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Accumulated Error

Small rounding errors can build up into a much larger total error when a calculation is repeated many times.

Small Errors Add Up

Example 1 — accumulated rounding error in a sum

10 measurements are each rounded to the nearest whole number (a maximum absolute error of 0.5 each). They are all added together. Find the worst-case accumulated error. 1 Each value can be off by up to 0.5 2 In the worst case, every error points the same direction (all too high, or all too low) 3 Multiply the individual error by the number of values: 0.5 \times 10 Answer: Worst-case accumulated error = 5

Example 2 — accumulated error in a real setting

A shop's till rounds every price to the nearest cent (a maximum error of 0.005 p er t r an s a c t i o n) . O v er 200$ transactions in a day, find the worst-case accumulated rounding error. 1 Each transaction can be off by up to $0.005 2 Multiply by the number of transactions: 0.005 \times 200 Answer: 1.00

Practice

15 lengths are each rounded to the nearest cm (a maximum error of 0.5 cm each). Find the worst-case accumulated error when they are all added together.

Your answer

Small Errors Add Up

Example 1 — accumulated rounding error in a sum

10 measurements are each rounded to the nearest whole number (a maximum absolute error of 0.5 each). They are all added together. Find the worst-case accumulated error. 1 Each value can be off by up to 0.5 2 In the worst case, every error points the same direction (all too high, or all too low) 3 Multiply the individual error by the number of values: 0.5 \times 10 Answer: Worst-case accumulated error = 5

Example 2 — accumulated error in a real setting

A shop's till rounds every price to the nearest cent (a maximum error of 0.005 p er t r an s a c t i o n) . O v er 200$ transactions in a day, find the worst-case accumulated rounding error. 1 Each transaction can be off by up to $0.005 2 Multiply by the number of transactions: 0.005 \times 200 Answer: 1.00

Practice

15 lengths are each rounded to the nearest cm (a maximum error of 0.5 cm each). Find the worst-case accumulated error when they are all added together.

Your answer