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### 10.37 Random Walk Index

The random walk index (RWI) by E. Michael Poulos is a measure of how much price ranges over N days differ from what would be expected by a random walk (randomly going up and down). A bigger than expected range suggests a trend.

The index is in two parts, an RWI high which looks at upward movement and an RWI low for downward movement. In Chart RWI high is shown in green, and RWI low in red. The RWI high looks at terms like

```High[today] - Low[K]     1
-------------------- * ------
Average TR [K]      sqrt(K)
```

which is the move from the low K days ago up to today’s high, scaled by an average of the true range (TR, see True Range). Such terms are calculated for each number of days 2, 3, etc, up to the given RWI parameter N, and the maximum is the RWI. The first term for instance is today’s high less yesterday’s low, compared to a two-day average of the true range (yesterday’s true range and the day before’s). RWI low is similar, but using High[K] - Low[today] for the movement down from past high to today’s low.

The factor sqrt(K) compares the movement to a random walk. If a random walk has a 50% chance of going up by one, or a 50% chance of going down by one, then it can be shown that on average the distance travelled after K steps is sqrt(K). So the formula compares observed distance in average day’s steps compared to the sqrt(K) steps which would be the expected move if it were random. Thus 1 is when movement is apparently random, and higher or lower if some apparently non-random trend or lack of trend (respectively) appears to be present.

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