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S1 Coding Using coding to make numbers easier to work with when data values are large Mean and Standard Deviation Calculate the mean and standard deviation of the following data set 3,10,15,7,8 Add 3 to each of the numbers and recalculate the mean and standard deviation Subtract 2 from each of the numbers and recalculate the mean and standard deviation Multiply each of the numbers by 10 and recalculate the mean and standard deviation Divide each of the numbers by 2 and recalculate the mean and standard deviation Solutions Mean Standard Deviation x x+3 x-2 10x x/2 3 6 1 30 1.5 10 13 8 100 5 15 18 13 150 7.5 7 10 5 70 3.5 8 11 6 80 4 8.6 11.6 6.6 86.0 4.3 4.39 4.39 4.39 43.93 2.20 What do you notice about the coded results compared to the original results? Why does this happen? x+a Mean +a, standard deviation does not change x-a Mean - a, standard deviation does not change ax Mean x a, standard deviation x a x/a Mean/a, standard deviation /a Example 1 A data set has been coded using y=x/10. The standard deviation is 1.41 Find the standard deviation of the original data. 1.41 x 10 = 14.1 Example 2 A data set has been coded using y=x-20. The standard deviation is 3.641 Find the standard deviation of the original data. 3.641 as the standard deviation does not change Example 3 A data set has been coded using y=x+100. 2 The standard deviation is 12.342 Find the standard deviation of the original data. 24.684 Adding 100 has no effect but the division by 2 has halved the standard deviation E.G. 4 Time taken to complete reading a paper Time take (secs) Freq (f) Midpoint (x) Coding Y=x-500 1000 fy fy² 0-3000 4 1500 1 4 4 3000-6000 16 4500 4 64 256 6000-8000 8 7000 6.5 52 338 8000-13000 2 10500 10 20 200 Σf=30 σ² Σfy=140 Σfy² =798 =798 – 140 ² = 4.82 30 30 Coded σ =2.19596 Original σ =2195.96

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