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Standard Deviation Forumla For Rostered and Asteriked Player

Rostered and Asterisked Players for the Westside Tennis League

(This document was drafted in the early years of the league and addresses the issue of weighting players performance when determining if a player should be required to play up a line or division. The practice of asterisking players in no longer done, players are only required to move up a division if they have achieved a win rate of 85% in the highest line in more than half the matches played in a season.)

• Introduction
The following is a definition for deciding whether a player should be:
• Asterisked
• Moved up a line
• Have asterisked removed
• Moved down a line
The impetus of this definition is to replace an intuitive somewhat arbitrary approach with a statistically sound approach. The goal is to provide a statistically based set of rules for evaluating players’ performances that is fair and symmetrical.

This definition is based upon standard statistical analysis, in particular the Standard Deviation. It is focused on the performance of individual players and not team performance since the purpose of this proposal is to address Rostered and Asterisked lines of individual players.

The idea is that tennis naturally lends itself to statistical analysis because of the nature of the game and its scoring. For example:
• Each player either wins or loses.
• On any given line, it is expected that on average a player should win 50% of their games
• Players that play up a line are expected to win less than the average and players that play down a line are expected to win more than the average
Keeping these things in mind, if a player wins a high enough percentage of their games then they are really playing at a higher level and should either be asterisked at their line or moved up.
Conversely, if a player is losing a high enough percentage of their games then they should either lose their asterisk or be moved down a line.
Also keep in mind that this statistical analysis applies across divisions.
• Statistical Analysis
There are many aspects to statistics but there are a few that are particularly useful for measuring and predicting performance.
• The Mean
One statistical measure is the “mean” or average. In tennis terms, this is the average number of wins a player has. Since in every match there is a winner and a loser, on average a player is expected to win 50% of their games.
• Standard distribution
• Overview
Another statistical measure is the Standard Deviation. This measure is a measure of prediction. That is, it statistically predicts levels of performance using a Normal Distribution, also known as a Bell Curve.
The Standard Deviation predicts what the expected deviation from the mean of the Normal Distribution is. That is, it predicts for a particular deviation (e.g., one standard deviation), what percentage of values should be above the mean at that deviation. The chart below shows the percentages for each standard deviation. The middle of the curve where s=0 is the mean.

The standard deviation also implies a differentiation of levels. Measurements at a standard deviation of +1 are a level above the norm and a standard deviation of –1 is a level below the norm.

A classical example is the analysis of student grades. Given that a test is fair, the distribution of grades will follow a bell shaped curve (i.e., a normal distribution). The average grade will be a C, a standard deviation of +1 will be a B, and a standard deviation of –1 would be a D. At the extremes, are the A’s and F’s at standard deviations of +2 and –2, respectively. Fractional deviations account for grades with pluses and minuses (e.g., C+, B-, etc.).
In this example, please note that the grades are typically based on a range of 50-100. That is a C is approximately a 75, a B (+1 standard deviation) is an 85, and a D (-1 standard deviation) is a 65. Teachers don’t necessarily strictly follow the Normal distribution percentages and they use an adjusted scale.
• Half Deviations
A half deviation from a score implies a range of expected deviation around that score. For example, there is an expected percentage of scores in the range of B- to B+. These grades might not strictly fall on the half deviations but they reflect the idea.
• Percentages
A standard deviation of 0.5 predicts that approximately 19% of the values will be above the mean and approximately 19% of the measurements will be below the mean. The standard deviation is symmetrical. The absolute values of the distribution are the same in either direction from the mean.
In terms of tennis wins and losses, if a player is winning between 41% and 69% of her matches, then that is the expected range at that particular line. If she is playing outside of this range then there should be an adjustment to indicate that a player is playing outside of the norm.
A standard deviation of 1.0 predicts that approximately 34% of the values will be above the mean and approximately 34% of the measurements will be below the mean. In terms of tennis wins and losses, if player is winning more than 84% of her matches then she is playing at the next level up. Conversely, if a playing is only winning 16% of her matches then she is playing at a level below.
• Adjustments
Statistically, if you are more than 0.5 standard deviations from the norm, then you are outside of the expected line of play and there should be an adjustment made.
• Adjustment Levels
In tennis terms, adjustments are made in one of the following ways:
• A player is “asterisked”
• A player is moved up a line
• A player is asterisked at the next line up
• A player has their asterisk taken away
• A player is moved down a line
• A player is asterisked at two lines down
• Adjustment Calculations
As previously mentioned, a player that is within 0.5 standard deviations from the norm is playing within the expected range.
If a player is playing at the higher line than the normal range then they are playing at a line that is superior to the mean and should be marked as so. If a player is playing so superior to the average such that they are at least a whole standard deviation from the norm then they should be forced to move up a line.
Since the normal distribution is symmetrical and there should be fairness in the adjustments, adjustments should also be made when a player performs below the expected normal range.
• Levels of Adjustments
A fair adjustment for lines and for being asterisked is as follows:
• Asterisk System
In the tennis league, a player performing at a level consistently above the average is marked as “asterisked”. In terms of standard deviations this would be in the range of greater than 0.5 and less than 1.0.
Symmetrically and fairly, if a player performs at a level less than –0.5 and greater than –1.0 then their asterisk should be taken away.
• Change of Level
If a player is playing at a level that is even more superior to the average than an asterisked player then they should be forced to move up a line. In terms of standard deviations, a player who is performing at a level grater than +1.0 standard deviation should be moved up one line.
Symmetrically and fairly, if a player performs at a level less than –1.0 standard deviation, then they should be moved down one line.
• Extreme adjustments
In cases where a player is performing way outside of the norm then a more extreme adjustment must be made. In these cases, if a player is playing at a level greater than +1.5 standard deviations (winning more than 93% of matches) from the norm then that player will become asterisked at the next line up.
In the case where a player is performing at a level less than –1.5 deviations (winning less than 7% of matches), then for symmetry, the player should be moved down two lines and be asterisked at that line.
• Adjustments for Across Line Play
When a test is given and the grade curve is calculated, it is based on a single test that all students take. Therefore, comparing scores is on an even ground. If some students took a harder test or some students took an easier test, then adjustments would have to be made when computing the scores. This is analogous to a student taking an AP exam, having it weighted, and then their GPA getting adjusted accordingly. This is why there are GPAs that are above 4.0.
The analog in Westside League tennis is when a player plays matches at a different line than the one they are rostered for, or in the case of subs, the line at which they play most of their matches.
This adjustment is explained in the next section.
• Calculation Adjustments
• Scores
For the purpose of scoring a player’s performance, it is natural to assign a value of 0 for a loss and a value of 1 for a win. This means on average a player, who is appropriately matched, would score 0.5 for the season.
• Adjusted Scores
If a player is playing at a different line, their performance score should be adjusted regardless of whether they win or lose. This is due to the expectations of playing at a different line. For example, if a player plays up one line and wins then that should be counted more than a win at the same line. Likewise, if a player wins at a line below then that player should be award a lesser score.
Conversely, if a player loses at a higher line they should not be penalized as much as a loss at their own line, and if they lose at a lower line then they should be penalized more. That is, if a player loses the majority of her matches at one line up then in calculating her percentages at the end of the season would be very low but should not force her down a level. In order to compensate for this, a win at a higher level will have a score greater than zero (see below).
Also, if a player loses at a lower line then they should be penalized more because of the expectation that they should win. Therefore a loss at a lower line carries a weighted score of less than zero.
Statistically, on average, a player playing at the next line up should win on average 34% more games. This is one standard deviation above the average at their line. In the case of playing down one line, on average if a player were playing at the lower level then they would be winning 34% less games.
Given this rationale, and the fact that on average a player should win half of the matches at their own line, the weighting should be as follows:
• A win at the same line is a score of 1
• A loss at the same line is a score of 0
• A win at a line above is a score of 1.34
• A loss at a line above is a score of 0.16
• A win at a line below is a score of 0.66
• A loss at a line below is a score of –0.34
• A win at two lines above is a score of 1.48
• A loss at two lines above is a score of 0.02
• A win at two lines below is a score of 0.52
• A loss at two lines below is a score of –0.48
• A win at three lines above is a score of 1.5
• A loss at three lines above is a score of 0
• A win at three lines below is a score of 0.5
• A loss at three lines below is a score of –0.5
• Season Scoring
Summing the specified values above for each game and then dividing by the number of games the player played calculate a player’s season score. So for example, if a player played 20 matches at her line and had 10 wins then her score would be (10 * 1.0) / 20 = 0.5.
If a player played 20 matches, won 10, but 2 were at a line above, then her score would be: ((8 * 1.0) + (2 * 1.34)) / 20 = 0.534.
• Percentiles
Since normal scoring is based on a value of 1 for a win and 0 for a loss, a player’s score if playing at their line will be between 0.0 (no wins) and 1.0 (all wins). This maps directly to a percentage scale of 0% to 100%. Players that play outside of their line will have their percentages adjusted per above.
The standard deviations of the normal distribution can be mapped to percentages. The percentages for the significant standard deviations, which come from a mathematical table, are as follows:

Standard
Deviation
Percentage
-2.0
2%
-1.5
7%
-1.0
16%
-0.5
31%
0
50%
+0.5
69%
+1.0
84%
+1.5
93%
+2.0
98%

• Application of Percentiles
Using the table above, once a player’s score has been calculated for the season, any adjustments that are indicated can be made.
Percentile
Range
Action to be taken
0 - 2
Player moves down two lines
3 - 7
Player moves down two lines and is asterisked
8 -16
Player moves down one line
17 -31
Asterisk removed, if already asterisked
32- 50
No action is taken
51 - 69
No action is taken
70 - 84
Player becomes asterisked
85 - 93
Player moves up one line
94 - 97
Player moves up one line and is asterisked
98 - 100+
Player moves up two lines

• Sampling Size
There needs to be a minimum number of games in order for the calculations to be applied with validity. Obviously if a player plays only two games in a season, their record does not accurately reflect their ability and they should not be adjusted based on the outcome of only two games.
Statistically, as the sample size increases, the deviation of a calculated mean from the expected mean decreases. In other words, the more matches a player plays the more accurate the calculated percentages will be. There needs to be enough matches played so that when the win/loss percentages are calculated they can be fairly compared to the Normal Distribution.
How much is enough? Based on a statistical formula that uses the expected mean and the calculated mean, the minimum number of matches needs to be 6.






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