EX 4EMEDIAN SMOOTHING

REVISION Smoothing involves replacing the original time

series with another one where most of the variation has been removed, in order to see if there is a secular trend. There are three basic smoothing techniques.

(a) Moving-average smoothing works best with an odd number of points. For a 3-point smooth, one point is lost at either end of the time series.

(b) Moving-average smoothing with an even number of points is a 2-step process. First perform a 4-point moving average, then centre by averaging pairs of the 4-point smooth. For a 4-point centred smooth, two points are lost at each end of the time series.

4-POINT MOVING AVERAGE

Time y-value

4-point average (smoothed value) 4-point average after centring

Calculation Result Calculation Result

2003   6        

           

2004 10        

    (6 + 10 + 14 + 12) ÷ 4 10.5    

2005 14     (10.5 + 11.75) ÷ 2 11.125

    (10 + 14 + 12 + 11) ÷ 4 11.75    

2006 12      (11.75 + 13) ÷ 2 12.375

    (14 + 12 + 11 + 15) ÷ 4 13    

2007 11       (13 + 13.5) ÷ 2 13.25 

    (12 + 11 + 15 + 16) ÷ 4 13.5    

2008 15        

           

2009 16        

MEDIAN SMOOTHING An alternative technique using the

median of each group. Use odd-point median smoothing as it

doesn’t require any calculations. Often id can be done directly on the

graph of a time series. Generally, the effect of median

smoothing is to remove some random fluctuations. It performs poorly on cyclical or seasonal fluctuations-unless the size of the range being used (3, 5, 7..points) is chosen carefully.

MEDIAN SMOOTHING FROM TABLE For a 3-point medians, we look at each

group of three points and choose the middle value.

Progress through the table one point at a time. As with other methods, points will be lost at the beginning and end of the table.

EXAMPLE Perform a 3-point median

smoothing on the data in the table below. The table shows the cost of an airline ticket between Perth and Melbourne over an 8-month period. Construct a time-series plot from the data.Time 1 2 3 4 5 6 7 8

Cost ($)

340

350

320

340 300 330 350 310

MEDIAN SMOOTHING FROM A GRAPH Perform a 3-point median

smoothing on the graph of a time series below. The 1st data points are 12, 18, 16 — so

median = 16.The 2nd data points are 18, 16, 8 — so median = 16.The 3rd data points are 16, 8, 12 — so median = 12.The 4th data points are 8, 12, 16 — so median = 12.The 5th data points are 12, 16, 12 — so median

= 12The 6th data points are 16, 12, 8 — so median = 12.The 7th data points are 12, 8, 10 — so median = 10.The 8th data points are 8, 10, 14 — so median = 10.

CLASSWORK/HOMEWORK Exercise 4E pg 178 Q’s 1-6.