I misidentified where the error was coming from. Here is my function in its entirety (sorry if some of the lines are obscure and confusing...)
def removeLines(input,CRVAL1,CDELT1): #Masks out the Balmer lines from the spectrum
    #Numbers 4060, 4150, 4300, 4375, 4800, and 4950 obtained from fit_RVs.pro.
    #Other numbers obtained from the Balmer absorption series lines
    
    for i in range(0,len(lineWindows),2):
        left = toIndex(lineWindows[i],CRVAL1,CDELT1)
        right = toIndex(lineWindows[i+1],CRVAL1,CDELT1)
        
        print "left = ", left
        print "right = ", right
        print "20 from right =\n", input[right:right+20]
        print "mean of 20 = ", numpy.mean(input[right:right+20])
        
        #Find the averages on the left and right sides
        left_avg = numpy.mean(input[left-20:left])
        right_avg = numpy.mean(input[right:right+20])   #<--- NOT here
        
        print "right_avg = ", right_avg
        
        #Find the slope between the averages
        slope = (left_avg - right_avg)/(left - right)
        
        #Find the y-intercept of the line conjoining the averages
        bval = ((left_avg - slope*left) + (right_avg - slope*right)) / 2
        
        for j in range(left,right):     #Redefine the data to follow the line conjoining
            input[j] = slope*j + bval   #the sides of the peaks
    left = int(input[0])
    left_avg = int(input[0])
    right = toIndex(lineWindows[0],CRVAL1,CDELT1)
    right_avg = numpy.mean(input[right:right+20])   #<---- THIS IS WHERE IT IS!
    slope = (left_avg - right_avg)/(left - right)
    bval = ((left_avg - slope*left) + (right_avg - slope*right)) / 2
    
    for i in range(left, right):
        input[i] = slope*i + bval
    return input
I have investigated the issue and found the answer, which is posted below (not in this post).
#left  = An index in the data (on the 'left' side)
#right = An index in the data (on the 'right' side)
#input = The data array
print "left = ", left
print "right = ", right
print "20 from right =\n", input[right:right+20]
print "mean of 20 = ", numpy.mean(input[right:right+20])
#Find the averages on the left and right sides
left_avg = numpy.mean(input[left-20:left])
right_avg = numpy.mean(input[right:right+20])
produced the output
left =  1333
right =  1490
20 from right =
[ 0.14138737  0.14085886  0.14038289  0.14045525  0.14078836  0.14083192
  0.14072289  0.14082283  0.14058594  0.13977806  0.13955595  0.13998236
  0.1400764   0.1399636   0.14025062  0.14074247  0.14094831  0.14078569
  0.14001536  0.13895717]
mean of 20 =  0.140395
Traceback (most recent call last):
...
  File "getRVs.py", line 201, in removeLines
    right_avg = numpy.mean(input[right:right+20])
  File "C:\Users\MyName\Anaconda\lib\site-packages\numpy\core\fromnumeric.py", line 2735, in mean
    out=out, keepdims=keepdims)
  File "C:\Users\MyName\Anaconda\lib\site-packages\numpy\core\_methods.py", line 59, in _mean
    warnings.warn("Mean of empty slice.", RuntimeWarning)
RuntimeWarning: Mean of empty slice.
It would appear that numpy.mean runs correctly when I print it, but differently when I assign it to a value. Any feedback would be very appreciated. Thank you for taking the time to read my question.
In short, I am writing a code to handle scientific data and part of the code involves taking the mean of about 20 values.
#left  = An index in the data (on the 'left' side)
#right = An index in the data (on the 'right' side)
#input = The data array
#Find the averages on the left and right sides
left_avg = numpy.mean(input[left-20:left])
right_avg = numpy.mean(input[right:right+20])
This code returns a numpy "Mean of empty slice." warning and annoyingly prints it in my precious output! I decided to try and track down the source of the warning as seen here, for example, so I placed
import warnings
warnings.simplefilter("error")
at the top of my code, which then returned the following snipped Traceback:
  File "getRVs.py", line 201, in removeLines
    right_avg = numpy.mean(input[right:right+20])
  File "C:\Users\MyName\Anaconda\lib\site-packages\numpy\core\fromnumeric.py", line 2735, in mean
    out=out, keepdims=keepdims)
  File "C:\Users\MyName\Anaconda\lib\site-packages\numpy\core\_methods.py", line 59, in _mean
    warnings.warn("Mean of empty slice.", RuntimeWarning)
RuntimeWarning: Mean of empty slice.
I omitted about 2/3 of the Traceback because it moves through about 5 difficult-to-explain functions that do not affect the readability or size of the data.
So I decided to print out the whole operation to see if right_avg really was attempting a numpy.mean of an empty slice... And that's when things got really weird.
I couldn't reproduce your error. Are you using the latest numpy version? However you could suppress the warnings by uding the keyword ignore( see https://docs.python.org/2/library/warnings.html#temporarily-suppressing-warnings)
This error normally means that an empty list was passed to the function.
>>> a = []
>>> import numpy
>>> numpy.mean(a)
/shahlab/pipelines/apps_centos6/Python-2.7.10/lib/python2.7/site-packages/numpy/core/_methods.py:59: RuntimeWarning: Mean of empty slice.
  warnings.warn("Mean of empty slice.", RuntimeWarning)
/shahlab/pipelines/apps_centos6/Python-2.7.10/lib/python2.7/site-packages/numpy/core/_methods.py:71: RuntimeWarning: invalid value encountered in double_scalars
  ret = ret.dtype.type(ret / rcount)
nan
>>> print numpy.mean(a)
nan
>>> import warnings
>>> warnings.simplefilter("ignore")
>>> numpy.mean(a)
nan
>>> a=[ 0.14138737, 0.14085886, 0.14038289, 0.14045525, 0.14078836, 0.14083192, 0.14072289, 0.14082283, 0.14058594, 0.13977806, 0.13955595, 0.13998236, 0.1400764,  0.1399636,  0.14025062, 0.14074247, 0.14094831, 0.14078569, 0.14001536, 0.13895717]
>>> numpy.mean(a)
0.140394615
>>> x = numpy.mean(a)
>>> print x
0.140394615
>>> numpy.__version__
'1.9.2'
Hope that helps.
I misidentified which line of code the error sat on. What I needed to do is write in code for the specific case in which the window (left and right sides) around the center-point being considered in the data is too close to the edge of the data array.
def removeLines(input,CRVAL1,CDELT1): #Masks out the Balmer lines from the spectrum
    
    for i in range(0,len(lineWindows),2):
        left = toIndex(lineWindows[i],CRVAL1,CDELT1)
        right = toIndex(lineWindows[i+1],CRVAL1,CDELT1)
        
        #Find the averages on the left and right sides
        left_avg = numpy.mean(input[left-20:left])
        right_avg = numpy.mean(input[right:right+20])
        
        #Find the slope between the averages
        slope = (left_avg - right_avg)/(left - right)
        
        #Find the y-intercept of the line conjoining the averages
        bval = ((left_avg - slope*left) + (right_avg - slope*right)) / 2
        
        for j in range(left,right):     #Redefine the data to follow the line conjoining
            input[j] = slope*j + bval   #the sides of the peaks
    left = 0
    left_avg = int(input[0])
    
    if toIndex(lineWindows[0],CRVAL1,CDELT1) < 0: right = 0
    else: right = toIndex(lineWindows[0],CRVAL1,CDELT1)
    right_avg = numpy.mean(input[right:right+20])
    slope = (left_avg - right_avg)/(left - right)
    bval = ((left_avg - slope*left) + (right_avg - slope*right)) / 2
    
    for i in range(left, right):
        input[i] = slope*i + bval
    return input
Simply change this
right = toIndex(lineWindows[0],CRVAL1,CDELT1)    #Error occurs where right = -10
right_avg = numpy.mean(input[right:right+20])    #Index of -10? Yeah, right.
to this
if toIndex(lineWindows[0],CRVAL1,CDELT1) < 0: right = 0    #Index 0, much better!
else: right = toIndex(lineWindows[0],CRVAL1,CDELT1)    #Leave it alone if it isn't a problem.
right_avg = numpy.mean(input[right:right+20])
Also, I was wrong about left = int(input[0]), so I changed it to left = 0.
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