Texas Mathematics Teacher Spring/Summer 2022 - 9

An Odyssey with Time Series: The Masters Golf Tournament
Figure 1 and Table 2 show several interesting features: (i) the winning scores for years 1934 - 1938 are higher as a group
than those for years 1939-1942, (ii) the lowest score for all these years is 279 strokes while 285 is the highest, and (iii) the
total winning score in each of the last three years is constant at 280 strokes. However, much more can be said if a time
series displays periods of relatively constant level with reasonable variation around that level, as this one does. A constant
level is typically described by the data average (or mean), which here is
Average (AVE) = (284 + 282 + 285 + 283 + 285 + 279 + 280 + 280 + 280) / 9 = 282 strokes
Time series variation can oftentimes be accurately described using statistics called moving ranges. A moving range is just
the absolute value of the difference between a data value and its predecessor in a time series. Of course, there will be always
be one fewer moving ranges than data values. The collection of moving ranges for Table 2 are shown in Table 3.
Moving ranges can be construed as " short-term "
variation, but they can be creatively used to
set reasonable limits on the overall, or " longterm " ,
variation seen in a time series. This is
accomplished using time series behavior charts
which provide visual perspectives of both the
overall (long-term) and instantaneous
(short-term) behaviors of a time series. The two
most common time series behavior charts are the
moving range chart and the individuals chart.
Moving Range Chart Construction:
A moving range chart provides a visual perspective on
the short-term variation seen in a time series, and it is
generally constructed as follows:
(a) Plot the moving ranges in a rectangular coordinate
system where the horizontal axis denotes a natural
sequence (1, 2, ...) and the vertical axis represents the
corresponding moving range at that natural sequence
value.
(b) Plot a horizontal line through the Average Moving
Range, or AMR.
(c) Plot a horizontal Upper Limit, UL, for moving ranges
at UL = 3.27 * AMR.
(d) Plot a horizontal Lower Limit, LL = 0 always, since
every moving range must be nonnegative.
(e) A versatile, unbiased estimate of the standard
deviation of the data is always given by
σ = AMR/1.128.
ˆ
The constant 3.27 is known as an unbiasing constant, and
it transforms the average moving range, by multiplication,
into an appropriate unbiased, upper three-standarddeviation
limit only for the moving range chart. Likewise,
the constant 1.128 is an unbiasing constant, however
it transforms the average moving range, by division,
into a robust, or versatile, estimate of the true standard
deviation, σ, of the data. Because the smallest moving
range attainable in any context is zero, points plotted in
a moving range chart cannot traverse this value, LL, by
design. So, the lower limit of zero is a natural boundary.
See Montgomery (2013) for greater detail concerning
unbiasing constants.
www.tctmonline.org
|282 - 284| = 2
|285 - 282| = 3
|283 - 285| = 2
|285 - 283| = 2
|279 - 285| = 6
|280 - 279| = 1
|280 - 280| = 0
|280 - 280| = 0
sum = 16
( |1935 value - 1934 value| )
( |1936 value - 1935 value| )
( |1937 value - 1936 value| )
( |1938 value - 1937 value| )
( |1939 value - 1938 value| )
( |1940 value - 1939 value| )
( |1941 value - 1940 value| )
( |1942 value - 1941 value| )
Average Moving Range
AMR = 16/8 = 2 strokes
Table 3. Moving ranges for Table 2 data
Using the moving ranges in Table 3 associated with the
winning scores for the first nine years of The Masters
Tournament, compute (i) AMR = 2.0 strokes, (ii) LL = 0,
and (iii) UL= 3.27 * 2.0 = 6.54 strokes. Figure 3 shows the
plotted results using all eight moving ranges.
The Moving Range Chart always shows " short-term " ,
or adjacent-point variation. Figure 2 has two striking
features. First, the fifth moving range value equals six
(See also Table 3), which is just barely below the upper
limit. This phenomenon is generally associated with a
noticeable break in the time series.
Figure 2. Moving range chart for Table 2 data
Spring/Summer 2022 | 9
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Texas Mathematics Teacher Spring/Summer 2022

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