Example 3.4 Interpretation of the Standard Deviation: Rules of Thumb

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | DOW.XLS n This file contains monthly closing prices for the Dow Jones Index from January 1947 through January n The monthly returns from the index are also shown starting with February Each return is the monthly percentage change (expressed) as a decimal) in the index. n How well do the rules of thumb work for these data?

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Rules of Thumb n Many data sets follow “rules of thumb”. n Approximately 68% of the observations are within one standard deviation of the mean. n Approximately 95% of the observations are within two standard deviations of the mean. n Approximately 99.7% - almost all - of the observations are within three standard deviations of the mean.

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Index Time Series Plot n A time series plot of the index show that the index has been increasingly fairly steadily over the period. n Whenever a series indicates a clear trend such as the index does, most of the measures we have been discussing are less relevant. n For example, the mean of the index for this period has at most historical interest. We are probably more interested in predicting the future of the Dow, and the historical mean has little relevance for predicting the future.

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Time Series Plot of Dow Closing Index

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Time Series Plot of Dow Returns

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Return Time Series Plot n A time series plot of the returns show no obvious trend over the period. n The measures we have been discussing are relevant in discussing the series of returns, which fluctuate around a stable mean. n We first calculate the mean and standard deviation of the returns by using the Excel functions AVERAGE and STDEV in cells B4 and B5. See the table on the next slide.

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Rules of Thumb for Dow Jones Data

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Returns -- continued n The average return is 0.59% and the standard deviation of about 3.37%. n Therefore, the rules of thumb (if they apply) imply, for example, that about 2/3 of all returns are within the interval 0.59% %, that is from -2.78% to 3.95%. n In order to determine if the rules of thumb apply to these returns, we can use a frequency table.

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Creating the Frequency Table n We first enter the upper limits of the suitable categories in the range A8:A15. n Any categories can be chosen but it is convenient to choose categories in which each breakpoint is one standard deviation higher than the previous one with the open-ended categories on either end are “more than 3 standard deviations from the mean”. n Next we use the FREQUENCY function to fill in column C. “=FREQUENCY(Returns,Bins)”

| 3.2 | 3.3 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Frequency Table continued n Finally, we use the frequencies in column C to calculate the actual percentage of return within k standard deviations of the mean for k=1, k=2 and k=3 and we compare these with the percentages from the rules of thumb. n The agreement between these percentages is not perfect - there are a few more observations within one standard deviation of the mean than the rule of thumb predicts - but in general the rules of thumb work quite well.