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Why two readings differ

Comparing a gap with the smallest division, and why a difference is never evidence that somebody lied.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

You will compare the gap between two readings with the smallest division of the instrument, decide whether they agree as closely as it can say, and know what to do in each case — without ever concluding that one of the readings is a lie.

2. What you already have

You can take several readings and turn them into a best estimate. This lesson is about the moment just before that: when the readings come out different and somebody has to decide what it means. Nothing here changes the arithmetic; it changes what you say about it.

You also know how to find what one small mark on a scale is worth. That number, the smallest division, is the ruler you will judge every gap against.

3. Words for this lesson

4. Different numbers are not disagreement about the truth

Two people measure the same pencil with the same ruler. One writes $148$ mm and the other writes $149$ mm.

Neither is lying, and neither is careless. The ruler is marked every millimeter, so 148 and 149 are the two nearest things it can be asked to say. Two readings one mark apart are as close to identical as that ruler allows — they agree.

So the useful comparison is never are the numbers the same. It is:

> how big is the gap, next to one mark on the instrument?

A gap of one mark: agreement. Take both, average them, move on.

A gap of eight marks: not agreement — but still not a lie. It means the two measurements were not really the same measurement, and there is something findable that was different about them.

5. Which verdict, and what each one asks you to do

The second row is the interesting one, because it is a lead, not a failure. When two groups time the same falling ball and get 1.4 s and 2.9 s on a stopwatch reading tenths, the gap is fifteen marks wide. Something real is behind that: one group started on the word go and the other when the ball left the hand; one dropped from the sill and one from head height.

Going and asking is how it gets found. Averaging the two would hide it — the average of two measurements of different things is a number about neither of them.

6. The same gap can mean different things

A gap on its own says nothing. It only means something next to the size of one mark.

Both gaps are five, and the verdicts are opposite. On a jug marked every five milliliters, a gap of five is one mark: the nearest two readings can be. On a tape marked every centimeter, a gap of five is five marks: far more than the tape wobbles. Always turn a gap into marks before you judge it.

7. Finding what was different

When a gap is too big to be wobble, there is a real difference to hunt for. The best way is to ask each person to show exactly how they measured.

• Different starting points. One measured from the end of the ruler, the other from the zero mark.
• Different things. One measured the tabletop, the other the frame underneath.
• Different timing. One started the stopwatch on go, the other when the ball moved.
• A slip that really happened. The ruler slid, or the stopwatch was started late.

Once the difference is found, both people can measure again the same way. Usually their new readings agree within a mark, which shows the difference was the method all along, and not anybody's honesty.

8. Why scientists share their methods

Real scientists hit this problem all the time. When two laboratories measure the same thing and get very different answers, nobody assumes a lie. They compare methods, line by line, until they find what was different.

That is why every scientific report includes a section called methods: exactly how the measurement was made, with what instruments, starting where. Anyone else can then repeat it. If they get the same result within a mark or two, everyone can be more confident. If they do not, the difference in method is a lead worth following.

Some of the most important discoveries in science started this way: two careful measurements that should have agreed and did not, and someone who asked why instead of deciding who was wrong.

9. How to check your verdict

Before you call two readings agreeing or not, run three checks.

1. Same thing? Were both readings of the same object, measured the same way? If not, they cannot be compared at all.
2. In marks? Did you divide the gap by the smallest division before judging?
3. No blame? Does your verdict say agree or check the method, and not someone is wrong?

The third check matters most in a classroom. The words you use about a difference change whether people feel safe writing down the readings they really got, and honest readings are the only kind worth having.

10. A finer instrument, a smaller wobble

If you need readings that agree more closely, blaming people will not help. What helps is an instrument with smaller marks, or a better method.

A tape measure marked in centimeters gives readings that wobble by about a centimeter. A ruler marked in millimeters wobbles by about a millimeter, ten times less. A stopwatch pressed by a thumb wobbles by a tenth or two of a second, because people react a little late; an electronic timer that starts and stops by itself wobbles by a thousandth.

So when two people's readings differ by one mark and you wish they agreed more closely, the fix is not to tell them to try harder. It is to choose an instrument whose marks are closer together, and to agree on a method before anyone starts. Engineers who build spacecraft measure parts with instruments that read to a thousandth of a millimeter, and even then two careful readings differ by about one of those tiny marks. The rule never goes away; the marks just get smaller.

That is the last lesson of this course, and the most useful habit it can leave you with. Whenever two numbers differ, ask how big the gap is next to one mark, and whether the two were measured the same way. Then you will know whether to average them or to go looking, and you will never need to call anybody wrong.

11. In the world: two thermometers in a classroom

A class has two thermometers on opposite walls. One reads $21$ °C and the other $22$ °C. Both are marked every degree, so the gap is one mark: they agree as closely as they can, and the room is about $21.5$ °C.

The next week one reads $21$ °C and the other $27$ °C: six marks apart. Nobody decides a thermometer is lying. The class goes to look, and finds the second one is hanging in a patch of sunlight from the window. Moved into the shade, it reads $22$ °C again.

Weather stations run by the National Weather Service follow the same rule. Their thermometers sit in white, shaded boxes, all set up the same way, so that when two stations disagree by several degrees, the difference is about the weather and not about the method.

12. In the world: measuring a runner at the doctor's

A school nurse measures a student's height with a wall chart marked in centimeters and gets $142$ cm. At the doctor's office a week later, the reading is $143$ cm. One mark apart: they agree, and the student is about $142.5$ cm tall.

If the doctor had read $148$ cm, six marks away, the right move would be to check the method, not to decide the nurse was careless. Perhaps the student kept their sneakers on at the doctor's, which can add several centimeters. Measured again barefoot, the readings agree.

Doctors track children's growth from measurements taken months apart, so they need to know whether a change is real growth or just a difference in method. Asking how many marks, and was it done the same way? is how they tell.

13. One of you must be wrong

This is the sentence to unlearn. It sounds like rigor and it is the opposite, because it treats a measurement as a thing you either get right or get wrong, like a spelling.

A measurement is a judgement made against marks a finite distance apart. It comes with a wobble the size of those marks, always, for everybody, however careful. Two readings that differ by about one mark are the expected result, and if a class of thirty all wrote exactly the same number, the thing to suspect would be copying.

Two related habits go with it. Quietly dropping the reading that does not fit — a reading may only be left out for a reason that happened, written down beside it; it did not match is not a reason. And choosing the reading that matches what you expected, which makes it impossible for a measurement ever to tell you something you did not already believe.

A last slip is judging the gap in units instead of marks. A gap of five sounds big and a gap of two sounds small, but five milliliters on a jug marked every five is agreement, and two seconds on a stopwatch reading tenths is twenty marks apart, which calls for a look at the method.

14. The pencil

1. Read the two readings.

   $148\ \text{mm and } 149\ \text{mm}$

   On a ruler marked in millimeters.

2. Find the gap.

   $149 - 148 = 1\ \text{mm}$

   The bigger minus the smaller.

3. Count it in marks.

   $1 \div 1 = 1 \text{ mark}$

   One mark on this ruler.

4. Give the verdict.

   $\text{agree}$

   As closely as the ruler can express.

5. Find the best estimate.

   $(148 + 149) \div 2 = 148.5\ \text{mm}$

   Average and carry on.

15. The table

1. Read the two readings.

   $122\ \text{cm and } 127\ \text{cm}$

   On a tape marked in centimeters.

2. Find the gap.

   $127 - 122 = 5\ \text{cm}$

   The bigger minus the smaller.

3. Count it in marks.

   $5 \div 1 = 5 \text{ marks}$

   Far more than the tape wobbles.

4. Give the verdict.

   $\text{check the method}$

   A lead, not a fault.

5. Look for the difference.

   $\text{tabletop or frame?}$

   Was one measuring the top and the other the frame underneath?

6. Measure again together.

   $\text{same starting point, same edge}$

   Now the readings should agree within a mark.

16. Three pairs at once

1. Read the first pair.

   $250 \text{ and } 255\ \text{mL, jug marked every } 5$

   Juice in a measuring jug.

2. Count its gap in marks.

   $5 \div 5 = 1 \text{ mark: agree}$

   One mark on this jug.

3. Read the second pair.

   $19 \text{ and } 24\ \text{s, stopwatch in whole seconds}$

   Ten swings of a pendulum.

4. Count its gap in marks.

   $5 \div 1 = 5 \text{ marks: check the method}$

   Far too big for the stopwatch.

5. Read the third pair.

   $84 \text{ and } 86\ \text{mL, beaker marked every } 2$

   Water in a beaker.

6. Count its gap in marks.

   $2 \div 2 = 1 \text{ mark: agree}$

   One mark on this beaker.

7. Compare the verdicts.

   $\text{same-sized gaps, different verdicts}$

   A gap only means something next to one mark.

17. Your turn: 250 mL and 255 mL in a jug marked every 5 mL. Agreement or not?

1. Find the gap.

   $255 - 250 = 5\ \text{mL}$

   The bigger minus the smaller.

2. Your turn: work this step out. Its working is at the end of the packet.

   Count it in marks.

3. Your turn: work this step out. Its working is at the end of the packet.

   Give the verdict.

18. Guided practice

Two people measure the length of a table. One writes $122$ and the other writes $127$ — five centimeters apart on a tape marked in centimeters. What should be said about the two readings?

• They agree as closely as this instrument can say, so average them and carry on
• The gap is far bigger than one mark, so something was different about how they measured, and it is worth finding out what
• One of them must have made it up

19. Guided practice

Complete the worked solution: two students measure the same pencil with a ruler marked in millimeters and write $146$ mm and $149$ mm. How far apart are they, how many marks is that, and what is the best estimate?

1. Find the gap.

   $149 - 146 =$ _a mm

   The bigger reading minus the smaller.

2. Count the gap in marks.

   $\text{gap} \div 1\ \text{mm} =$ _b marks

   More than one mark: check how each of them measured.

3. Find the best estimate for now.

   $(146 + 149) \div 2 =$ _c mm

   Halfway between, until the method is checked.

20. Guided practice

Two readings of the water in a beaker: $84$ and $86$. How far apart are they?

Answer: mL

21. Guided practice

Two students measure the same pencil with the same ruler and write $148$ mm and $149$ mm. Mark the sentence that describes what happened.

This task has no paper form; do it on a device.

22. Guided practice

Match each thing you might find to what it calls for next.

23. Practice

Two pairs of readings. Work out how far apart each pair is, then look at the last column before you judge either of them.

24. Practice

Two readings of the mass of a stone: $76$ and $77$. Mark the best estimate from the two.

72 |——————————| 81

Mark the position with a cross, then write the value:

25. Practice

Three pairs of readings, each with the smallest division of its instrument. For each pair, fill in the gap, how many marks that gap is, and the verdict: *agree* or *check the method*.

26. Somewhere new

Two groups drop the same ball from the same window and time the fall with the same stopwatch, which reads tenths of a second. One group gets $1.4$ s, the other $2.9$ s. What is the sensible next move?

• Compare exactly how each group did it, because a gap that big means the two were not measuring the same thing
• Average the two and write down 2.15 s
• Keep the group whose answer looks more sensible
• Decide that one group is not to be trusted

27. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

28. Test question

Two pairs of readings. Work out how far apart each pair is, then look at the last column before you judge either of them.

29. What you can do now

You can judge whether two readings agree by comparing their gap with one mark on the instrument. Tell someone why two different numbers for the same pencil is the expected result.

Working for the steps left to you

17. Your turn: 250 mL and 255 mL in a jug marked every 5 mL. Agreement or not?, step 2

17. Your turn: 250 mL and 255 mL in a jug marked every 5 mL. Agreement or not?, step 3