2. Chapter 3 Lesson 6 Scale Changes of Data

3. Vocabulary  Scale Change- A transformation that maps each data value x i in a set of data {x 1, x 2, x 3 ……x n } to ax i, where a is a nonzero constant.  Scale Factor- The nonzero constant by which each data value is multiplied in a scale change.  Scale Image- The result of a scale change, or the point it represents.  Scaling- Applying a scale change to a data set, also called rescaling.

4. Mean Median Mode 1 1 2 4 6 7 7 7 8 9 9 11  Mean?  Median?  Mode?

5. Now I will Multiply all numbers by 3 3 3 6 12 18 21 21 21 24 27 27 33  Mean?  Median?  Mode?

6. What Happened to the Stats?  Mean?  Median?  Mode?

7. Theorem  Multiplying each element in a data set by the factor a multiplies each of the mode, mean, and median by a.

8. Range, IQR, Variance, Standard Deviation 1 1 2 4 6 7 7 7 8 9 9 11  Range?  IQR?  Variance?  Standard Deviation?

9. Now I will multiply all numbers by 3 3 3 6 12 18 21 21 21 24 27 27 33  Range?  IQR?  Variance?  Standard Deviation?

10. What happened to the stats?  Range?  IQR?  Variance?  Standard Deviation?

11. What would happen if  I multiplied all the data values by -2?

12. Theorem  If each element of a data set is multiplied by a, then the variance is a 2 times the original variance, the standard deviation is ∣a∣ times the original standard deviation, the range is ∣a∣ times the original range, and the IQR is ∣a∣ times the original IQR.

13. Using CPI  On page 194  Current baseline time period is 1982-1984  CPI calculates percent increase in cost from baseline time period to the given year, in this section it uses 1994.  To calculate the percentage increase when the CPI is given, simply subtract 100 from the CPI and that is the percentage increase from the baseline time period  Example: if CPI is 132, then percent increase is 32%  If CPI is 114.6, then percent increase is 14.6%

14. Range  Suppose A is the maximum value in a data set and B is the minimum value.  Write an expression for the range of the data set.  If we multiply all the elements in the set by a scale factor of C, what are the maximum and minimum values of the image data set?  Write and simplify an expression for the range of the image data set.

15. Formula for transformations Original123456 Scaled61218243036 Notice how each original element is being multiplied by 6 to get the new scaled element. The way we write this is as follows: S:x i →6x i

16. S:x i → Ax i  Every part except the A stays the exact same.  A gets replaced by whatever constant is being multiplied by each element in the data set.  If you are dividing by a constant, write it as:  S:x i → 1 / A x i

17. Reminder on Standard Deviation  The standard deviation of a data set tells you how spread out the values are  The smaller the standard deviation is, the closer the data values are to each other  The larger the standard deviation is, the farther apart the data values are

18. Homework  Worksheet 3-6