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What is a weighted moving average and how is it calculated?

Weighted moving average explanation with chart examples and formula

Weighted average of world population from 1982 to 2010

You can see the world population data below together with the alarming growth rate. Below it is the 4 period weighted moving average of it.

 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 World population (Billions) 4.59 4.75 4.92 5.09 5.27 5.43 5.6 5.76 5.92 6.07 6.23 6.38 6.53 6.68 6.84 Four period Weighted Average - - - 4.92 5.09 5.26 5.43 5.60 5.76 5.92 6.07 6.23 6.38 6.53 6.68

The colour coding is used to show how the calculations are made.  The first four periods are 4.59, 4.75, 4.92, 5.09 now we will look at the calculation.

It is more complex than the calculation of the simple moving average as it "weights" more on the most recent data and "fades" the older data.

The calculation is shown below.

Period 1 (4.59 x 1 ) + Period 2 (4.75 x 2) + Period 3 (4.92 x 3) + Period 4 ( 5.09 x 4)  = 49.21 then we divide this by the sum of the data x multiplier. Which is 1+2+3+4 = 10 which gives us 4.92 as a four period weighted average.

So to explain what is happening I will start at the beginning. The first period used is always = to itself which is the same as multiplied by 1.

The second period is multiplied by 2

The third period is  multiplied by 3

The fourth period is multiplied by 4.

Once we have them multiplied up, we add all the multiplier used up together thus 1 + 2 +3 +4 = 10. So the number 10 will be our divisor which we use to divide the total of Period1 x 1 + Period2 x 2 + Period 3 x 3 + Period 4 x 4.

The correct mathematic statement for this would be ( (4.59 x 1 ) + (4.75 x 2) + (4.92 x 3)  ( 5.09 x 4)) / 10

Please note there are double brackets around the formula above.  Basically this means you will do the small calculations first and then the final division by 10 is the last step.

The next stage in creating a weighted moving average is to roll forward to the right by 1 number. So for the next value we will use Period  2 ,3, 4, 5, 6

Key points to remember

• Weighted Moving averages produce a faster response than a simple moving average

• Weighted average give more weight to recent data and less weight to older data.

• Weighted averages need 4 days to compute the first value ( if a four day average is required )

• A weighted moving average will usually be more volatile than a simple moving average as it is weighted towards recent data.

In the next example we shall use a 6 period weighted moving average.

Weighted average of USA unemployment figures from 1998 to 2012

 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 USA unemployment (Percent) 4.50 4.20 3.90 4.6 5.8 6.2 5.4 5 4.2 4.4 5.8 9.5 9.7 9 8.2 6 Period Weighted Average - - - - - 5.20 5.35 5.35 5.08 4.85 5.03 6.27 7.40 8.14 8.45

As before we are going to start by multiplying period 1 by 1 and then move across from left to right adding 1 to the multiplier in each step. Thus...

Period 1 (4.50 x 1 ) + Period 2 (4.20 x 2) + Period 3 (3.90 x 3) + Period 4 (4.6 x 4) + Period 5 ( 5.8 x 5 ) + Period 6 ( 6.2 x 6 )  = 109.2  then we divide this by the summation of all multipliers. Which is 1+2+3+4+5+6 = 21 which gives us 5.20 as a six  period weighted average.

The calculation is then shifted one data point to the right so for the 2nd value we use the following...remembering that the first value is multiplied by 1.

Period 2 (4.20 x 1 ) + Period 3 (3.90 x 2 ) + Period 4 (4.6 x 3 ) + Period 5 ( 5.8 x 4 ) + Period 6 ( 6.2 x 5 ) +Period 7( 5.4 x 6) = 112.40

Again we divide this by he total of all the multipliers which is 1+2+3+4+5+6 = 21 which gives us 112.40 divided by 21 = 5.35.

• The 6 day weighted average is clearly making a smoother indication of the data price.

• Note the delay (LAG) in the response time of the yellow average line. It turns up on 2 periods later than the underlying data price

• Moving averages can be used to smooth erratic data so as to be clearer to understand and interpret.

• The more days ( periods ) used in the calculation the smoother they will become, and the more lag they will have.

Mathematical symbol for weighted average

Other types of moving average and formulas

 Exponential moving average Triangular moving average Hull moving average

• A very advanced type of moving average is shown below. PLA Dynamical has over 1000 lines of computer code used to generate its output

PLA Dynamical is the fastest moving average on the planet ( If overshoot = True and Speed = 100)

• PLA Dynamical shown below has a faster response time than other moving averages due to its complex algorithms

• The formula has over 1000 lines of computer code to generate it.

PLA Dynamical response times compared to standard type MA type filters Tested at length 30

 Filter Length Speed Overshoot Response time to new price range PLA Dynamical 30 100 True 3  Bars PLA Dynamical 30 100 False 18 Bars Weighted MA 30 - - 24 Bars Triangular MA 30 - - 25 Bars Exponential MA 30 - - 43 Bars Filter Length Speed Overshoot Response time to 1 standard deviation from mid point PLA Dynamical 30 100 True 19 Bars PLA Dynamical 30 100 False 23 Bars Weighted MA 30 - - 25 Bars Triangular MA 30 - - 27 Bars Exponential MA 30 - - 61 Bars

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