurement does not reduce a systematic error. So how is it removed? By correcting the instrument, or by correcting the method. Hold on to this; it is the key to our puzzle. Got it?

2:23Random errors: up and down

Alright, now random errors. These behave very differently. They vary irregularly from reading to reading: one reading a little high, the next a little low, no fixed direction. Where do they come from? Small changes in the conditions, and the observer's judgment, which is not exactly the same every time. Okay? Now tell me. How would you reduce a random error? Yes, exactly! Take many readings, and average them. So averaging does work, but on random errors, not on systematic ones. Clear?

2:57Topper's trick: one question decides it

Now let me share a trick that toppers use. When a question describes an error in words, ask just one question: does it push every reading in the same direction? If yes, the double arrow on screen takes you to systematic, and the fix is the instrument or the method. If it is irregular, up and down with no pattern, it is random, and the fix is averaging. That's it, one question, and you know: systematic or random. Really simple, right?

3:26Least-count errors and blunders

Two more kinds, and then we solve problems. First, the least-count error. It comes from the resolution of the instrument, how finely it can read. And how big can it be? At most one least count. Remember this; we will use it in Example two. Last, gross errors, also called blunders. These are simply mistakes: you read the wrong value, or you copy a number wrongly. And honestly, it happens to everyone in a hurry! They are avoided by care. And if one reading stands far away from all the others, that is your signal: check it again. Okay?

4:02Example 1: classify five errors as systematic or random

Now, Example one. Classify each error as systematic or random. One: a wall clock that gains two seconds every hour. Two: small fluctuations of the mains voltage during an experiment. Three: a metre scale whose end is worn off by one millimetre. Four: always reading a thermometer from slightly above the mercury level. And five: the unpredictable reaction time in starting a stopwatch. Pause the video, and try it first. Done? Okay, let's check. The clock gains two seconds every hour, every single hour. Always the same direction! So it is systematic, from the instrument. The mains voltage fluctuates, small ups and downs with no pattern. Clearly random. Now the worn-off scale. Remember our puzzle from the start? With the end worn off, every reading is off in the same way. So it is systematic, from the instrument. And that means averaging many readings will not remove it. Puzzle solved! The cure is to correct the instrument. Now number four, the thermometer. It is the student's own habit, something personal. So it is random, right? No, no, careful! This is the trap. Look at the word, always. Always from slightly above, so every reading is pushed in the same direction. That makes it systematic: personal, but systematic. Got it? And five, the reaction time in starting a stopwatch. The question itself calls it unpredictable: sometimes quicker, sometimes slower. So this one is random.

5:44Example 2: least-count error in a pendulum's period

Now a harder one, JEE level. Here, a stopwatch of least count zero point one second times twenty oscillations of a pendulum as thirty point zero seconds. Find the period, with its least-count error. Then, how many oscillations must be timed, so that the error in the period is at most zero point zero zero two second? Pause and try. Done? Okay, let's check. The period is the time for one oscillation. So, thirty point zero seconds divided by twenty: one point five zero zero seconds. Easy. Now the error. The least count is zero point one second, and the error is at most one least count. So the error in the period is zero point one second, right? No, no! That is the trap. See, that zero point one second is the error in the total time, the time of all twenty oscillations. So the error in one period is zero point one divided by twenty: zero point zero zero five second. The delta T you see on screen means the error in the period, okay? So the answer: the period is one point five zero zero, plus or minus zero point zero zero five, seconds. The plus minus sign shows the error on both sides. Now the second part. Suppose you time n oscillations instead of twenty. What happens to the error in the period? Yes, it gets smaller! The error in the period becomes zero point one over n, and we want it less than or equal to zero point zero zero two, the sign you see on screen. Zero point one divided by zero point zero zero two is fifty. So time at least fifty oscillations. Same stopwatch, smaller error. See how fast that was?

7:38Quick revision: systematic, random, least-count, gross

So, quick revision. Systematic errors push every reading the same way. Repeating does not help, so correct the instrument or the method. Random errors are irregular, so take many readings and average. The least-count error is at most one least count, and a blunder is caught by checking the odd reading again. Now go and try the practice problems on this page. Ask the one question, same direction or not, and you will find them easy. All the best!