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Towards the end of part 1 of "Harrington on Hold'em" in the section titled "A Sample Hand," Harrington writes the following:

I thought I was one of the better players remaining at the table, so I wanted to reduce, rather than increase, my volatility on the hand. (Volatility is a mathematician's word for the size of the money swing on the hand.) A weaker player in the same situation should be looking to increase volatility; hence he would definitely want to throw in a bet.

What exactly does "volatility on hand" mean? How do you calculate it? And why should a strong player attempt to reduce it, and a weak player attempt to increase it?

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In this instance, it looks like the author is using the term "volatility" as a way of describing "variance" (or as a replacement for the word). A short definition is included in your quote, but another way I would put it is "the amount in which the outcome in a particular hand could vary". The calculation comes from statistics, I'll link the wikipedia page (https://en.wikipedia.org/wiki/Variance) but there are lots of other articles on google that you can check out for calculations relating specifically to poker.

To me, it is a bit more important to understand the logic behind variance rather than outright calculate it (especially at the table) so I'll give a few examples.

High Variance Hand: you are on the button with 7s8s and MP raises 3x the BB. You 3-bet 3.5x the MP raise and MP calls. Flop comes 5h6cAd and MP checks to you. You C-bet half the pot and MP calls.

Low Variance Hand: Same situation as above, you are on the button with 7s8s and MP raises 3x the BB you call and the blinds call behind you. Flop comes the same, 5h6cAd and MP checks to you and you check, action is on the blinds.

Without going through all of the action in these hands, there are some big differences between the high variance and the low variance hands that directly relate to the way that you played the hand.

  • There are only 2 players in the high variance hand because your 3-bet discouraged the blinds from calling (not a guarantee they will call but more likely if you do not 3-bet). The low variance hand is a 4 player hand.
  • The pot is much bigger in the high variance hand before you act on the flop (22.5BB) vs the low variance hand (12BB)

A hand that involves less players is ideal for weaker players simply because there is less to keep track of, reducing the possibility for errors in decision making. Strong players can be confident that they will make the correct decision or possibly outplay their opponents depending on the action. Strong players can take advantage of heads up play as well, but it is especially important for weaker players to focus on heads up pots.

Bigger pots are one of the key points on variance, more money in the pot means that you will be increasing the amount that you win or lose once the hand is complete. It is a powerful tool for weaker players because playing this way forces the tough decisions on the other players. This does not mean to 3-bet every suited connector or to spew off chips no matter what they have, but it does mean that weaker players should lean towards betting and increasing the size of the pot when they have a significant amount of equity. This leaves the outcome more in the hands of the other player(s) in the hand, and depending on the player they have a decent chance of making a big mistake and awarding a large pot without the weaker player getting an opportunity to make mistakes.

Strong players want to reduce variance because they will rarely make mistakes. Stronger players would rather play lots of smaller pots that they have a 60% chance of winning, rather than put their whole stack on the line with 60% equity in one big pot. This will prevent them from getting stacked when they end up losing a hand that they had the advantage in.

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Your opponent is bad, and is playing a single heads up match against you for $1000, then will go home. In the first hand your opponent open shoves and flips over his hand. You look at your hand and calculate you have a 51% to win the tournament if you call. Should you?

The quick answer is yes, calling is clearly +EV. However, it is probably more +EV to fold. The chance of you winning the tournament given how good you are (and how bad your opponent is) is far greater than 51%. You want to decrease variance because you are the better player.

What if you are getting 49% to win but are significantly the worse player? The quick answer is to fold, but it is probably your best option to call. Right now you can win the tournament 49% of the time! Given your skill difference, you will probably only win 30% of the time. (Also, why are you playing when you admit you are the fish ;) ).

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