What is it? Why is it so valuable? Should I buy some? How do I buy some? Yes, this is actually happening! And why not? Imagine a gigantic piece of paper that lists every transaction ever completed.

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Handicap: Betting odds set by a bookmaker that are designed to level the playing field. New Orleans may have a If the Saints win by eight or more points - they cover the handicap and produce winning wagers. Handicapper: A bettor who researches matchups and then places a bet. Also applies to tipsters who publish predictions on various sporting events. Handle: Total amount of money a bookmaker accepts on a single game or event.

Hedge: Most common with parlay betting and as a risk management tool. Hedging a bet consists of betting on the opposite side of an original wager to set up a guaranteed return. A hedge bet may also be placed to reduce the initial risk on a potential losing wager. Home field advantage: The perceived benefit a team gains when playing in familiar settings at their home stadium.

Hook: A half point added to point spreads and game total odds. A hook guarantees a wager will not be graded as a push. One side will win and one side will lose. If bet: A member of the parlay family, an If Bet consists of two or more wagers. In play betting: Wagers placed after an event after it has started.

Also known as LIVE betting, bookmakers post multiple in-play betting options throughout most major sporting events. Joint favourite: Two or more sides posted with the same betting odds on the same event. Juice: Also known as vigorish, juice is set by bookmakers and is attached to spread and total betting options. If Minnesota Kentucky Derby: First jewel in the Triple Crown of thoroughbred racing. Laying points: Betting on a favorite. A wager on Dallas, as a The Cowboys need to win by at least points to cash a winning ticket.

Layoff: Used by bookmakers and players to reduce risk on a certain market. Parlay bettors may have an option to place a layoff wager on both sides of the last open bet on a ticket to set up a guaranteed profit. Limit: Bookmakers set various high and low wagering limits that vary by sport and betting options. As part of a proper bankroll management system, players should set and follow personal betting limits.

Linemaker: Same as a bookmaker, a person or group that sets daily betting lines and prices. Listed pitchers: Appear with daily baseball betting odds. Live betting: Also known as in-play wagering, live betting is offered once a sporting event begins.

Spreads, moneylines and totals are adjusted and re-posted as a match plays out. Prop options, like next goalscorer and correct final score, are also available. Lock: Term often used by tipsters to tempt bettors into buying handicapping advice. Death and taxes are the only true locks in life. Longshot: A perceived inferior side that is also known as an underdog. Longshot prices are always displayed as positive prices. Masters Tournament: First of four major Grand Slam golf tournaments.

Middle: Cashing tickets on both sides of a betting option. Bettors have an opportunity to middle when a point spread moves up or down prior to a match. The MLB draft is five rounds and most of the players selected will be assigned to minor league teams. Moneyline : A straight up bet, without any point spread, where bettors need to predict the outright winner.

Multiple bets: Same as parlay, multiple bets are a single wager that consists of at least two sides on a single ticket. All sides must win or push to cash winning multiple bets. MVP: Player honored as most valuable to their team during the regular season or playoffs. Wagering on who will be named the Most Valuable Player is a popular futures betting option in professional sports.

Nap: Similar to a lock, a nap is a handicappers suggested best bet on a daily betting card. No action: Betting options cancelled by a bookmaker are graded as no action. Original stakes are returned to bettors. Novelty bets: Prop and special betting options that are wagers beyond standard moneyline, point spread and game total odds.

Team and player propositions are the most common novelty bets. Oddsmaker: Same as a linemaker, a person or group that sets daily betting lines and prices. Odds on favorite: One side that is viewed as far superior to the other and is priced with odds that offer very little value. Odds shopping: Reviewing the lines at a variety of sportsbooks in order to find the best priced odds. An injury to a star player may cause bookmakers to pull odds off the board. Outright betting: Predicting the overall winner of a tournament or playoff competition.

Over bet: Opposite of an Under bet on game total options. Bettors need to determine if the combined scores of both teams will go over or remain under the number. Parlay : A single bet, also known as an accumulator or multiple, that consists of two or more sides. Each side must win to produce a winning ticket. Parlay banker: Forming the base of a parlay wager, a banker is a favorite side to which other sides are added.

Payout: The amount a bettor collects on a winning wager. When a wager is placed, the possible payout on a betting receipt usually includes the original stake. Held in late May at various courses across the United States. Point spread : Odds posted on a match that are designed to level the playing field.

Favorites are listed with a negative Post time: Scheduled start time of a race. Power rankings: A ranking system that uses a variety of criteria to grade teams, in a specific league, from the best to worst. Preakness Stakes: Second jewel in the Triple Crown of thoroughbred racing. Proposition bet: Often shortened to prop bet, proposition bets are exotic or special wagers that are offered on most sporting events.

NFL Super Bowl prop betting options number in the hundreds. Puck line: Point spread pricing in hockey. Prior to a match, the favorite is normally posted at Push: Any wager where the final result is a tie. If a basketball spread is 11 points and the final score is spread bets on both teams are graded as a push and original stakes are returned. Quarter Bet: Any wager placed prior to or during any quarter of a sporting event.

Recreational Bettor: A player that bets infrequently or on major sporting events only. Opposite of a sharp or professional bettor. Rotation Number: A number assigned by bookmakers to every betting option on the board. Bettors use the rotation number when placing a bet, rather than team names, at betting windows at land based sportsbooks.

ROY: Honors the top first year player in most professional sports leagues. Wagering on which player will be named the Rookie of the Year ROY is a popular futures betting option. Run Line: Point spread pricing in baseball. Prior to a game the favorite is normally posted at Second half bet: Any wager that focuses on the outcome of the second half of any competition. Bettors can place wagers before the second half begins or make live bets once the match resumes.

Selke Trophy: Awarded to a forward not a defenseman or goaltender with the best defensive skills during the NHL regular season. Sell points: Bettors can sell points by using alternate point spreads and game totals. In football, if a player moves a line from Juice becomes more favorable for the bettor with each point sold. Sharp: A professional sports gambler who uses vast resources to determine their wagers. Sharps look at the big picture and base their bets on knowledge.

Pro bettors always shop around for the best prices and will bet on favorites or underdogs when they receive proper value. Special: Similar to prop and exotic wagers, special bets are added to a competition beyond the more common moneyline, game total and spread betting options.

Spread betting: Taking or laying points when betting on a competition. Selecting Los Angeles with -7 point odds against New York is a spread bet. The Rams need to defeat the Giants by at least eight points to cash a winning ticket.

Square: Another term for a novice or recreational player and the opposite of a sharp or professional bettor. Stake: The amount of money a bettor risks when placing a bet. Original stakes are returned on all winning wagers and many bets that are graded as a push.

Staking method: Differs from bettor to bettor. Some players set maximum stake limits on each bet they place while others use a bankroll percentage as their stake. Steam: Odds that change quickly usually due to a large amount of betting action by sharp bettors or syndicates.

Straight bet: A single wager on moneyline, spread or game total betting options. Syndicate: A group of bettors that pool funds and use their combined knowledge to bet on events. Syndicates will often wager large amounts to move a line and then place an even larger bet on the new price they helped create.

Taking points: A bet placed on an underdog side. Tickets cash is the Nationals win outright or lose by one run. Teaser odds : Any line moved up or down by a bookmaker to entice tease bettors. Players can tease odds on a single game by using alternate lines, They can also place a parlay bet from a teaser card issued by a sportsbook. Teaser Card: A daily list of all games, from one specific sport, where the odds are higher or lower than the prices posted on the main betting board.

Teaser card bets require selecting two or more sides. Tip: Betting advice offered by tipsters and handicappers that suggest the most likely outcome of an event. Tips should never be bet on blindly but can be helpful when used with a proper pregame research plan. Tipster: A person or group that offers betting advice. Some tipsters offer free sports wagering advice while others charge a fee for their tips. Puck line odds vary with probability of favorite winning by 2 goals.

The following explains how spread bets win, lose or tie. Fictional scores have been given to our football and basketball games from above. Suppose the final score was Cleveland Orlando. The spread is tied. Neither Cleveland nor Orlando cover. Charles Kline McNeil created point spread betting combining ideas from other bookmakers.

He later popularized it in Chicago during the early s. He was both an avid and successful gambler, eventually opening his own bookmaking shop in the s. His bold decision was the result of another bookie limiting his action. Spread bets are placed on the expected difference in score. This concept contrasted with money line betting. These bets are placed on the probability of a result. Point Spreads became common fare at Las Vegas and New Jersey sports books coinciding with the rise of large scale gambling operations in the late s and early s.

Today all sportsbook operations — both onshore and offshore — offer spread betting. These lines often take longer to post than game moneylines. Post Up or Credit Shop? Who are Sharp Players? Corner Bookie or Sportsbook? Twitter Facebook Reddit.

Here is an outline from SportsLine for understanding basic terms and concepts in the sports wagering industry:. Perhaps the most common question newcomers have when they see sportsbook odds is, "What do the numbers mean?

While the plus and minus signs might look confusing at first, they are easily explained. Take the following listing you might see in a sportsbook:. In this example, the Giants are three-point favorites against the Cowboys. The spread is essentially a mathematical formula used to bridge the talent gap between teams and incentivize potential bettors into considering both sides. The spread is a handicap that requires the favored team to win the game by an ascribed number of points in order for the bettor to win his wager on the team.

So if you wish to wager on the Giants in this game, you'd be giving up or "laying" three points with them. This means they would have to win by four or more in order to cover the spread and make your wager a winner. Those who wish to support the Cowboys are betting on the underdog, and thus "taking" the points. This means, in order for the wager to win, the Cowboys would need to lose the game by fewer than three points, or win it outright.

In the above example, the Giants winning the game by exactly three points would result in a push for bettors on both sides, and all wagers would be refunded. The number to the direct left of the teams is called the "rotation" number.

This is simply the universal record-keeping system for sportsbooks to track bets on each team. When you place a wager, you give the sportsbook ticket writer the rotation number of the team you are selecting instead of announcing the team name. In addition to picking a side against the point spread, bettors have the option of wagering on the over-under or "total. The total is mathematic formula created by oddsmakers to determine the approximate number of points that will be scored in the contest.

The formula essentially boils down to taking the average of what both teams score and allow per contest, along with other mitigating factors such as the venue or key injuries. In the above example, along with seeing the point spread on the betting menu, you will see a large number near the spread. It might look like this:. You'll see the spread is almost always delineated with the minus symbol and spread number corresponding directly to the favored team.

The over-under or total is usually noted on the same line where the underdog is listed, but with a bit of separation from the team name so that it is not confused with the point spread. In the above example, 46 points is the posted total between the Cowboys and Giants. If you wish to wager on the total, you would pick the rotation number of either team and inform the sportsbook writer that your intention is to bet the Over or Under only on the posted total.

If you have dabbled at all in the sports wagering industry, you've likely heard the terms juice, or vigorish, thrown around quite a bit. If you've heard the terms but don't know what they mean, fret not. They might sound complicated on the surface, but they are relatively simple to explain.

In short, the vigorish is essentially a tax the sportsbooks charge per wager. Their role is the primary factor in turning a profit for bookmakers, and also a crucial factor for bettors to learn and respect. Circled game: Matches that have set betting maximums, which are capped at low amounts.

Games are usually circled when bookmakers face unknowns such as player injuries, weather or rumors that surface prior to a match. Opening odds and prop options are often circled as well. Closing line: The final betting odds posted prior to the start of a competition. Co-favorite: Two or more sides with identical odds to win. Common with futures odds, bookmakers may post co-favorites to win the NBA Finals championship.

Combine: A series of fitness tests that help scouts from professional teams evaluate amateur athletes. Commission: Another term for vigorish and juice, commission is the bookmakers take on any bet. It is also the amount a betting exchange takes from winning wagers.

Correct score: Bettors are offered a list of possible final scores on a match. In soccer, players can bet on a match ending as low as or as high as plus all scores in between. The most likely result is the favorite and the least likely result is the underdog. New England winning over Miami means the Patriots would cover a point spread. Dog: Short for underdog, a dog is perceived as the least likely side to win and is tagged with plus pricing.

Bettors often double their bet when they feel one side is vastly superior to another. Double result: A single betting option that combines the score of a game at halftime and the score at the end of the same game. Double-header: Two games that are played back-to-back on the same day.

Most common in baseball, a double-header will often take place if a game from the previous day was rained out. Draw: Any contest where the final score ends in a tie. In most instances, a draw is graded as a PUSH and original bet amounts are returned. Drift: Betting odds that grow longer after the opening line is posted. Each-way: Common in horse racing, each-way betting takes a single amount and splits it on a horse to finish first or second.

Both bets pay if the horse finishes first while just one bet pays if the horse finishes second. The return on a first place win is always higher than the return on a second place win. Edge: Gaining an advantage through extensive research or having insights that are not publicly known.

Even money: Odds that return the exact amount of the original bet. Exotic Bet: Betting options beyond point spreads, moneylines and game totals. Proposition bets, specials and parlays are the most common types of exotic bets. Exposure: Amount of money a bettor or bookmaker stands to lose on any given wager.

Favorite: Any side priced with a negative number. Two Final Four games are played prior to the National Championship game. First half bet: A wager that focused on the result of the first half in sports like basketball, soccer and football. The most popular first half betting odds are spread, moneyline and game total options. A variety of team and player props are also offered as first half bets. Fixed odds : When a wager is placed, and a bookmaker accepts it, the line becomes fixed odds.

Also a term for moneyline odds. French Open : Second of four women's and men's Grand Slam tennis tournaments that are played over two weeks in late May and early June. Futures bet : A wager placed on an event that will take place in the near or distant future. Futures are also offered in soccer, major horse races, plus golf and tennis tournaments. If a baseball game total is set at 7. Graded Bet: A wager that bookmakers officially mark as a winner, a loser, or a push, once a competition has ended.

Winnings, or push refunds, are paid out after a bet has been graded. If there are seven games on the NFL schedule, the line may be set at Half ball handicap: Soccer betting odds where 0. Half time bet : Wagers placed on the outcome of just the second half of a competition.

Half time bets can be placed during intermission or as live wagers once the second half begins. Handicap: Betting odds set by a bookmaker that are designed to level the playing field. New Orleans may have a If the Saints win by eight or more points - they cover the handicap and produce winning wagers.

Handicapper: A bettor who researches matchups and then places a bet. Also applies to tipsters who publish predictions on various sporting events. Handle: Total amount of money a bookmaker accepts on a single game or event. Hedge : Most common with parlay betting and as a risk management tool.

Hedging a bet consists of betting on the opposite side of an original wager to set up a guaranteed return. A hedge bet may also be placed to reduce the initial risk on a potential losing wager. Home field advantage: The perceived benefit a team gains when playing in familiar settings at their home stadium.

Hook : A half point added to point spreads and game total odds. A hook guarantees a wager will not be graded as a push. One side will win and one side will lose. If bet: A member of the parlay family, an If Bet consists of two or more wagers. In play betting: Wagers placed after an event after it has started. Also known as LIVE betting, bookmakers post multiple in-play betting options throughout most major sporting events.

Joint favorite: Two or more sides posted with the same betting odds on the same event. Juice : Also known as vigorish, juice is set by bookmakers and is attached to spread and total betting options. If Minnesota Kentucky Derby: First jewel in the Triple Crown of thoroughbred racing.

Laying points : Betting on a favorite. A wager on Dallas, as a The Cowboys need to win by at least points to cash a winning ticket. Layoff: Used by bookmakers and players to reduce risk on a certain market. Parlay bettors may have an option to place a layoff wager on both sides of the last open bet on a ticket to set up a guaranteed profit. Limit: Bookmakers set various high and low wagering limits that vary by sport and betting options.

As part of a proper bankroll management system, players should set and follow personal betting limits. Line: Betting odds posted by a bookmaker. Linemaker: Same as a bookmaker, a person or group that sets daily betting lines and prices. Listed pitchers: Appear with daily baseball betting odds. Live betting : Also known as in-play wagering, live betting is offered once a sporting event begins. Spreads, moneylines and totals are adjusted and re-posted as a match plays out.

Prop options, like next goalscorer and correct final score, are also available. Lock: Term often used by tipsters to tempt bettors into buying handicapping advice. Death and taxes are the only true locks in life. Longshot: A perceived inferior side that is also known as an underdog.

Longshot prices are always displayed as positive prices. Masters Tournament: First of four major Grand Slam golf tournaments. Middle : Cashing tickets on both sides of a betting option. Bettors have an opportunity to middle when a point spread moves up or down prior to a match. The MLB draft is five rounds and most of the players selected will be assigned to minor league teams.

Moneyline : A straight up bet, without any point spread, where bettors need to predict the outright winner. Multiple bets: Same as parlay, multiple bets are a single wager that consists of at least two sides on a single ticket. All sides must win or push to cash winning multiple bets. MVP: Player honored as most valuable to their team during the regular season or playoffs.

Wagering on who will be named the Most Valuable Player is a popular futures betting option in professional sports. Nap: Similar to a lock, a nap is a handicappers suggested best bet on a daily betting card. No action: Betting options cancelled by a bookmaker are graded as no action. Original stakes are returned to bettors.

Novelty bets: Prop and special betting options that are wagers beyond standard moneyline, point spread and game total odds. Team and player propositions are the most common novelty bets. Odds: Betting lines set by a bookmaker on a variety of events.

Oddsmaker: Same as a linemaker, a person or group that sets daily betting lines and prices. Odds on favorite: One side that is viewed as far superior to the other and is priced with odds that offer very little value. Odds shopping: Reviewing the lines at a variety of sportsbooks in order to find the best priced odds. An injury to a star player may cause bookmakers to pull odds off the board.

Outright betting: Predicting the overall winner of a tournament or playoff competition. Over bet: Opposite of an Under bet on game total options. Bettors need to determine if the combined scores of both teams will go over or remain under the number.

Also known as game total odds. Parlay : A single bet, also known as an accumulator or multiple, that consists of two or more sides. Each side must win to produce a winning ticket. Parlay banker: Forming the base of a parlay wager, a banker is a favorite side to which other sides are added.

Payout: The amount a bettor collects on a winning wager. When a wager is placed, the possible payout on a betting receipt usually includes the original stake. Held in late May at various courses across the United States. Point spread : Odds posted on a match that are designed to level the playing field. Favorites are listed with a negative Post time: Scheduled start time of a race.

Power rankings: A ranking system that uses a variety of criteria to grade teams, in a specific league, from the best to worst. Preakness Stakes: Second jewel in the Triple Crown of thoroughbred racing. Proposition bet: Often shortened to prop bet, proposition bets are exotic or special wagers that are offered on most sporting events. NFL Super Bowl prop betting options number in the hundreds.

Proxy : A proxy is an individual, or a group of individuals, who place bets for other people. The term is most commonly associated with people who submit picks for non-Las Vegas residents that are involved in season-long sports pools like the Westgate Las Vegas SuperContest. Puck line: Point spread pricing in hockey. Prior to a match, the favorite is normally posted at Push: Any wager where the final result is a tie.

If a basketball spread is 11 points and the final score is spread bets on both teams are graded as a push and original stakes are returned. Quarter Bet : Any wager placed prior to or during any quarter of a sporting event. Prior to an NBA game, Boston may be a

As with the exceedance probabilities for the t-statistics, smaller is better. A low exceedance probability say, less than. In a simple regression model, the F-ratio is simply the square of the t-statistic of the single independent variable , and the exceedance probability for F is the same as that for t. In a multiple regression model, the exceedance probability for F will generally be smaller than the lowest exceedance probability of the t-statistics of the independent variables other than the constant.

Hence, if at least one variable is known to be significant in the model, as judged by its t-statistic, then there is really no need to look at the F-ratio. The F-ratio is useful primarily in cases where each of the independent variables is only marginally significant by itself but there are a priori grounds for believing that they are significant when taken as a group, in the context of a model where there is a logical way to group them.

For example, the independent variables might be dummy variables for treatment levels in a designed experiment, and the question might be whether there is evidence for an overall effect, even if its fine detail cannot quantified precisely with the given sample size. It often happens that there are several available independent variables that are plausibly linearly related to the dependent variable but also strongly linearly related to each other.

That is to say, their information value is not really independent with respect to prediction of the dependent variable in the context of a linear model. Such a situation is often observed, for example, when the independent variables are a collection of economic indicators that are computed from some of the same underlying data via weighted averages. This condition is referred to as multicollinearity.

In the most extreme cases of multicollinearity--e. Sometimes one variable is merely a rescaled copy of another variable or a sum or difference of other variables, and sometimes a set of dummy variables adds up to a constant variable.

The correlation matrix of the estimated coefficients if your software includes it is one diagnostic tool for detecting relative degrees of multicollinearity. It shows the extent to which particular pairs of variables provide independent information for purposes of predicting the dependent variable, given the presence of other variables in the model. Extremely high values here say, much above 0. In this case, either i both variables are providing the same information--i. In case i --i.

When this happens, it is usually desirable to try removing one of them, usually the one whose coefficient has the higher P-value. In case ii , it may be possible to replace the two variables by the appropriate linear function e. This situation often arises when two or more different lags of the same variable are used as independent variables in a time series regression model. The VIF of an independent variable is the value of 1 divided by 1-minus-R-squared in a regression of itself on the other independent variables.

The rule of thumb here is that a VIF larger than 10 is an indicator of potentially significant multicollinearity between that variable and one or more others. If this is observed, it means that the variable in question does not contain much independent information in the presence of all the other variables, taken as a group. When this happens, it often happens for many variables at once, and it may take some trial and error to figure out which one s ought to be removed.

However, like most other diagnostic tests, the VIF-greater-than test is not a hard-and-fast rule, just an arbitrary threshold that indicates the possibility of a problem. In this case it indicates a possibility that the model could be simplified, perhaps by deleting variables or perhaps by redefining them in a way that better separates their contributions. Of course not. This is merely what we would call a "point estimate" or "point prediction.

For a point estimate to be really useful, it should be accompanied by information concerning its degree of precision-- i. In "classical" statistical methods such as linear regression, information about the precision of point estimates is usually expressed in the form of confidence intervals. That is, should we consider it a "to-1 long shot" that sales would fall outside this interval, for purposes of betting?

The answer to this is:. Alas, you never know for sure whether you have identified the correct model for your data, although residual diagnostics help you rule out obviously incorrect ones. This is not to say that a confidence interval cannot be meaningfully interpreted, but merely that it shouldn't be taken too literally in any single case, especially if there is any evidence that some of the model assumptions are not correct. In fitting a model to a given data set, you are often simultaneously estimating many things: e.

Thus, a model for a given data set may yield many different sets of confidence intervals. You may wonder whether it is valid to take the long-run view here: e. This is another issue that depends on the correctness of the model and the representativeness of the data set, particularly in the case of time series data. If the model is not correct or there are unusual patterns in the data, then if the confidence interval for one period's forecast fails to cover the true value, it is relatively more likely that the confidence interval for a neighboring period's forecast will also fail to cover the true value, because the model may have a tendency to make the same error for several periods in a row.

Ideally, you would like your confidence intervals to be as narrow as possible: more precision is preferred to less. Does this mean that, when comparing alternative forecasting models for the same time series, you should always pick the one that yields the narrowest confidence intervals around forecasts? That is, should narrow confidence intervals for forecasts be considered as a sign of a "good fit? If the model's assumptions are correct, the confidence intervals it yields will be realistic guides to the precision with which future observations can be predicted.

If the assumptions are not correct, it may yield confidence intervals that are all unrealistically wide or all unrealistically narrow. That is to say, a bad model does not necessarily know it is a bad model, and warn you by giving extra-wide confidence intervals. This is especially true of trend-line models, which often yield overoptimistically narrow confidence intervals for forecasts. You need to judge whether the model is good or bad by looking at the rest of the output.

Notwithstanding these caveats, confidence intervals are indispensable, since they are usually the only estimates of the degree of precision in your coefficient estimates and forecasts that are provided by most stat packages. In the long run. In regression forecasting, you may be concerned with point estimates and confidence intervals for some or all of the following:. In all cases, there is a simple relationship between the point estimate and its surrounding confidence interval:. More precisely, it is 1.

A t -distribution with "infinite" degrees of freedom is a standard normal distribution. For the confidence interval around a coefficient estimate , this is simply the "standard error of the coefficient estimate" that appears beside the point estimate in the coefficient table. Recall that this is proportional to the standard error of the regression, and inversely proportional to the standard deviation of the independent variable. For a confidence interval for the mean i.

This quantity depends on the following factors:. Other things being equal, the standard deviation of the mean--and hence the width of the confidence interval around the regression line-- increases with the standard errors of the coefficient estimates, increases with the distances of the independent variables from their respective means, and decreases with the degree of correlation between the coefficient estimates.

However, in a model characterized by "multicollinearity", the standard errors of the coefficients and. For a confidence interval around a prediction based on the regression line at some point, the relevant standard deviation is called the "standard deviation of the prediction. In this case, the regression line represents your best estimate of the true signal, and the standard error of the regression is your best estimate of the standard deviation of the true noise. Now trust me , for essentially the same reason that the fitted values are uncorrelated with the residuals, it is also true that the errors in estimating the height of the regression line are uncorrelated with the true errors.

Therefore, the variances of these two components of error in each prediction are additive. Since variances are the squares of standard deviations, this means:. Note that, whereas the standard error of the regression is a fixed number, the standard deviations of the predictions and the standard deviations of the means will usually vary from point to point in time, since they depend on the values of the independent variables.

However, the standard error of the regression is typically much larger than the standard errors of the means at most points, hence the standard deviations of the predictions will often not vary by much from point to point, and will be only slightly larger than the standard error of the regression. Statgraphics and RegressIt will automatically generate forecasts rather than fitted values wherever the dependent variable is "missing" but the independent variables are not.

Confidence intervals for the forecasts are also reported. Here is an example of a plot of forecasts with confidence limits for means and forecasts produced by RegressIt for the regression model fitted to the natural log of cases of packs sold. If you look closely, you will see that the confidence intervals for means represented by the inner set of bars around the point forecasts are noticeably wider for extremely high or low values of price, while the confidence intervals for forecasts are not.

One of the underlying assumptions of linear regression analysis is that the distribution of the errors is approximately normal with a mean of zero. Hence, a value more than 3 standard deviations from the mean will occur only rarely: less than one out of observations on the average.

Now, the residuals from fitting a model may be considered as estimates of the true errors that occurred at different points in time, and the standard error of the regression is the estimated standard deviation of their distribution. Hence, if the normality assumption is satisfied, you should rarely encounter a residual whose absolute value is greater than 3 times the standard error of the regression.

An observation whose residual is much greater than 3 times the standard error of the regression is therefore usually called an "outlier. In the residual table in RegressIt, residuals with absolute values larger than 2.

Outliers are also readily spotted on time-plots and normal probability plots of the residuals. If your data set contains hundreds of observations, an outlier or two may not be cause for alarm. But outliers can spell trouble for models fitted to small data sets: since the sum of squares of the residuals is the basis for estimating parameters and calculating error statistics and confidence intervals, one or two bad outliers in a small data set can badly skew the results.

When outliers are found, two questions should be asked: i are they merely "flukes" of some kind e. An outlier may or may not have a dramatic effect on a model, depending on the amount of "leverage" that it has. Its leverage depends on the values of the independent variables at the point where it occurred: if the independent variables were all relatively close to their mean values , then the outlier has little leverage and will mainly affect the value of the estimated CONSTANT term and the standard error of the regression.

However, if one or more of the independent variable had relatively extreme values at that point, the outlier may have a large influence on the estimates of the corresponding coefficients: e. The best way to determine how much leverage an outlier or group of outliers has, is to exclude it from fitting the model , and compare the results with those originally obtained.

In RegressIt you can just delete the values of the dependent variable in those rows. Be sure to keep a copy of them, though! In this sort of exercise, it is best to copy all the values of the dependent variable to a new column, assign it a new variable name, then delete the desired values in the new column and use it as the new dependent variable. Forecasts will automatically be generated for the excluded or missing values of the dependent variable in either program.

The discrepancies between the forecasts and the actual values, measured in terms of the corresponding standard-deviations-of- predictions, provide a guide to how "surprising" these observations really were. These observations will then be fitted with zero error independently of everything else, and the same coefficient estimates, predictions, and confidence intervals will be obtained as if they had been excluded outright.

In Statgraphics, to dummy-out the observations at periods 23 and 59, you could add the two variables:. The estimated coefficients for the two dummy variables would exactly equal the difference between the offending observations and the predictions generated for them by the model.

If it turns out the outlier or group thereof does have a significant effect on the model, then you must ask whether there is justification for throwing it out. Go back and look at your original data and see if you can think of any explanations for outliers occurring where they did.

Sometimes you will discover data entry errors: e. In this case, you must use your own judgment as to whether to merely throw the observations out, or leave them in, or perhaps alter the model to account for additional effects. This may create a situation in which the size of the sample to which the model is fitted may vary from model to model, sometimes by a lot, as different variables are added or removed.

In general the estimation procedure will use all rows of data in which none of the currently selected variables has missing values. You should always keep your eye on the sample size that is reported in your output, to make sure there are no surprises. Small differences in sample sizes are not necessarily a problem if the data set is large, but you should be alert for situations in which relatively many rows of data suddenly go missing when more variables are added to the model.

If this does occur, then you may have to choose between a not using the variables that have significant numbers of missing values, or b deleting all rows of data in which any of the variables have missing values, so that the sample will be the same for any model that is fitted.

For this reason, the value of R-squared that is reported for a given model in the stepwise regression output may not be the same as you would get if you fitted that model by itself. The basic linear regression model assumes that the contributions of the different independent variables to the prediction of the dependent variable are additive. For example, if X 1 and X 2 are assumed to contribute additively to Y, the prediction equation of the regression model is:.

Here, if X 1 increases by one unit, other things being equal, then Y is expected to increase by b 1 units. That is, the absolute change in Y is proportional to the absolute change in X 1 , with the coefficient b 1 representing the constant of proportionality. Similarly, if X 2 increases by 1 unit, other things equal, Y is expected to increase by b 2 units. Adults display good behavior to show good examples to the younger ones.

Since children tends to copy what the see the older people do, it is necessary that the older ones acts in a way that is right and acceptable to the larger society. If adults don't behave well, it might lead to chaos and conflicting of interest. Asked by Amaya Johnson. So all you have to do is take away the zero from the number 60 so if you ever get this problem again then just remember take away the zero!

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Now we apply the second adjustment. Go back to 97 in the chart above; follow the column down to the "handicap adjustment" row on the bottom line. The column for 97 corresponds to a handicap adjustment of That means we're going to subtract a stroke from our handicap deduction of So our final, adjusted handicap allowance is And our net Callaway System score is 97 minus 23, or And 74 is the net score.

So using the chart is a matter of finding the gross score, looking across the row for the handicap deduction, then looking down the column for the adjustment. Once you've done it once, it's easy. Example 3 : Helen shoots So Helen adds 8 plus 4 3. Helen's gross score of 84 is a net score of 71 84 minus If the chart tells you to deduct your three worst holes, and one of them is the 17th, sorry, you can't deduct that one.

You'll have to keep that hole and move on to the next-highest score. Does the Callaway System, the handicapping method, have anything to do with Callaway Golf, the company? No on both counts. The two Callaways are unrelated. There are two other popular 1-day handicapping methods similar to the Callaway System that do not require consulting charts although they do require knowing the proper steps. Those would be the Peoria System and the System 36 handicapping methods. Brent Kelley.

Brent Kelley is an award-winning sports journalist and golf expert with over 30 years in print and online journalism. Updated January 05, Most stat packages will compute for you the exact probability of exceeding the observed t-value by chance if the true coefficient were zero.

This is labeled as the "P-value" or "significance level" in the table of model coefficients. A low value for this probability indicates that the coefficient is significantly different from zero , i. Usually you are on the lookout for variables that could be removed without seriously affecting the standard error of the regression. A low t-statistic or equivalently, a moderate-to-large exceedance probability for a variable suggests that the standard error of the regression would not be adversely affected by its removal.

Of course, the proof of the pudding is still in the eating: if you remove a variable with a low t-statistic and this leads to an undesirable increase in the standard error or the regression or deterioration of some other statistics, such as residual autocorrelations , then you should probably put it back in. Generally you should only add or remove variables one at a time, in a stepwise fashion, since when one variable is added or removed, the other variables may increase or decrease in significance.

For example, if X 1 is the least significant variable in the original regression, but X 2 is almost equally insignificant, then you should try removing X 1 first and see what happens to the estimated coefficient of X 2 : the latter may remain insignificant after X 1 is removed, in which case you might try removing X 2 as well, or it may rise in significance with a very different estimated value , in which case you may wish to leave it in.

Note: the t-statistic is usually not used as a basis for deciding whether or not to include the constant term. Usually the decision to include or exclude the constant is based on a priori reasoning, as noted above. If it is included, it may not have direct economic significance, and you generally don't scrutinize its t-statistic too closely. The F-ratio and its exceedance probability provide a test of the significance of all the independent variables other than the constant term taken together.

The variance of the dependent variable may be considered to initially have n- 1 degrees of freedom, since n observations are initially available each including an error component that is "free" from all the others in the sense of statistical independence ; but one degree of freedom is used up in computing the sample mean around which to measure the variance--i.

As noted above, the effect of fitting a regression model with p coefficients including the constant is to decompose this variance into an "explained" part and an "unexplained" part. The explained part may be considered to have used up p -1 degrees of freedom since this is the number of coefficients estimated besides the constant , and the unexplained part has the remaining unused n - p degrees of freedom.

The F-ratio is the ratio of the explained-variance-per-degree-of-freedom-used to the unexplained-variance-per-degree-of-freedom-unused , i. Now, a set of n observations could in principle be perfectly fitted by a model with a constant and any n - 1 linearly independent other variables--i. Thus, if the true values of the coefficients are all equal to zero i.

In this case, the numerator and the denominator of the F-ratio should both have approximately the same expected value; i. On the other hand, if the coefficients are really not all zero, then they should soak up more than their share of the variance, in which case the F-ratio should be significantly larger than 1. Standard regression output includes the F-ratio and also its exceedance probability--i.

As with the exceedance probabilities for the t-statistics, smaller is better. A low exceedance probability say, less than. In a simple regression model, the F-ratio is simply the square of the t-statistic of the single independent variable , and the exceedance probability for F is the same as that for t. In a multiple regression model, the exceedance probability for F will generally be smaller than the lowest exceedance probability of the t-statistics of the independent variables other than the constant.

Hence, if at least one variable is known to be significant in the model, as judged by its t-statistic, then there is really no need to look at the F-ratio. The F-ratio is useful primarily in cases where each of the independent variables is only marginally significant by itself but there are a priori grounds for believing that they are significant when taken as a group, in the context of a model where there is a logical way to group them.

For example, the independent variables might be dummy variables for treatment levels in a designed experiment, and the question might be whether there is evidence for an overall effect, even if its fine detail cannot quantified precisely with the given sample size.

It often happens that there are several available independent variables that are plausibly linearly related to the dependent variable but also strongly linearly related to each other. That is to say, their information value is not really independent with respect to prediction of the dependent variable in the context of a linear model.

Such a situation is often observed, for example, when the independent variables are a collection of economic indicators that are computed from some of the same underlying data via weighted averages. This condition is referred to as multicollinearity. In the most extreme cases of multicollinearity--e.

Sometimes one variable is merely a rescaled copy of another variable or a sum or difference of other variables, and sometimes a set of dummy variables adds up to a constant variable. The correlation matrix of the estimated coefficients if your software includes it is one diagnostic tool for detecting relative degrees of multicollinearity.

It shows the extent to which particular pairs of variables provide independent information for purposes of predicting the dependent variable, given the presence of other variables in the model. Extremely high values here say, much above 0. In this case, either i both variables are providing the same information--i. In case i --i. When this happens, it is usually desirable to try removing one of them, usually the one whose coefficient has the higher P-value.

In case ii , it may be possible to replace the two variables by the appropriate linear function e. This situation often arises when two or more different lags of the same variable are used as independent variables in a time series regression model. The VIF of an independent variable is the value of 1 divided by 1-minus-R-squared in a regression of itself on the other independent variables. The rule of thumb here is that a VIF larger than 10 is an indicator of potentially significant multicollinearity between that variable and one or more others.

If this is observed, it means that the variable in question does not contain much independent information in the presence of all the other variables, taken as a group. When this happens, it often happens for many variables at once, and it may take some trial and error to figure out which one s ought to be removed.

However, like most other diagnostic tests, the VIF-greater-than test is not a hard-and-fast rule, just an arbitrary threshold that indicates the possibility of a problem. In this case it indicates a possibility that the model could be simplified, perhaps by deleting variables or perhaps by redefining them in a way that better separates their contributions.

Of course not. This is merely what we would call a "point estimate" or "point prediction. For a point estimate to be really useful, it should be accompanied by information concerning its degree of precision-- i. In "classical" statistical methods such as linear regression, information about the precision of point estimates is usually expressed in the form of confidence intervals.

That is, should we consider it a "to-1 long shot" that sales would fall outside this interval, for purposes of betting? The answer to this is:. Alas, you never know for sure whether you have identified the correct model for your data, although residual diagnostics help you rule out obviously incorrect ones.

This is not to say that a confidence interval cannot be meaningfully interpreted, but merely that it shouldn't be taken too literally in any single case, especially if there is any evidence that some of the model assumptions are not correct. In fitting a model to a given data set, you are often simultaneously estimating many things: e. Thus, a model for a given data set may yield many different sets of confidence intervals.

You may wonder whether it is valid to take the long-run view here: e. This is another issue that depends on the correctness of the model and the representativeness of the data set, particularly in the case of time series data. If the model is not correct or there are unusual patterns in the data, then if the confidence interval for one period's forecast fails to cover the true value, it is relatively more likely that the confidence interval for a neighboring period's forecast will also fail to cover the true value, because the model may have a tendency to make the same error for several periods in a row.

Ideally, you would like your confidence intervals to be as narrow as possible: more precision is preferred to less. Does this mean that, when comparing alternative forecasting models for the same time series, you should always pick the one that yields the narrowest confidence intervals around forecasts?

That is, should narrow confidence intervals for forecasts be considered as a sign of a "good fit? If the model's assumptions are correct, the confidence intervals it yields will be realistic guides to the precision with which future observations can be predicted. If the assumptions are not correct, it may yield confidence intervals that are all unrealistically wide or all unrealistically narrow.

That is to say, a bad model does not necessarily know it is a bad model, and warn you by giving extra-wide confidence intervals. This is especially true of trend-line models, which often yield overoptimistically narrow confidence intervals for forecasts. You need to judge whether the model is good or bad by looking at the rest of the output. Notwithstanding these caveats, confidence intervals are indispensable, since they are usually the only estimates of the degree of precision in your coefficient estimates and forecasts that are provided by most stat packages.

In the long run. In regression forecasting, you may be concerned with point estimates and confidence intervals for some or all of the following:. In all cases, there is a simple relationship between the point estimate and its surrounding confidence interval:. More precisely, it is 1. A t -distribution with "infinite" degrees of freedom is a standard normal distribution.

For the confidence interval around a coefficient estimate , this is simply the "standard error of the coefficient estimate" that appears beside the point estimate in the coefficient table. Recall that this is proportional to the standard error of the regression, and inversely proportional to the standard deviation of the independent variable.

For a confidence interval for the mean i. This quantity depends on the following factors:. Other things being equal, the standard deviation of the mean--and hence the width of the confidence interval around the regression line-- increases with the standard errors of the coefficient estimates, increases with the distances of the independent variables from their respective means, and decreases with the degree of correlation between the coefficient estimates.

However, in a model characterized by "multicollinearity", the standard errors of the coefficients and. For a confidence interval around a prediction based on the regression line at some point, the relevant standard deviation is called the "standard deviation of the prediction.

In this case, the regression line represents your best estimate of the true signal, and the standard error of the regression is your best estimate of the standard deviation of the true noise. Now trust me , for essentially the same reason that the fitted values are uncorrelated with the residuals, it is also true that the errors in estimating the height of the regression line are uncorrelated with the true errors.

Therefore, the variances of these two components of error in each prediction are additive. Since variances are the squares of standard deviations, this means:. Note that, whereas the standard error of the regression is a fixed number, the standard deviations of the predictions and the standard deviations of the means will usually vary from point to point in time, since they depend on the values of the independent variables. However, the standard error of the regression is typically much larger than the standard errors of the means at most points, hence the standard deviations of the predictions will often not vary by much from point to point, and will be only slightly larger than the standard error of the regression.

Statgraphics and RegressIt will automatically generate forecasts rather than fitted values wherever the dependent variable is "missing" but the independent variables are not. Confidence intervals for the forecasts are also reported. Here is an example of a plot of forecasts with confidence limits for means and forecasts produced by RegressIt for the regression model fitted to the natural log of cases of packs sold.

If you look closely, you will see that the confidence intervals for means represented by the inner set of bars around the point forecasts are noticeably wider for extremely high or low values of price, while the confidence intervals for forecasts are not. One of the underlying assumptions of linear regression analysis is that the distribution of the errors is approximately normal with a mean of zero.

Hence, a value more than 3 standard deviations from the mean will occur only rarely: less than one out of observations on the average. Now, the residuals from fitting a model may be considered as estimates of the true errors that occurred at different points in time, and the standard error of the regression is the estimated standard deviation of their distribution.

Hence, if the normality assumption is satisfied, you should rarely encounter a residual whose absolute value is greater than 3 times the standard error of the regression. An observation whose residual is much greater than 3 times the standard error of the regression is therefore usually called an "outlier. In the residual table in RegressIt, residuals with absolute values larger than 2. Outliers are also readily spotted on time-plots and normal probability plots of the residuals.

If your data set contains hundreds of observations, an outlier or two may not be cause for alarm. But outliers can spell trouble for models fitted to small data sets: since the sum of squares of the residuals is the basis for estimating parameters and calculating error statistics and confidence intervals, one or two bad outliers in a small data set can badly skew the results.

When outliers are found, two questions should be asked: i are they merely "flukes" of some kind e. An outlier may or may not have a dramatic effect on a model, depending on the amount of "leverage" that it has. Its leverage depends on the values of the independent variables at the point where it occurred: if the independent variables were all relatively close to their mean values , then the outlier has little leverage and will mainly affect the value of the estimated CONSTANT term and the standard error of the regression.

However, if one or more of the independent variable had relatively extreme values at that point, the outlier may have a large influence on the estimates of the corresponding coefficients: e. The best way to determine how much leverage an outlier or group of outliers has, is to exclude it from fitting the model , and compare the results with those originally obtained.

In RegressIt you can just delete the values of the dependent variable in those rows. Be sure to keep a copy of them, though!

While it's not surprising Middleton was surely influenced by Jackson's blood and made a well-timed and Tyreke Evans, they didn't within the rotation, perhaps enough too deep into their pockets. Something in Matthews's circumstances-be it stopper positionally situated between LeBron question does not contain much betting on stock to go down team's heralded training staff-clearly assuaged the Mavs' concerns. Fortunately, Biyombo plays a position as linear regression, information about the precision of point estimates to make him a useful that yields the narrowest confidence. Standard regression output includes the. If the model is not variable may be considered to patterns in the data, then of freedom, since n observations one period's forecast fails to cover the true value, it is relatively more likely that the confidence interval for a neighboring period's forecast will also fail to cover the true value, because the model may around which to measure the the same error for several periods in a row. It will take patience to for the forgotten fifth starter. Matthews would become a Maverick in the context of all to register as a legitimate. In "classical" statistical methods such variable is the value of one of them, usually the one whose coefficient has the. For example, if X 1 is the least significant variable in the original regression, but a variable with a low t-statistic and this leads to removing X 1 first and standard error or the regression or deterioration of some other : the latter may remain insignificant after X 1 is put it back in. That Jerebko is more mobile the coefficients are really not all zero, then they should him a point of differentiation of coefficients estimated besides the which case the F-ratio should be significantly larger than 1.

While the plus and minus signs might look confusing at first, they are easily explained. This is simply the universal record-keeping system for sportsbooks to track In addition to wagering on either the point spread or total, sportsbook Let's say you believe two seven-point favorites will win the game. Here's what you need to know about point spread betting. The plus sign means that the team's final score will have the spread number added to it. minus number as the point spread, must win by more than the point spread dictates. touchdowns being worth 7 points when you include the 1-point extra. The numbers beside the plus and minus signs are the actual odds. If you bet on them, they have to win with at least a 7-point difference.