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3. Error in Measurement     cm 12 3 0 a b c d e e Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. The lowest value that a measurement can take is sometimes referred to as the lower bound. The highest value that a measurement can take is sometimes referred to as the Upper bound. Lower bound = 1.5 cm Upper bound = 2.5 cm (Note that this is really 2.49999… cm) In earlier work on errors we discussed which of the pieces of wood below would be measured as 2cm correct to the nearest cm.

4. 100 ml 200 ml 300 ml 400 ml 500 ml Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Questions Error in Measurement 1 Weight of an egg 62 g (nearest g) Amount of liquid 320 ml (nearest 20 ml) 2 3 Time of a race 19.4 seconds (nearest tenth of a second) Find the upper and lower bounds for the situations shown below. Upper bound 62.5 g Lower bound 61.5 g Upper bound 19.45 seconds Lower bound 19.35 seconds Upper bound 330 ml Lower bound 310 ml

5. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations 100 m 50 m Example Question 1. The sports pitch below has the dimensions shown measured to the nearest metre. Calculate the maximum and minimum areas for this sports pitch. The maximum area (Upper bound) = 100.5 x 50.5 = 5075.25 m 2 The minimum area (Lower bound) = 99.5 x 49.5 = 4925.25 m 2

6. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations Example Question 2. All the 110 marbles shown below are identical. A marble is taken and weighed as 15.6 g to the nearest tenth of a gram. Find the upper and lower bounds for the weight of all the marbles. Upper bound = 110 x 15.65 = 1721.5 g Lower bound = 110 x 15.55 =1710.5 g

7. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations Example Question 3. The numbers below are all measured to 1 dp. Find (a) an upper bound (b) a lower bound for the calculation. So lower bound is given by (a) The upper bound is when the numerator is a maximum and the denominator a minimum. So upper bound is given by (b) The lower bound is when the numerator is a minimum and the denominator a maximum.

8. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations Example Question 4. A cyclist travelled 750 metre along a road and was timed at 38.6 seconds. If the road length was measured to the nearest metre and the stop watch was accurate to 1/10 of a second, find (a) the upper bound for his average speed (b) the lower bound for his average speed. Upper bound for speed is given by: (a) The upper bound is when the distance is at a maximum and the time a minimum. Lower bound for speed is given by: (b) The lower bound is when the distance is at a minimum and the time a maximum.

9. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations Question 1. Find the maximum and minimum perimeter of the snooker table if its length and width are measured to the nearest cm. Maximum perimeter = 2(90.5 + 210.5) = 602 cm Minimum perimeter = 2(89.5 + 209.5) = 598 cm 210 cm 90 cm

10. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations Question 2. All of the coloured discs are identical. One of them was weighed as 46.8 g to the nearest tenth of a gram. Find the upper and lower bounds for the weight of all the discs. Upper bound = 57 x 46.85 = 2670.45 g Lower bound = 57 x 46.75 =2664.75 g

11. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations Question 3. The numbers below are all measured to 2dp. Find (a) an upper bound (b) a lower bound for the calculation. Lower bound is given by: Upper bound is given by:

12. Remember: All measurements are approximations. They are accurate to  1/2 the stated degree of accuracy centred on the measurement. Error in Measurement Errors in Calculations Question 4. An athlete ran a cross country race of 3000 m in a time of 15 minutes 28.3 seconds. If the course was measured to the nearest 100 m and the stopwatch was accurate to 1/10 of a second, find (a) an upper bound for the average speed and (b) a lower bound for the average speed. Upper bound for speed is given by: Lower bound for speed is given by: