Does confidence interval of the mean fully correct for size? Say I have heartrate data at every minute during the day.
I suspect the person was exercising at different points during the day (based on some other data), and now I want to examine my hypothesis.
I mark these periods as "Active" and the others as "Inactive", and calculate the mean and confidence interval of the heart rate. They are significantly different.
Now, there are much more "inactive" points than "active". The confidence interval takes this into account by dividing by the square root of n, but is this enough?
To make this point a bit clearer, lets say most of the day the heart rate was 60, for 30 minutes it was 100 and for 15 more minutes it was 150. Now, I can mark the 150bpm as either "active" (and join them to the 100bpm), or inactive and join them to the 60 bpm. if I mark them as inactive, I would still arrive at the same conclusion since they would hardly affect the mean and sd.
Reproducible example:
Case 1: I WRONGLY classify the 150bpm as inactive (say I had another source of information which recognized this period as inactive):
df <- data_frame(HeartRate = c(rep(60, 300), rep(100, 10), rep(110, 10),
rep(150,10)), IsActive = c(rep(FALSE, 300), rep(TRUE, 10), rep(TRUE, 10), 
rep(FALSE, 10)))

And calculate the mean and CI:
df %>% group_by(IsActive) %>% summarise(MeanHR = mean(HeartRate), 
low = (-1*qnorm(0.975) * sd(HeartRate)/sqrt(length(HeartRate)))/12, 
high = (qnorm(0.975) * sd(HeartRate)/sqrt(length(HeartRate)))/12)

I get:
# A tibble: 2 x 4
  IsActive MeanHR    low  high
  <lgl>     <dbl>  <dbl> <dbl>
1 FALSE      62.9 -0.148 0.148
2 TRUE      105   -0.187 0.187

Case 2: I now classify the 150 bpm as active, and get the same statistical behaviour:
# A tibble: 2 x 4
  IsActive MeanHR    low  high
  <lgl>     <dbl>  <dbl> <dbl>
1 FALSE        60  0     0    
2 TRUE        120 -0.655 0.655

So although I initially classified the 150bpm as inactive, due to the size of the larger group this error gets "swallowed".
Would there be a way to correct for the fact that the 60 bpm appear many times? 
 A: I have used your data files 
transfer.sh/xuf6M/HR.csv
transfer.sh/kW6lb/Steps.csv

and in order to match the different time stamps I took averages over 3 minute intervals.

This can be turned into the following graphs:

You could say then:


*

*No activity: HR of 80 or lower only occur below 5 steps per minute

*Low activity: HR of 80-100 do not occur much in activities with above 20 steps per minute

*Medium activity: HR of 100-140 occur in activities with above 20 steps per minute

*High activity or anomalies: HR above 140 seem to not occur in activities with above 20 steps. These might be other activities than walking that are harder than walking (e.g. running, cycling)


So in this way you calibrate heart rates according to stepsizes. And you should not consider the heart rates above > 150 as inactive just because you did not measure steps during that period. Logically you should have 
$$HR_{\text{no activity}} < HR_{\text{low activity}} < HR_{\text{medium activity}} < HR_{\text{high activity}}$$

In your case using confidence intervals seems not right to me. You could mathematically express something like the average heart rate for a certain activity and express something like a confidence interval for it, but the question is whether it makes sense (aside from your question about the proper categorization of activity and it's influence on the confidence intervals). (1) The heart rates do not follow an ordinary distribution for which you can express the confidence intervals of the mean/average (e.g. using a t-distribution for the mean when the data is Gaussian distributed) (2) The average/mean heart rate may not be the relevant parameter.

Possibly plots like below may help you as well:

Here the points are connected and the lines are a path in time (succeeding points in time are connected). So you can see that the high >140 heart-rates are due to the day 1 and day 2. You can also see that these points are in a loop with some walking or running, so it seems that there has been some mixture of walking and other activities. 
