STANDARD ERROR v. STANDARD DEVIATION?

Question

What are the various confidence intervals for 2 Standard Errors v. 2 Standard Deviations from a Linear Regression Trendline or Simple Moving Average?

For instance, if I determine the value that is Two Standard Errors from a 50-day Linear Regression Trendline and I determine the value that is Two Standard Deviations from that same 50-day Linear Regression Trendline, they can't both have a Confidence Interval of 95% if they are dramatically different values; can they? By way of example, in most stock trading software, Standard Error Bands are used most often in combination with a Linear Regression Trendline and Standard Deviation Bands are used most often in conjunction with a Simple Moving Average. Can someone explain why this might be and what the different confidence intervals represent?

Answer 1

Imagine that in 2005, someone sampled the entire population of students in a school--1000 to be exact--and determined that their mean test score was 85, with a standard deviation of 5.

Now you want to do the same thing, but you don't have the time. So, you sample 25 students. Their mean is 76. You sample another 25. Their mean is 85. You sample another 25. Their mean is 80. You do this for 10 different samples. If you add up all 10 samples' means and divide by 10, you will have the "mean of the means". (Let's assume in this case it is 81).

Also, if you were to use the standard deviation formula and use those 10 scores to determine a SD, you would have calculated the standard error. Standard error is the standard deviation of the sample means. Let's say in this case it is 7 points).

The standard error, as you pointed out above, will not always have a 95% confidence interval. Let's assume for a moment that the data found in 2005 did represent a 95% confidence interval. You can calculate whether or not this data does by applying the formula:

95% = Mean (of the means) +/- (z * standard error)
z is 1.96 since that's the z score for a 95% confidence.

So we get:

a) 81 + (1.96 x 7) and b) 81 - (1.96 x 7)

So a 95% confidence interval would be 94.7 - 67.2.

If all we plotted was the error bar, you'd see a student's score at say 81 +/- 7, which doesn't represent 95% confidence.

As far as stocks go:

A linear regression trendline plots a straight line over prices, helping investors make an educated guess about what the price movement will do *in the future*. Since this price hasn't been discovered, the bars will show a standard error.

On the other hand, a simple moving average is (usually) the average of 15 days worth of prices. In this case, 15 days is the entire population, so standard deviation bars are appropriate.

Good luck and hope it helped!

Regards,

Mysstere