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To compute variance of a large measurement sequence data, recursive timevariance formula (4) below give more efficient computation and data storage. You don't need to keep all the measurement data. All you need is to compute the current variance _{ } is the current measurement data _{ } , current timeaverage _{ } and the previous variance _{ } . The formula is as follow
_{ } (4)
Click here to see the proof of the formula above
For time _{ } , it is defined that _{ } . Thus, the computation using equation (4) starts at _{ } .
Using the previous measurement data of 4, 6, 12, 9 we can get exactly the same result of variance as the usual computation method, but in more efficient way. Table below show the example of computation using the recursive timevariance formula above.
Time ( _{ } ) 
Measurement ( _{ } ) 
Average ( _{ } ) 
Variance ( _{ } ) 
1 
_{ } 
_{ } 
_{ } (by definition). 
2 
_{ } 
_{ } 
_{ } 
3 
_{ } 
_{ } 
_{ } 
4 
_{ } 
_{ } 
_{ } 
To give you better understanding about recursive mean and variance, I made a simple program below. You can try with your own data or test it with the example above. To use the program you need to set JavaScript enabled in your browser and you need to type the input data one at a time (I include 4 as an initial current input data), then press "Input" button to compute the time average and time variance. Alternatively, click "Random Input" button repeatedly to create simulated data. You need at least two data points to plot the chart. To restart with an empty data, click "Reset" button. If you use Internet Explorer, the chart may not work properly to show the effect of variance, please use FireFox or Google Chrome instead.
If you click Random input for about more than say 25 points, you will notice that not only the time average is moving and varies over time, even the time variance is also fluctuated. Compare this result with the constant average and constant variance of traditional mean and average .
The next section we will do something reverse: to get the data from the statistics
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These tutorial is copyrighted .
Preferable reference for this tutorial is
Teknomo, Kardi. (2006) Recursive Average and Variance.
http://people.revoledu.com/kardi/tutorial/RecursiveStatistic/index.html