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Repeated readings and a best estimate

Why one reading is not the measurement, how several of them become one answer, and how that answer is used.

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 take several readings of the same thing, record every one of them, combine them into a best estimate, and use that estimate in a calculation.

2. What you already have

From Grade 3 you can read a value off a scale by working out what one small mark is worth. Everything in that lesson still holds. What this one adds is what to do when you read the same thing twice and get two different answers — which, if you look carefully enough, is almost always.

You can also work out a speed from a distance and a time. Real times come from stopwatches, and stopwatch readings wobble, so this lesson ends by combining the two ideas.

3. Words for this lesson

4. Measure the same leaf five times

Take a ruler and one leaf, and measure its length. Write the number down. Now put the ruler down, pick it up again, and measure the same leaf from the start. Five times in all.

Here is what one class got, and you will get something like it:

The leaf did not grow. The ruler did not change. Nobody was careless — the same person did all five, slowly.

So, before reading on, decide two things and be ready to defend them.

Which of those five numbers is the length of the leaf? If you want to say the one that came up twice, notice that two of them came up twice. If you want to say the first, ask what makes it better than the fifth.

Was somebody wrong? Four of the five readings are not 31, and if one number is the truth then four of them are mistakes. Is that what happened here? Look at a ruler and say how far apart its smallest marks are, then look at the spread of the five readings, and say whether the two are the same size as each other.

The next part says what to do with five numbers like these, and it is not pick one.

5. One reading is not the measurement

Nothing went wrong with those five readings. A ruler has a smallest mark, the leaf has a soft edge, and each time you have to decide where the edge is. Every careful measurement wobbles like this, and none of the five is a mistake.

So you take several readings and combine them:

$$\text{best estimate} = \frac{\text{all the readings added up}}{\text{how many there were}}$$

Here: $31 + 30 + 32 + 30 + 32 = 155$, and $155 \div 5 = 31$ mm. Notice that nobody ever read $31$ on that particular occasion — and notice that the answer is still $31$ mm, because a best estimate is a measurement and measurements carry units.

6. Writing the readings down, not just the answer

A record of a measurement looks like this:

Every reading is there, including the ones that look odd. That is not tidiness for its own sake: a record with only the final number in it cannot be checked by anybody, and cannot be checked by you in a week's time either.

The order matters too. Take all the readings first, then add, then divide. Working out the average as you go along tempts you to stop when the number looks right, which is a way of getting the answer you expected.

7. Why averaging helps

Each reading is a little too high or a little too low, and nobody can tell which. When you add several readings, the highs and lows start to cancel. A reading $1$ mm too long and another $1$ mm too short add to exactly the right total for two.

So the average of five readings is usually closer to the true value than most of the single readings. The more readings you take, the more the wobbles cancel. That is why a doctor may take your blood pressure more than once, and why a science fair judge wants to see several trials, not one.

Averaging cannot fix a mistake that pushes every reading the same way, though. If the ruler's zero is worn off and every reading is $2$ mm too long, the average is $2$ mm too long as well. Checking the instrument is a separate job from averaging the readings.

8. The spread tells you how steady the readings are

The spread is the largest reading minus the smallest. For the leaf, $32 - 30 = 2$ mm. A small spread means the readings agree closely; a large spread means they wobble a lot.

Compare the spread with the smallest mark on the instrument. A ruler marked in millimeters can easily give readings $1$ or $2$ mm apart, so a spread of $2$ mm is normal. A spread of $20$ mm on the same ruler would mean something else was going on, and the next lesson is about how to tell.

Scientists always report the spread along with the best estimate. About $31$ mm, give or take $1$ says much more than $31$ mm on its own.

9. Using a best estimate in a calculation

A best estimate is a measurement, so you can use it just like any other. If a toy car rolls $4$ m and five timings give a best estimate of $2$ s, then its speed is $4 \div 2 = 2$ m/s.

Always average first and calculate second. Working out five speeds and averaging them usually gives nearly the same answer, but not always, and it takes five times as much work. More importantly, averaging the readings first keeps the record honest: you can see the five times, their total and their average, and anyone can check each step.

Scientists measuring the speed of sound, the time for a ball to fall, or the length of a day on another planet all work this way: repeat, average, then calculate.

10. How to check a best estimate

Run three checks on any best estimate.

1. It lies among the readings. It can never be smaller than the smallest reading or larger than the largest.
2. It has the readings' unit. Readings in seconds give a best estimate in seconds.
3. Multiplying back gives the total. Five times the best estimate should equal the sum of the five readings.

The first check catches the most common slip: dividing by the wrong number, or forgetting to divide at all. A best estimate of $155$ mm from readings near $31$ mm is simply the total, not the average.

11. In the world: timing a track race

At a middle school track meet, three officials time each runner in the $100$ m dash with separate stopwatches. For one runner they read $14.2$, $14.4$ and $14.3$ seconds. Nobody made a mistake: each official pressed the button a fraction of a second earlier or later.

The official time is worked out from all three. They add to $42.9$ s, and $42.9 \div 3 = 14.3$ s. The runner's speed is then $100 \div 14.3 \approx 7.0$ m/s.

Big competitions now use electronic timing that starts with the starting gun and stops when a runner breaks a beam of light, because it wobbles far less than a human thumb. But even electronic timers are checked against each other, and the rule is the same: more than one reading, and every reading recorded.

12. In the world: a science fair experiment

A student wants to know whether a heavier ball rolls down a ramp faster than a lighter one. She times each ball five times.

The light ball takes $2.1$, $1.9$, $2.2$, $1.8$ and $2.0$ s: a total of $10.0$ and a best estimate of $2.0$ s. The heavy ball takes $2.0$, $2.1$, $1.9$, $2.0$ and $2.0$ s: also a best estimate of $2.0$ s. The spreads overlap completely, so the honest conclusion is that her experiment found no difference.

If she had timed each ball only once, she might have got $1.8$ s for one and $2.2$ s for the other and wrongly concluded that one was faster. Repeating the readings is what stopped her from reporting a difference that was only wobble. Judges at science fairs look for exactly this: several trials, every reading written down, and a conclusion that respects the spread.

13. The odd reading is not a mistake to be thrown away

Faced with $31$, $30$, $32$, $30$, $32$, learners often want to pick the one that came up most often, or the one that looks neatest, or simply the first. All three throw away information that was collected on purpose.

The second habit is worse: quietly dropping a reading because it does not fit. If a reading is kept out of the sum, the reason has to be something that happened — the stopwatch was started late, the ruler slipped — and it has to be written down beside it. It did not match the others is not a reason; it is the thing you were measuring.

And the average is not more precise than the instrument pretends. Five readings on a ruler marked in millimeters can give a best estimate of $31$ mm. They do not turn it into a ruler that reads thousandths.

One more slip: forgetting to divide. Adding five readings of about $31$ mm gives $155$, and $155$ mm is not the length of the leaf; it is five leaves laid end to end. The check that catches it is the first one in the list above: a best estimate must sit among the readings. If yours is bigger than every single reading, you have found the total, not the average, and one division is still missing from your work.

14. Ten swings of a pendulum

1. Read the five timings.

   $71, \ 69, \ 70, \ 72, \ 68\ \text{s}$

   A spread of four seconds: wobbly, and normal.

2. Add all five readings.

   $71 + 69 + 70 + 72 + 68 = 350$

   Every reading goes in.

3. Share the total between five.

   $350 \div 5 = 70\ \text{s}$

   The best estimate.

4. Check it lies among them.

   $68 \le 70 \le 72$

   Inside the set, as it must be.

5. Notice something odd.

   $70 \text{ was read once}$

   The estimate need not be the most common reading.

15. A handful of rice

1. Read the five masses.

   $48, \ 52, \ 50, \ 49, \ 51\ \text{g}$

   Five scoops of the same size.

2. Add all five readings.

   $48 + 52 + 50 + 49 + 51 = 250$

   The total.

3. Share the total between five.

   $250 \div 5 = 50\ \text{g}$

   The best estimate.

4. Find the spread.

   $52 - 48 = 4\ \text{g}$

   Largest minus smallest.

5. Report the estimate and spread.

   $50\ \text{g, give or take } 2$

   Half the spread either side.

6. Check by multiplying back.

   $5 \times 50 = 250$

   It equals the total.

16. A toy car's speed

1. Read the distance and the times.

   $6\ \text{m}; \ 1.4, \ 1.6, \ 1.5, \ 1.3, \ 1.7\ \text{s}$

   One distance, five timings.

2. Add the five times.

   $1.4 + 1.6 + 1.5 + 1.3 + 1.7 = 7.5$

   Every timing goes in.

3. Share the total between five.

   $7.5 \div 5 = 1.5\ \text{s}$

   The best estimate of the time.

4. Divide distance by time.

   $6 \div 1.5 = 4$

   Speed from the best estimate.

5. Write the speed with its unit.

   $4\ \text{m/s}$

   Meters per second.

6. Check by multiplying back.

   $4 \times 1.5 = 6\ \text{m}$

   It gives the distance again.

7. Explain the order.

   $\text{average first, calculate second}$

   One clean calculation from one honest estimate.

17. Your turn: 12, 14, 13, 13, 13 cm. What is the best estimate?

1. Add all five readings.

   $12 + 14 + 13 + 13 + 13 = 65$

   Every reading goes in.

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

   Share the total between five.

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

   Check it lies among them.

18. Guided practice

Five measurements of the mass of a handful of rice give $48$, $52$, $50$, $49$ and $51$. What is the best estimate?

Answer: g

19. Guided practice

Complete the worked solution: a ball is dropped five times from the same height, and its bounces reach $26$, $29$, $25$, $30$ and $25$ cm. Find the total, the best estimate and the spread.

1. Add the five readings.

   $\text{all five added} =$ _a cm

   Every reading goes in.

2. Share the total between five.

   $\text{total} \div \text{five} =$ _b cm

   That is the best estimate.

3. Find the spread.

   $30 - 25 =$ _c cm

   Largest minus smallest shows how much the readings wobble.

20. Guided practice

Why is it better to measure the same thing five times and average, rather than measuring once carefully?

• Each reading is a little out one way or the other, and averaging lets those cancel
• Because one reading on its own might be a lie
• Because the thing being measured keeps changing
• Because scientists always use five of everything

21. Guided practice

Put the steps of taking a best estimate in the right order.

Number the steps in order (write the number in the box):

• Measure it once and write the number down.
• Measure it again in exactly the same way, and write that down too.
• Carry on until there are five readings written down.
• Add the five readings together.
• Divide the total by five to get the best estimate.

22. Guided practice

Finish the sentence about the five readings of the water a dripping tap wastes in a minute: $8$, $9$, $10$, $9$, $9$.

The five readings add up to _tot, and sharing that between five gives a best estimate of _avg mL.

23. Practice

Here is somebody's record of measuring the water a dripping tap wastes in a minute five times. Finish the last two lines.

24. Practice

The five readings of the height a ball bounces back are $12$, $14$, $13$, $13$ and $13$. Mark the best estimate on the scale.

10 |——————————| 16

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

25. Practice

A class times a toy car rolling $8$ m along a floor, five times. The stopwatch readings are $4.1$, $3.8$, $4.2$, $3.9$ and $4.0$ seconds. Using the best estimate of the time, what is the car's speed?

Answer: unit: m/s / km/h

26. Somewhere new

Five people stand at the finish line of a school race and time the same runner with their own stopwatches. They get $16$, $19$, $18$, $21$ and $16$ seconds. What is the best estimate of the runner's time?

Answer: unit: h / s / min

27. Lesson test

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

28. Test question

Here is somebody's record of measuring the height a ball bounces back five times. Finish the last two lines.

29. What you can do now

You can turn a set of repeated readings into a best estimate. Tell someone why the odd-looking reading still goes in the sum.

Working for the steps left to you

17. Your turn: 12, 14, 13, 13, 13 cm. What is the best estimate?, step 2

17. Your turn: 12, 14, 13, 13, 13 cm. What is the best estimate?, step 3