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From Wikipedia

Round-off error

For the acrobatic movement, roundoff, seeRoundoff.

A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits). This is a form of quantization error.

When a sequence of calculations subject to rounding error are made, errors may accumulate in certain cases known as ill-conditioned, sometimes to such an extent as to dominate the calculation and make the result meaningless.

Representation error

The error introduced by attempting to represent a number on the computer is called representation error. Some examples:

1/70.0.142 8570.000 000 
ln 2 0.693 147 180 559 945 309 41...  0.693 147 0.000 000 180 559 945 309 41...
log10 2 0.301 029 995 663 981 195 21...  0.3010 0.000 029 995 663 981 195 21...
1.259 921 049 894 873 164 76...  1.25992 0.000 001 049 894 873 164 76...
√ 1.414 213 562 373 095 048 80...  1.41421 0.000 003 562 373 095 048 80...
e 2.718 281 828 459 045 235 36...  2.718 281 828 459 045   0.000 000 000 000 000 235 36...
Ï€ 3.141 592 653 589 793 238 46...  3.141 592 653 589 793 0.000 000 000 000 000 238 46...
Increasing the number of digits allowed in a representation reduces the magnitude of possible round-off errors, but any representation limited to finitely many digits will still cause some degree of round-off error for uncountably many real numbers. This kind of error is unavoidable for conventional representations of numbers, but can be reduced by the use of guard digits.

Double-rounding can increase the round-off error. For example, if the numeral 9.945309 is rounded to two decimal places (9.95) for data entry purposes, and then rounded again to one decimal place (10.0) for display purposes, the apparent round-off error is 0.054691. If the original number was rounded to one decimal place in one step (9.9), the round-off error is only 0.045309.

There are five standard ways of performing the rounding in IEEE standard arithmetic:

  • truncation: simply chop off the remaining digits; also called rounding tozero.
0. ≈ 0.142 (dropping all significant digits after the third).
  • round to nearest: round to the nearest value, with ties broken in one of two ways. The result may round up or round down.
0. ≈ 0.143 (rounding the fourth significant digit. This is rounded up because 8 ≥ 5).
0. ≈ 0.14 (rounding the third significant digit. This is rounded down because 2 < 5).
  • round to −∞: always round to the left on the number line
  • round to +∞: always round to the right on the number line

If the final digit of a decimal number is 5, then it doesn't matter whether it is rounded up or down because a rounding error of the same magnitude will occur. A common rule is to round toward the nearest even number; since 5 is odd, this rule prevents cascading of rounding errors.

Construction Estimating Software

Construction estimating software is computer software designed for contractors to estimate construction costs for a specific project. A contractor will typically use estimating software to estimate his bid price for a project owner, which will ultimately become part of a resulting construction contract. Some architects and engineers may also use estimating softare, but usually only to provide a budgetary cost estimate to an owner prior to construction.


Traditional Estimating Methods

Contractors review a project's plans and specifications to produce a takeoff (a list of item and material quantities needed for the project). This is traditionally done by analyzing the project plans and, utilizing knowledge of required construction methods, producing an itemized list of the project requirements. Then, based on this list, a contractor will tabulate the various resources and costs for every aspect of construction. There are many ways to estimate with so many different programs. These resource costs include labor, equipment, materials, subcontractors, and any other related costs.

The Rise of Spreadsheets

With the advent of computers in business, contractors began using spreadsheet applications like VisiCalc, Lotus 1-2-3, and Microsoft Excel to duplicate the traditional tabular format, while automating redundant mathematical formulas.

As the popularity of spreadsheets has grown, exchanges are starting to appear where professional estimators can buy or sell construction estimating spreadsheets that they have created. see http://www.estimatingtemplates.com for example.

Database Applications Emerge

As more and more contractors came to rely on spreadsheets, and the formulas within the spreadsheets became more complex, spreadsheet errors became more frequent. These were typically formula errors and cell-reference errors. Hard-coded formulas in database applications were originally created to overcome these errors. As these applications became more and more popular over the years, additional features, such as saving data for reuse and trade-specific calculations, have become available.

Many of these software applications are specific to different construction markets, such as residential building, remodeling, masonry, electrical, and heavy construction. Today most contractors use Microsoft Project and Primavera. For example, programs like Sage Timberline Office and MasterBuilder, that are designed for building construction, include libraries and program features for traditional builders. In sharp contrast, programs like HCSS HeavyBid and SharpeSoft Estimator, that are designed for civil construction, include libraries and program features for roadway, utility, and bridge builders.

Like most other business applications, estimating programs typically run on Microsoft Windows-based computers, but some, such as Turtlesoft Goldenseal, will also run on Macintosh computers.

Typical Software Features

  • Item or Activity List: All estimating software applications will include a main project window that outlines the various items or activities that will be required to complete the specified project. More advanced programs are capable of breaking an item up into subtasks, or sublevels. An outline view of all of the top-level and sub-level items provides a quick and easy way to view and navigate through the project.
  • Resource Costs: Resources consist of labor, equipment, materials, subcontractors, trucking, and any other cost detail items. Labor and equipment costs are internal crew costs, whereas all other resource costs are received from vendors, such as material suppliers, subcontractors, and trucking companies. Labor costs are usually calculated from wages, benefits, burden, and workers compensation. Equipment costs are calculated from purchase price, taxes, fuel consumption, and other operating expenses.
  • Item or Activity Detail: The detail to each item includes all of the resources required to complete each activity, as well as their associated costs. Production rates will automatically determine required crew costs.
  • Calculations: Most estimating programs have built-in calculations ranging from simple length, area, and volume calculations to complex industry-specific calculations, such as electrical calculations, utility trench calculations, and earthwork cut and fill calculations.
  • Markups: Every program will allow for cost mark-ups ranging from flat overall mark-ups to resource-specific mark-ups, mark-ups for general administrative costs, and bonding costs.
  • Detailed Overhead: Indirect costs, such as permits, fees, and any other overall project costs can be spread to project items.
  • Closeout Window: Many estimating programs include a screen for manually adjusting bid prices from their calculated values.
  • Reporting: Project reports typically include proposals, detail reports, cost breakdown reports, and various charts and graphs.
  • Exporting: Most software programs can export project data to other applications, such as spreadsheets, accounting software, and project management software.
  • Job History: Storing past projects is a standard feature in most estimating programs.

Point estimation

In statistics, point estimation involves the use of sampledata to calculate a single value (known as a statistic) which is to serve as a "best guess" for an unknown (fixed or random) population parameter.

More formally, it is the application of a point estimator to the data.

In general, point estimation should be contrasted with interval estimation.

Point estimation should be contrasted with general Bayesian methods of estimation, where the goal is usually to compute (perhaps to an approximation) the posterior distributions of parameters and other quantities of interest. The contrast here is between estimating a single point (point estimation), versus estimating a weighted set of points (a probability density function). However, where appropriate, Bayesian methodology can include the calculation of point estimates, either as the expectation or median of the posterior distribution or as the mode of this distribution.

In a purely frequentist context (as opposed to Bayesian), point estimation should be contrasted with the specific interval estimation calculation of confidence intervals.

Routes to deriving point estimates directly

Routes to deriving point estimates via Bayesian Analysis

Properties of Point estimates

Margin of error - Wikipedia, the free encyclopedia

The true p percent confidence interval is the interval [a, ... minus 1.96 standard errors is a 95% confidence interval, and a 99% confidence interval runs .... any statistical calculator may be used to calculate the probability that a ...

From Yahoo Answers

Question:I've fitted a set of data with a non-linear equation. Now i'm trying to find the 95% confidence interval for a particular point. Is calculating the standard error of the estimate the right approach?

Answers:Yes, if you can calculate the standard error you are almost home. It doesn't matter if the model is linear or not. The principle that the 95% confidence interval is roughly the estimate of the parameter +/- 2SD depends on the assumption that the error follows a normal distribution. This is approximately true for most models but sometimes you must apply the formula to the log of the estimate rather than to the estimate itself. For example, if the estimate is 1 and the SD is 0.5 and the parameter is known to be positive, you must look at the log of the parameter instead.

Question:The circumference of a sphere was measured to be 70 cm with a possible error of 0.9 cm. Use differentials to estimate the maximum error in the calculated surface area. _____? Estimate the relative error in the calculated surface area. ______? Hint: The circumference of a sphere of radius r is C = 2 \pi r, and its surface area is A = (4pi)(r^2). Eliminate r first!

Answers:S=4pir^2 so dS = 8pir*dr but 2pi r= C and 2pidr =dC so dS = 4 Cdr and dr dC/2pi so dS= 2/pi C*dC So the maximum error is dS=2/pi*70*0.9cm^2 =40.11 cm^2 Relative error = dS/S = 40.11/480 =8.36%

Question:Calculate the percent error made over one mile of distance by the "5 second rule" for estimating the distance from a lightning strike for each of the following temperatures. The "5 second rule" says that for every 5 seconds between seeing a lightning strike and hearing the associated sound, the lightning is 1 mile distant. Assume 1 mile equals 1610 meters. (a) 42 C _________% (b) 18 C _________%

Answers:Find the velocity of sound c for each case. You can use the approximate formula c = 331.3+0.606*T, where T is temperature in deg C. See ref. Then multiply c by 5 to get the number of meters sound travels in 5 s. Comparing the result to 1610 m allows you to find the error: Error % = 100*(1610/c-1).

Question:Market researchers use the number of sentences per advertisement as a measure of readability for magazine advertisements. Research shows that the population standard deviation is 6.0 sentences per advertisement. (1)A random sample of 54 advertisements had a sample mean of 12.4 sentences. Find a 90% confidence interval for the population mean number of sentences in advertisements. (2)What size sample, n, should be used to have 90% confidence that the sample mean is within 1 sentence of the population mean? (3)Suppose a 90% confidence interval for the mean number of sentences is computed from a sample of 54 advertisements. Which of the following would produce a confidence interval with a smaller margin of error? a)Using a confidence level of 95% b)Using the sample standard deviation value instead of the population standard deviation. c)Using a sample of 100 advertisements. d)None of the above will produce a smaller margin of error

Answers:Since we are not given any raw data, we must assume normality and use the population standard deviation. The 90% CI for the population mean is given by: (xbar-z(.05)*sigma/sqrt(n), xbar+z(.05)*sigma/sqrt(n)) Here we are told xbar = 12.4, sigma = 6, and n = 54. Using a table of normal probabilities, we find z(.05) = 1.645. Simple calculation shows the CI is: (11.0569, 13.7431) For part (2), we want our margin or error to be 1 with a 90% confidence level. Our population standard deviation is still 6 and z(.05) is still 1.645. Thus, we want: 1.645*6/sqrt(n) = 1 9.87/sqrt(n) = 1 sqrt(n) = 9.87 n = 97.4169 Round up to get n = 98. For your additional comments: Increasing the confidence level also increases the margin of error--not (a). The sample standard deviation could be higher or lower than the population standard deviation, so it could increase or decrease the margin of error, but there is no guarantee--not (b). Increasing the sample size to 100 makes for a better estimate of the mean and would definitely decrease the margin of error. Thus, (c) is the correct choice.

From Youtube

Standard error of estimate (SEE) :A linear regression gives us a best-fit line for a scatterplot of data. The standard error of estimate (SEE) is one of the metrics that tells us about the fit of the line to the data. The SEE is the standard deviation of the errors (or residuals)

Excel Statistics 75: Point Estimates & Sample Error :Topics for: 1.See how to take a sample from a data set using the INT, RAND, and VLOOKUP functions 2.Calculate our Point estimates: Mean, Standard Deviation and Proportions 3.Calculate Sampling Error for our Point Estimates 4.Calculate Sample Proportions using COUNTIF function 5. Busn 210 Business Statistical Using Excel Highline Community College taught by Mike Gel excelisfun Girvin