Volleyball Coaching Life-Assessing Service Quality By Using the Expected Value

I spent some time thinking about the subject of making service errors and its correlation with serving aggressively. Just taking my volleyball brain out for a spin with my math background.

The topic of aggressive serving has been front and center amongst coaches and fans of volleyball in recent years. The concept under scrutiny is the idea that if the server is to reach an acceptable level of serve aggression, they must be untethered from the fear of making a service error. This concept became accepted first in the men’s game because it is at a point in its evolution where an unaggressive serve will inevitably result as an easy counterattack. One only need to watch the men’s competition in the Olympics or in the NCAA men’s competition to understand the verity of the premise.

The thinking is to not only to mitigate the fear by de-emphasizing the service error, but the men’s team coaches also had deliberately made the number of a team’s number of service errors a measured indicator of how a team’s serving aggression.

John Speraw, the USA men’s NT coach at the time, mandated a minimum number of service errors for his teams to ensure that they are attacking the service game aggressively so that they attain a level of aggressive serving that he feels is necessary to be competitive internationally. He intends the rule to encourage the servers to not hold back their swings, to obviate the natural tendency to avoid the service error by “just getting the serves in”. It gives the server’s carte blanche to make errors without concern. The idea of having a pre-set minimum limit on the number of service errors to ensure aggressive serving created discussion amongst coaches. Those who coached the younger and less experienced players were aghast, while those who coached highly skilled and experienced players saw a reason for the mandate, but most are still not completely sold on the idea because the idea of committing errors is contrary to the coaching mindset. I had poked around in the idea qualitatively here. https://polymathtobe.blogspot.com/2022/01/volleyball-coaching-life-hard-limits-on.html

Earlier in the year, people watching the Olympics also did not understand why the best of the best, the Olympic level competitors, were missing serves at the rate that they were. More recently, the commentators during the Division I NCAA women’s tournament were troubled that Kevin Hambly, Stanford’s head coach, stated: “we have not found through analysis a number of service errors that are too much, in fact we have found missing more is better but… what I really want is freedom from the service line, psychologically I don’t want them thinking about missing, I want them to be aggressive.” (Hambly, 2025) Which contradicts the common coaching paean to NOT MISS SERVES. Ever.

Hambly had generalized Speraw’ edict of making a minimum number of service errors to ensure aggressive serving to ignoring service error as a metric to focus on. In view of what had happened in the NCAA match between Northern Iowa and Louisville, where Louisville was able to recover from being down in the fifth set because of six straight missed serves by Northern Iowa, the discussion became even more heated, many people citing the existence of streaks, fully embracing the Gambler’s fallacy: if an event (whose occurrences are independent and identically distributed) has occurred less frequently than expected, it is more likely to happen again in the future (or vice versa). (From Wikipedia). The fact is that each serve is an independent event.

However, I noticed that these very commentators who complain about the service errors, particularly those who have played the game at a very high level, are also adamant about swinging aggressively at each set to win. They incessantly proselytized about the importance of aggressively letting it rip when attacking a set. They equate attacking the set as a necessary condition to play to win and equate deliberately tipping and rolling the set with the sufficient case to claim playing to not lose. All sound advice, yet it is interesting that they reverse themselves when discussing serving aggressiveness. Two similar situations with two very different conclusions. Why is that?

We, as coaches and players, have been taught that the serve has a unique distinction out of all the skill in volleyball: first, the serve is the inception of play, and second, it is the only skill performed when the ball is not moving; for the rest of the skills in volleyball, the ball is played while in motion and on the rebound, with both the ball and the players moving. Since the serve is the exception as compared with the rest of the skills, players and coaches automatically infer that serving is the only skill where the players can and should be in complete control, from that initial inference, the coaches and players also generalized that belief to infer that the server need to use that control the best way possible, which is to not make a service error or squander a serving opportunity. A contrary interpretation of the server having control over the ball is that the servers need to take advantage of that control and serve as aggressively as possible to gain an advantage over the opponent by making sure that the served ball create maximum disruption and chaos on the opponent’s side of the net, to force the receiving team to err. This is where the seeming contradiction comes from: attacking the opponent so that they are playing on their heels while also not allowing the server to be aggressive while taking risks to make mistakes.

Scoring in volleyball is done by termination scoring, as with most sports: a point is awarded at each dead ball, replays and penalties notwithstanding. A summary of these termination points are listed here. https://thecuriouspolymath.substack.com/p/stats-for-spikes-termination-scoring

There are four categories of termination points. The officials determine the other points through sanctions. There are our earned points, our errors — which are gift points to the opponents, the opponents earned points, and the opponents’ errors — which are gift points to our team.

The points earned are:

· Kills,

· Blocks, and

· Service aces.

The error points are:

· Service errors,

· Passing errors,

· Ball handling errors,

· Attacking errors,

· Blocking errors, and

· Digging errors.

Feeling the way I feel about setting a hard limit on a free flowing game and the infinite number of variety of situations that could happen and the attendant changes in context, I thought that rather than calculating a minimum number of services errors to serve as a metric for service aggression, it would be better, and probably easier to calculate a metric to determine the value of service effectiveness.

The first thought was to compare the total number of service errors versus aces, but that was too simplistic as the ratio does not take into account the amount of chaos aggressive serving wreaks on the receiving team and the indirect disruptions on the receiving team as a consequence of the aggressive service. The after effect of the aggressive serve manifests itself in receiving team errors: passing errors, ball handling errors, and attacking errors; all can result in serving team points.

The nature of the continuity of action in volleyball can only be modelled statistically by using the statistical model known as the Markov Chains. This means that the statistical model for creating complete statistical models tends to need exponentially more specific situational data because of chain of action resulting from just the serve. I tried to qualitatively explain how the Markov chain modelling can be laid out qualitatively. https://polymathtobe.blogspot.com/2021/03/stats-for-spikes-markov-chains.html. This continuity of the action needs to be kept in mind when thinking about the metric for the serve because the results of the serve does not necessarily result in a termination point, i.e. the rally can continue without a point being directly scored as a consequence of the serve, but the effect of the aggressive will carry through to the receiving team attack from the serve, the serving team defensive response to the attack, and the serving team counterattack stemming from the receiving team attack from serve.

The question is: how to assess the relative success of aggressive serving while also taking into account the downstream effects of an aggressive serve, even when it does not directly result in an ace or a service error for the serving team?

The expected value in probability (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of an weighted average. Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. It is important to understand that the expected value is NOT the value you would “expect” to get, instead, it is summary of all the data points on the team’s scoring from the serve, weighted by the team’s history in certain situations. I thought it might be useful to assess the effectiveness of the serve by using the termination point statistics that many teams record as a matter of course, for both teams and for individuals. The calculations for the expected value is also relatively simple, given that the match statistics are well kept and accurate, although infinitesimal accuracy and precision is not required as their will be other sources of noise will be prevalent. “Perfection is the enemy progress.” “Perfection is the enemy of the good.” Both quotes apply.

The Expected Value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes (The value of a serve.) In the case of a continuum of possible outcomes, the expectation is defined by integration or summation. (We will use the summation.)

Consider a random variable X with a finite list x₁, ..., x_k of possible outcomes, each of which (respectively) has probability p₁, ..., p_k of occurring. The expectation of X is defined as

E[X]=x₁p₁+x₂p₂+x₃p₃+…+x_kp_k

it is natural to interpret E[X] as a weighted average of the x_i values, with weights given by their probabilities p_i.

We can investigate the expected value of the result of the serve by listing all the possible values that are the most basic results from the serve:

· x₁=+1, Points for the serving team, -1 for the receiving team:

o Ace

o Receiving team passing error.

o Receiving team ball handling error

o Receiving team hitting error.

o Receiving team attack, blocked ball.

· x₂=-1, point for the serving team, +1 for the receiving team:

o Serving team service error.

o Receiving team kill.

o Serving team blocking error.

o Serving team digging error.

· x₃=0, neutral. Opponent sends the ball over to server’s side

o Dug ball from Opponent attack. (Includes free ball and down balls)

The first primitive form of the service expected value could be calculated by just taking into account the serving team statistics for earning a point, losing a point, or having no termination point scored. These three probabilities can be roughly obtained from total game statistics:

p₁=Total points gained on serve/Total number of serves.

p₂=Total points lost on serve/Total number of serves.

p₃=Total number of events where no points are scored on serve/Total number of serves.

Where p₁ is the probability for the serving team to gain a point, p₂ is the probability for the serving team to lose a point, and p₃ is the probability for neither team gains a point, or the situation where the receiving team is unable to get a kill after serve receive, which results in the receiving team sends the ball back over to the serving team for the continuation of the rally. It is important to remember that p₁ + ⋅⋅⋅ + p_k = 1 must necessarily be true or else the metric is meaningless. The expected value can also be calculated either for the team or for each server. This simple calculation assumes that the difference in the probabilities due to factors like: the opponents, opponent rotation, and variation in personnel, and any other mitigating factors are ignored.

E[X]=(+1)p₁+(-1)p₂+(0)p₃

The main problem with calculating probability is in obtaining enough data points to properly represent the probability of accurately depicting the team tendencies. It is also difficult to obtain a good estimate of the probabilities at the beginning of the season, although the statistics for the previous season can be used after subtracting out the statistics for players that have left the program. This problem is not new because the amount of data that is needed for obtaining adequately accurate depiction of the team tendencies is always far greater than the amount of data that can be easily collected. We just need to hold our noses and keep the problem regarding the amount of samples in the back of our minds, fully knowing the uncertainties associated with the probabilities. This calculation just gives the coaches and players a gauge to assess the serving game, a measure of the quality of the service game. The measure can also be used to compare the service aggression of each team as the coaches and players increase their aggression over time.

What the Expected Value for the service results gives is an indicator of the service game quality for a team or a player. A positive Expected Value for the service means that the serve should result in a positive outcome for the serving team, a negative Expected Value for the service means that the serve should result in a negative outcome for the serving team, and an Expected Value hovering around zero means that the serve will more likely result in a tossup, which indicates that there is no serving advantage.

This measure can also allow the coach to experiment with throttling up the aggressiveness on the serves and see where that takes the Expected Value for the serve. The Expected Value after the adjustment will take time to settle into a somewhat steady state value because the service aces and service error probabilities will increase, commensurate with the number of aces and opponent errors.

If the expected value is close to zero or negative, the natural next step for a coach and player is to take a deeper dive into the granularities of the component results of probabilities to see which of the component probabilities are causing the skew.

The probability p₁ can be further broken down into their component probabilities of: aces, receiving team passing error, receiving team ball handling error, receiving team hitting error, and blocked ball from a receiving team attack. These component probabilities should add up to p₁. The practice of breaking down into the component probabilities might give better indication as to where the points come from for the serving team and where they need more work. Equivalently, p₂ can be broken down into: serving team service error, receiving team kill, serving team blocking error, and serving team digging error. Generally, coaches and players will know where the problems are in the team play but teasing out the probabilities will give some quantitative sense to the instincts.

Note that the sum of the component probabilities discussed above should sum to p₁ and p₂, so there is no reason to re-calculate the Expected Value.

There is also a way to extend the Expected Value into the next step in the Markov Chain by converting some of the x₃=0 result to either a +1 or -1. Some of the x₃=0 can be converted to x₃=+1 if the serving team scores after receiving a defended free ball, down ball, or a roll shot from the receiving team after the initial pass. Similarly, some of the x₃=0 can be converted to x₃=-1, if the serving team makes an error on the counterattack. The probabilities can be calculated accordingly. This takes the frame of reference slightly past the serve and the immediate result of the serve. There will still be x₃=0 results, but probability will be smaller.

The process of extending the chain could continue endlessly, but the effects of a serve on the action should decrease exponentially after the initial serve and pass and extending it beyond that point would not add any more insight, the amount of data would also be decreasing exponentially.

I have not applied this idea, but it might be interesting to apply this to a team with a good sample space of data to see how this works out. It would be interesting to see whether the Expected Value metric for the serving is useful as a metric to assess a service quality or whether it can help coaches and players better determine how service aggression affects their team’s ability to score positive points from service.

I would love to hear about anyone applying this to their team stats.

References

Hambly, K. (2025, January 15). By email: Missed serve statistics and its effect on scoring. (P. Wung, Interviewer)