Section 2.1 Part 2: Transforming Data, Density Curves

Objectives Describe the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data. Know the basic properties of a density curve. Approximately locate the median (equal-areas point) and the mean (balance point) on a density curve.

Example 1 Suppose that the teacher was nice and added 5 points to each test score. How would this change the shape, center, and spread of the distribution?

Example 1 Continued From both the graph and summary statistics, we can see that the measures of center and measures of position all increased by 5. However the shape of the distribution did not change nor did the spread of the distribution.

Example 1 Continued Suppose that the teacher in the previous alternate example wanted to convert the original test scores to percents. Since the test was out of 50 points, he should multiply each score by 2 to make them out of 100. How would this change the shape, center, and spread of the distribution?

Example 1 Continued From the graphs and summary statistics we can see that the measures of center, location, and spread all have doubled, just like the individual observations. But even though the distribution is more spread out, the shape hasn’t changed. It is still skewed to the left with the same clusters and gaps.

Example 2 In 2010, Taxi Cabs in New York City charged an initial fee of $2.50 plus $2 per mile. In equation form, fare = (miles). At the end of a month a businessman collects all of his taxi cab receipts and calculates some numerical summaries. The mean fare he paid was $15.45 with a standard deviation of $ What are the mean and standard deviation of the lengths of his cab rides in miles? Mean = miles St. Dev.= 5.10 miles

Density Curves Density Curves trap an area of 1 between them and the x-axis AND never dip below the x-axis.

Find the density curve(s) [1 block =.10 units 2 ] A: yes B: no C: no D: yes E: yes