DS280 - Intro to Statistics

Lecture Review #13 - Central Tendency

REVIEW QUESTIONS:

1) Discuss key differences between sample and population in statistics.

2) When is the median a better representation of the "middle" of the data than the mean?

3) What is the relationship between the mean, median, and mode when the data are from a normal distribution? When they are skewed right? Skewed left?

4) What is a geometric mean and when is it used?

Computational exercises:

1) Find the sample mean and median for the following data sets.

a) 3, 1, 5, 2, 4

b) 25, 30, 10, 9000, 42

2) Clorinda Cragdingle's portfolio increased in value by 30% in 2007. It lost half its value in 2008. She tripled her money in 2009. What is her average annual rate of return, over the three years?

3) Tuition and fees at Stetson University totaled $7675 in 1988. They were $31,770 in 2009.

a) What has been the average annual rate of increase?

b) If current trends continue, when will tuition hit $100,000/year?

c) How long will it take tuition to double?

SOLUTIONS:
1a) mean = 3, median = 3      1b) mean = 1821.4, median = 30
2) Geometric mean. Third root of (1.3 * .5 * 3) = 1.249, or 24.9% average annual increase.

3a) Geometric mean. (31770/7675)^(1/21) = 1.070, or 7.0% average increase.
3b) If it keeps growing, using the annual multiplier of 1.070,
      then after n years tuition will be (31770)*(1.07)ⁿ.
      Set this equal to $100,000, and solve for n:

(31770)*(1.07)ⁿ = 100,000

(1.07)ⁿ = 100,000/31770 = 3.1476

n * log(1.07) = log(3.1476)

n = log(3.1476)/log(1.07) = 16.95

So tuition will be $100,000 in just under seventeen years - that is, around 2027.

3c) By similar calculations:

(1.07)ⁿ = 2

n * log(1.07) = log(2)

n = log(2)/log(1.07) = 10.24

Or (by Rule of 72): 72/7.0 = 10.29 years.