Make order of magnitude calculations (to include: estimating the mean test…

The idea

An order-of-magnitude calculation estimates the scale of a value using powers of ten and provides a quick plausibility check. To estimate a mean, divide the overall total by the number of participants after rounding both to convenient values. For example, 4,850 words recalled by 97 participants is approximately 5,000 divided by 100, or 50 words per participant. In scientific notation this is $5 × 10^1$, so its order of magnitude is $10^1$. Compare the estimate with the precise calculation: a large discrepancy signals that the arithmetic, decimal position or units should be checked. An estimate is deliberately approximate, so report it with an appropriate unit and limited precision.

What earns the marks

• State the method: the estimated mean is the rounded overall total divided by the rounded number of participants.
• Round sensibly to make the division easy, for example treating 4,850 as about 5,000 and 97 as about 100 to obtain about 50 per participant.
• Giving the answer with its unit and to a sensible accuracy, such as about 50 words recalled per participant, rather than an unlabelled figure.
• Use scientific notation to identify scale: 50 is $5 × 10^1$, so its order of magnitude is $10^1$.
• Use the estimate to check a calculated result and investigate discrepancies in arithmetic, decimal position or units rather than assuming one cause.

Common mistakes and how to put them right

• Dividing the number of participants by the total score instead of the other way round, which gives a fraction of a word per person and is never a sensible mean.

• Rounding so hard that the estimate stops being informative, such as calling 97 participants one hundred and a total of 4,850 ten thousand, then reporting a mean of one hundred.

• Writing the estimate with no working, which hides whether the total was divided by the participant count in the correct direction.

• Dropping the unit, so a mean reported as 50 could be seconds, words recalled or percent and means nothing on its own.

Exam tips

• Read the table heading carefully to decide whether the total belongs to one participant or to the full sample before estimating the mean.
• Show the rounded values and the division so the method can be followed and checked.
• If the command says estimate, use sensible rounding and avoid reporting a long string of calculator decimals.
• Use the estimate to check the precise result, then reconsider the operation, decimal position and units if their scales differ substantially.