Calculating RMSE Values

I'm helping a friend with a paper on field data collection using smart phone apps. He wants me to perform some accuracy comparisons using internal phone gps, external bluetooth gps (consumer grade), external mapping grade bluetooth gps, all the way up to my survey-grade Trimble R2. Using my R2, I plan on establishing a bench mark at my office using a 6-hour static observation and process through Opus. Then with each device, I will connect to my smart phone via bluetooth and use the Esri Collector app to collect a streaming line feature at 1s intervals for 30-minutes each. I'll convert the line vertices to points and get state plane coordinates for each point. Then I'll bring the data in Excel to calculate RMSE values compared to the benchmark. Here's where I need some help. What is the proper way to calculate RMSE (horizontal only)? Here's one way:

1. Calculate north error for each point (measured_n - benchmark_n)
2. Calculate east error for each point (measured_e - benchmark_e)
3. Calculate horizontal error for each point (SQRT(x^2+y^2))
4. Square the horizontal error for each point
5. Calculate the average of the squared errors
6. Calculate the square root of the average of the squared errors to obtain the RMSE error

Here's another way I saw:

1. Calculate north and east error as above
2. Calculate standard deviation for north error
3. Calculate standard deviation for east error
4. Calculate the square root of the sum of the squared standard deviations to obtain RMS

Which way is correct?