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Variance and Standard Deviation (2) Scaling

Variance and Standard Deviation (2) Scaling. Fred tries out Laxo Laxative - and finds his visits increase by 5 a day. Mean = 8.1. Mean = 13.1. 5 more visits per day, Sum is 12x5 (60) more = 157 mean is 5 more. 13.1. Sum = 97. Standard Deviation.

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Variance and Standard Deviation (2) Scaling

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  1. Variance and Standard Deviation (2)Scaling

  2. Fred tries out Laxo Laxative- and finds his visits increase by 5 a day

  3. Mean = 8.1 Mean = 13.1 5 more visits per day, Sum is 12x5 (60) more = 157 mean is 5 more 13.1 Sum = 97

  4. Standard Deviation The mean will be increased by the same value The new mean becomes x + b Standard Deviation = (xi - x)2 n Sx =((xi+b) - (x+b))2 =(xi + b- x - b)2 n n If each piece of data (xi) is increased by a value, say b Each value becomes xi + b The ‘b’s just cancel and you are left will the original formula

  5. Fred’s Example Deviations from mean will be the same before and after - so the Standard Deviation and Variance must be the same

  6. Fred tries out Quixo Laxative- and finds his visits doubling

  7. Mean = 8.1 Mean = 16.2 Twice the visits per day, Sum is doubled = 194 So, the mean is doubled Sum = 97

  8. Standard Deviation The mean will be increased by the same factor The new mean becomes ax Standard Deviation = (xi - x)2 n Sx =(axi- ax)2 =a2 (xi - x)2 =a(xi - x)2 n n n If each piece of data (xi) is increased by a factor, say a Each value becomes axi The Standard Deviation is Scaled by the factor a

  9. Fred’s Example The Standard Deviation has increased by a factor of 2

  10. Summary The new mean y = ax + b • If all values are increased by adding the same value • The mean increases by that value • The Standard Deviation remains the same • If all values are multiplied by the same value • The mean is multiplied by that value • The Standard Deviation is also multiplied by this value In general, if a variable x is transformed using the linear transformation ‘y = ax + b’ The new Standard Deviation Sy = aSx

  11. Example The linear transformation ‘y = ax + b’ and Sy = aSx y = ax + b x = 21.5 , Sx = 6.4 • A test is marked out of 30 • The mean mark is 21.5 • The Standard Deviation is 6.4 • To make it into a percentage the teacher decides to multiply the marks by 3 and add 10. • What are the new mean and Standard Deviation? New mean = 3 x 21.5 + 10 = 74.5 s.d. = 3 x 6.4 = 19.2

  12. Activity Page 28 of your Statistics 1 book and try … • Exercise 1F • Scaling

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