DS350 - Quantitative Methods

Lecture Review #2 - Normal Distribution

REVIEW QUESTIONS:

1) What is the normal distribution? What are some examples of quantities that follow the normal distribution?

2) For a normal distribution, what percentage of the observations will fall within one standard deviation of the mean? Within two standard deviations? Within three standard deviations?

COMPUTATIONAL EXERCISES:

1) Scores on each component of the SAT test are calibrated to be normally distributed with a mean of 500 and a standard deviation of 100.

a) What percentage of scores on the SAT-Verbal are between 500 and 620?

b) What percentage of scores on the SAT-Math are between 450 and 550?

c) What percentage of scores on the SAT-Verbal are between 200 and 800?

d) What score do you need, to be in the top 10% on the SAT-Math?

2) Daily stock market returns are approximately normally distributed. (It’s not an exact fit – but it’s pretty close.) Historically, the average daily return on the Dow Jones Industrial Average has been 0.02526%. That’s pretty small – less than three hundredths of a percent per day. (Of course, after a lot of days even a small percentage increase adds up significantly.) The standard deviation of these returns is much larger, however, amounting to 1.162% per day. (These figures are based upon real data, from the inception of the Dow in 1928 through the end of 2009.) Use the normal distribution to find the following quantities.

a) What percentage of all days will the stock market gain in value? (That is, what is the probability that the Dow will have a daily return greater than 0?)

b) What is the chance that the stock market (as measured by the Dow) will decline in value by 2% or more, on a given day?

c) What is the chance that the Dow will decline in value by 5% or more, on a given day?  (This is "off the chart" so the answer is "approximately zero".  But use a spreadsheet to find a more exact value.)

c) The stock market is open approximately 250 days a year. (Weekends and holidays account for the remaining days.) Thus, the very best day of the year will happen with probability 1-in-250, or 0.004 (i.e., 0.4%). What returns can we expect, on the best 0.4% of the days in the stock market?

SOLUTIONS:
1a) 0.3849     1b) 0.3930      1c) virtually all     1d) 628
2a) 0.5080     2b) 0.0409      1c) 0.00000076     1d) 2.326%

NOTE on 2c: In actual fact, the Dow has experienced drops of 5% or more 70 times out of 20403 market days since 1928. While that’s pretty rare – only about one-third of one percent of the time – it’s still not the "approximately 0" that the normal distribution would predict. The normal distribution is a reasonably good predictor of "normal" market behavior, but tends to underestimate the frequency of extremes (both high and low). Stock market returns are actually somewhat heavier-tailed than the normal distribution would indicate.