190 ◆AUTHORING A PHD
changes or transitions that can occur in the central level of the
data series. Smoothing data is a way of doing this. It essentially
works as follows. You put the actual data numbers you have in
one column, and then next to it you generate a new column of
numbers. Here you substitute for each actual data number a new
number which is an average of that observation and the obser-
vations immediately before and after it. You can do this very
easily on a spreadsheet by writing a formula that will take the
mean of the three observations. For instance, if you have a
series of numbers like 52, 56, 74, 60, 58 then the smoothed
number for the 74 here would be 56 74 60 210, divided
by 3, which is 63. This technique is called mean-smoothing and
it will eliminate ‘normal’ fluctuations in data series. But if you
have some very unusual one-off observations (either high or
low) then they may still push the mean-smoothed figure up or
down a lot. For instance, if we revise the series of numbers
above by changing the 74 to a very unusual 124 we get the
series: 52, 56, 124, 60, 58. Here the mean-smoothed figure
for the 124 will be 80, which still sticks out well above the level
of the surrounding numbers, despite being a solitary unusual
observation.
Median-smoothing works in the same way but this time you
replace an observation with the medianof that observation, the
one before and the one after. Take the series above, 52, 56, 74,
60, 58. The median-smoothed number for the 74 is the middle
one of 56, 60 and 74, which is 60. This technique is much more
powerful than mean-smoothing in screening out one-off,
unusual observations. For instance, if we again replace the 74
by 124 to get the series 52, 56, 124, 60, 58 then the mean-
smoothed figure for the 124 will still be 60, meaning that the
unusual observation has been completely discarded and has no
impact on the median-smoothed numbers. You will need to
repeat the median-smoothing operation a second time, by
median-smoothing your first-smoothed numbers again into a
third column. This is necessary to get to a fully stable smoothed
series, and one that places real enduring changes in the trend
line of your data at the right place. (Median-smoothing a data
series only once may misplace such real changes up or down by
one period, for instance suggesting that a real change which
took place in May of a given year actually occurred in June.