Dear sammy and Jon,

I would imagine that the data being examined is the SAM Puttlab data mostly collected by Christian Marquardt, which is mostly from European players. Jon notes that the measurements indicate that the "mean (+/- SD) shaft angle at impact for the group was 22.6 +/- 2.8 degrees". SD is "standard deviation" determined statistically.

The

interpretation of the SD's significance is shown in this passage: "In practice, one often assumes that the data are from an approximately normally distributed population. This is frequently justified by the classical central limit theorem, which says that sums of many independent, identically-distributed random variables tend towards the normal distribution as a limit. If that assumption is justified, then about 68 % of the values are within 1 standard deviation of the mean, about 95 % of the values are within two standard deviations and about 99.7 % lie within 3 standard deviations. This is known as the 68-95-99.7 rule, or the empirical rule."

The "

variance" is defined as: "The variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. While the expected value shows the location of the distribution, the variance indicates the variability of the values. A more understandable measure is the square root of the [average or mean] variance, called the standard deviation. As its name implies it gives in a standard form an indication of the usual deviations from the mean."

A "

standard deviation" (SD) is the square root of the average variance, so the average variance is seen by squaring the SD. Hence, in this case, if the SD is +/- 2.8, the average variance is 2.8 x 2.8 or 7.84 degrees about a "mean". This indicates that in the SAM Puttlab sample of 150 players, the lie angle -- on average -- varies from about 56.4 degrees flat to 75.4 degrees upright, using 67.4 degrees as the mean from the data set. And even under the "empirical rule" (assuming a normal pattern of distribution of the data points), 95% of the lie angle measurements in this data set lie within the range of two SDs, or +/- 5.6 degrees of this "mean", which is from 28.2 (61.8) degrees up to 17.0 (73.0) degrees using 22.6 (67.4) degrees as the "mean". In other words: "all over the map"!

With this large of an SD, you are basically not learning anything about "usual" from the data sample. That is because: "The standard deviation of a random variable X is defined as:

SD sigma = SQR [E((X - E(x))^2)] = SQR [E(X^2) - E(X))^2]

where E(X) is the

**expected value** of X.

** Not all random variables have a standard deviation, since these expected values need not exist.**" To the same effect: A

Cauchy distribution pattern indicates that there is no "expected value" in the data set. "Various results in probability theory about expected values, such as the strong law of large numbers, will not work in such cases." In sum, the SD in this case is so large in relation to the "expected value" (if there is one) or to the "mean", that we are not getting a very clear picture of exactly what constitutes "elite" dynamics.

In addition, what cannot be determined from this data is a) the actual lie of the putters being used, and b) the lie that any one golfer in the sample should be using. Making guesses about this is a hazardous procedure for science.

Granted, most putters come with 71 degrees (19 degrees off vertical, or 90 degrees) of lie angle, but we know nothing of the sizes and setup postures of the sample golfers, nor do we know anything about the length and lie of the putters they were using when the data was collected.

Even so, I would assume that the so-called "elite" players usually putt with their toes stuck up in the air. Phil Mickelson certainly does!

Scotty Cameron putters typically have what he terms "high toe": "High Toe The toe flows upwards to give the appearance of a more upright lie cheating the eye. The studio series putters have a high flowing toe." This tricks the golfer into thinking the toe is "up" when in fact the sole is more closely flattened to the surface, so it's a bandaid approach that "corrects" what so-called "elite" players do "naturally."

There are a couple of speculative explanations for why lots of tour players have this weird dynamic in their putting, aside from the general observation that they are not aware of it (which is a fact that should by itself dissaude other golfers from basing their putting on what so-called "elite" players do!).

First, they may have their hands lower than "average" golfers using "off-the-rack" putters. When one player sets up with his hands lower than another player of equal stature, this flattens the lie angle. Standard putters are made for standard golfers -- the most numerous group -- and standard golfers set up with hands too high on the putter, and not with arms and hands hanging naturally. In my observation, although pros are not really hanging their arms fully and naturally as a group (a few do, but by far most have some crook and tension in their elbows held back inside the line from shoulder sockets to balls of feet), the pros are certainly setting up with hands lower than the amateurs. That's why Lee Janzen uses a 33-inch putter and Phil Mickelson uses a 34.5-inch putter, although both are well over 6 feet tall.

Second, the head-eye positioning of pros is not good, with the gaze that results from the head posture in relation to the ball position out from the stance (according to Scotty Cameron) typically positioning the eyes about 1.25 inches inside the ball. (The eyes are usually riding about 54 inches above the surface.) And the last 1 inch of the toe end of the putter is farther away than the center of the ball and sweetspot of the putter head by about 1.5 more inches. (most putters are about 4.5 inches from heel to toe, with this last 1 inch segment of the toe centered 1.5 inches out from the sweetspot and hence 2.75 inches out from the position of the eyes.) The very end of the toe of the putter is usually about 0.75 inches out farther than the center of this toe area. So the eyes are "inside" the last 1 inch of toe of the putter by about 2.75 or more inches for these same players and the very end of the toe is 3.5 out. This corresponds to an angle of the gaze outward at the last 1 inch of the toe out of vertically below the eyeballs of 2.9 degrees and to the very end of the toe of 3.7 degrees. Depending on where the golfer is looking at "the toe", the top surface of the toe is not flush to the line of sight unless the toe is tilted up by these degrees, roughly an average up-tilt of 3.3 degrees more or less.

Incidentally, a player who putts with the toe of his putter in the air has a mismatch between where the putter face aims and where the player rolls the ball, with the plane of the lofted and up-tilted putter face having a "normal" or aim direction out of the center of the face that is biased somewhat to the inside (left for a right hander), causing the player's stroke to be more in-to-out in a sort of a "push" action required to compensate for the face's odd attitude in space. In other words, toe-up corrupts the stroke.

All of which is to say that pros -- in general, by far most -- are not very aware of what they are doing in their setup and stroke and also that pros are using putters that are not well fitted to them in terms of their setup and stroke. That being the case, plus the problematic interpretation of a data set with a standard deviation this huge, tells me that we don't know much of value or precision from this particular collection of pro data, even if we wanted to model our putting after this group.

Cheers!

Geoff Mangum

Putting Coach and Theorist

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