Mathematics
Standard Error
The standard error measures the variability of sample means around the true population mean. It quantifies the precision of the sample mean as an estimate of the population mean. A smaller standard error indicates that the sample mean is likely closer to the true population mean, while a larger standard error suggests greater uncertainty in the estimate.
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9 Key excerpts on "Standard Error"
eBook - PDFDealing with Data
The Commonwealth and International Library: Physics Division
- Arthur J. Lyon, W. Ashhurst(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
an estimate of some parameter of a population—such as the mean, or the median, or the standard deviation—which can be derived from the data contained in a sample or set of samples drawn from the population. By regarding a set of readings as a random sample from a hypothetical infinite population of similar readings, we can use the definition to estimate the Standard Error of the mean, which in such cases will usually be adopted as the result of the measurement. Measurement procedures are, however, not always as simple as this, and involve special considerations which do not arise in ordinary statistical estimation. Standard ErrorS 111 It is therefore desirable to recast the definition of Standard Error so that it applies specifically to measurements. In doing so we shall assume, as in § 21.1, that the measurement consid-ered is carried out in accordance with some definite prescribed procedure under clearly specified conditions. The prescribed procedure is also assumed to include any corrections to be applied or special conventions or methods of analysis to be adopted. The Standard Error of an experimental result is then defined as the standard deviation of the random errors which would occur in a series of similar results obtained by repeating the measurement, in accordance with the prescribed procedure, an indefinite number of times. This definition is in essence identical with the earlier one except that experimental result replaces estimate, and no explic-it reference is made to samples from a population. It is appli-cable to any measurement—whether a single reading, a mean of several readings, or a determination based on a number of subsidiary measurements, and using a mathematical for-mula including, perhaps, such parameters as the gradient of a fitted straight line. 24.1. Standard Error and replication error A distinction was drawn in § 8.1 between the replication error and the occasional bias which are both components of the total random experimental error.
- John Townend(Author)
- 2012(Publication Date)
- Wiley(Publisher)
o C Ql ::J CT Ql t.t sample mean ± Standard Error Population mean s.e. s.e. s.e.is.e . Figure 2 Frequency distribution for the means of a series of similar random samples taken from a population. Most means lie within one Standard Error (s.e.) of the population mean. Also, given a particular sample mean, in most cases the population mean will lie within one Standard Error of it However, in a practical experiment or survey, we usually only take one sample of individuals from a population, so we cannot calculate the stan- dard deviation of the sample means using the same formula that we would use to calculate the standard deviation of a set of individual measurements. Instead we use the relationship standard deviation of the individuals III the sample Standard Error = -----------;=====--------=--- J sample size For example: Sample values 8,12,10,7,7,11,8 mean = 9.0 standard deviation = 2.0 sample size = 7 Standard Error = 2.o;j7 = 0.76 2.5 The basis of statistical tests 19 The Standard Error gives the range in which we can be approximately 68% confident that the population mean really lies. In this case there is approxi- mately a 68% chance that the true population mean lies in the range 9.0 ± 0.76, i.e. between 8.24 and 9.76. Even if the population mean is really outside this range, it will probably not be far outside. It is possible to calculate ranges for other degrees of certainty. People sometimes quote the range in which we can be 95% confident that the true population mean lies. This is called the 95% confidence interval: 95% CI = Standard Error x tdf.o.os The value idLO.OS is called a t-value and must be obtained from t-tables or a computer. The tables have rows for different numbers of degrees offreedom (df). When using the formula above, the rule for calculating the df is df = sample size - 1 In the tables.
eBook - ePubMedical Statistics
A Textbook for the Health Sciences
- Stephen J. Walters, Michael J. Campbell, David Machin(Authors)
- 2020(Publication Date)
- Wiley-Blackwell(Publisher)
Figure 5.2 clearly shows that the spread or variability of the sample means reduces as the sample size increases. In fact, in turns out that sample means have the following properties.Properties of the Distribution of Sample Means
The mean of all the sample means will be the same as the population mean. The standard deviation of all possible sample means is known as the Standard Error (SE) of the mean or SEM. Given a large enough sample size, the distribution of sample means, will be roughly Normal regardless of the distribution of the variable.The Standard Error or variability of the sampling distribution of the mean is measured by the standard deviation of the estimates. If we know the population standard deviation, σ, then the Standard Error of the mean is given by . In reality, an investigator will only complete a study once (although it may be repeated for confirmatory purposes by others) so this single study provides a single sample mean, , and this is our best (and only) estimate of μ. The same sample also provides s, the standard deviation of the observations, as an estimate of σ. So, with a single study, the investigator can then estimate the standard deviation of the distribution of the means by without having to repeat the study at all.Properties of Standard Errors
The Standard Error is a measure of the precision of a sample estimate. It provides a measure of how far from the true value in the population the sample estimate is likely to be. All Standard Errors have the following interpretation:- A large Standard Error indicates that the estimate is imprecise.
- A small Standard Error indicates that the estimate is precise.
- The Standard Error is reduced, that is, we obtain a more precise estimate, if the size of the sample is increased.
Worked Example – Standard Error of a Mean – Birthweight of Preterm Infants
Simpson (2004 ) reported the birthweights of 98 infants who were born prematurely, for which n = 98, = 1.31 kg, s = 0.42 kg and the SEM often written SE(
eBook - PDF- L S Blake(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
1/30 Mathematics and statistics frequently saves effort. However, this method is not recom-mended for use on computers because of the danger of loss of accuracy when n is large and x, has several significant figures. 1.12.1.5 Coefficient of variation The coefficient of variation is the standard deviation expressed as a percentage of the mean. This is useful for dealing with properties whose standard deviation rises in proportion to the mean, for instance the strengths of concrete as measured by compressive tests on cubes. 1.12.1.6 Standard Error The Standard Error is the standard deviation of the mean (or of any other statistic). If in repeated samples of size n from a population the sample means are calculated, the standard deviation calculated from these means is expected to have a value: Sm = alvn (1.71) where σ is the standard deviation of the population. An important result is that whatever the distribution of the parent population (normal or not) the distribution of the sample mean tends rapidly to normal form as the sample size increases. 1.13 Samples and population 1.13.1 Representations 1.13.1.1 Frequency The number of observations having values between two speci-fied limits. It is often convenient to group observations by dividing the range over which they extend into a number of small, equal, intervals. The number of observations falling in each interval is then the frequency for that interval. This allows a convenient representation of the information by means of a histogram. These parameters of the distribution are estimated by the sample mean JC and standard deviation s. It has been found that a great many frequency distributions met with in practice fit quite closely to the normal distribution. However, one should beware of thinking that there is any law which says that this shall be so; it is simply a matter of experience.
eBook - PDF- L S Blake(Author)
- 1994(Publication Date)
- CRC Press(Publisher)
1/30 Mathematics and statistics frequently saves effort. However, this method is not recom- mended for use on computers because of the danger of loss of accuracy when n is large and x~ has several significant figures. 1.12.1.5 Coe~cient of variation The coefficient of variation is the standard deviation expressed as a percentage of the mean. This is useful for dealing with properties whose standard deviation rises in proportion to the mean, for instance the strengths of concrete as measured by compressive tests on cubes. 1.12.1.6 Standard Error The Standard Error is the standard deviation of the mean (or of any other statistic). If in repeated samples of size n from a population the sample means are calculated, the standard deviation calculated from these means is expected to have a value: Sm=a/,dn (1.71) where tr is the standard deviation of the population. An important result is that whatever the distribution of the parent population (normal or not) the distribution of the sample mean tends rapidly to normal form as the sample size increases. 1.13 Samples and population 1.13.1 Representations 1.13.1.1 Frequency The number of observations having values between two speci- fied limits. It is often convenient to group observations by dividing the range over which they extend into a number of small, equal, intervals. The number of observations falling in each interval is then the frequency for that interval. This allows a convenient representation of the information by means of a histogram. 1.13.1.2 Histogram or bar chart A diagram in which the observations are represented by rec- tangles or bars with one side equal to the interval over which the observations occurred and the other equal to the frequency of occurrence of observations within that range (Figure 1.45). 1.13.1.3 Distribution curve The result of refining a histogram by reducing the size of the intervals and correspondingly increasing the total number of observations.
eBook - PDFUncertainty Analysis for Engineers and Scientists
A Practical Guide
- Faith A. Morrison(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
For the distribution under considera- tion ¯ w = 0, and thus we obtain σ 2 = e R −e R 1 2e R w 2 dw = 1 2e R w 3 3 e R −e R = 1 6e R e 3 R + e 3 R = 1 3 e 2 R Variance of a rectangular distribution of half-width e R σ 2 = e 2 R 3 (3.11) The rectangular distribution in Equation 3.8 is the sampling distribution of reading error. The Standard Error is defined as the standard deviation of the sampling distribution; therefore, the standard reading error is Standard reading error (rectangular distribution) e s,reading = e R √ 3 (3.12) With this result, we now have a standardized quantity by which we can report reading error. The significance of this is that we used the same mathematics 122 3 Reading Error here as we used for standard replicate error: Standard Error is standard deviation of the sampling distribution of the quantity. Thus, the standard reading error in Equation 3.12 may appropriately be combined with standard replicate error and with other similarly standardized errors to determine combined standard uncertainty (we combine Standard Errors by following how variances of independent stochastic variables combine, in quadrature [21]). We discuss the statistical reasoning behind combining Standard Errors in the next section. We can practice with our new reading-error tools by revisiting Example 3.2. In that example we were presented with a device with a fluctuating signal. Fluctuations are part of reading error, and with the framework just developed, we are now better able to address the issues raised by fluctuations. Example 3.6: Error on a single fluctuating measurement of electric current, revisited. A digital multimeter (Fluke Model 179, DMM) is used to measure the electric current in an apparatus, and for a particular measurement the signal displayed by the DMM fluctuates between 14.5 mA and 14.8 mA. What value should we report for the electric current? We wish to use our value in a formal report.
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
sample standard deviation 5 s 5 Ï s 2 5 Î SS n 2 1 (4.8) Sample variance is represented by the symbol s 2 and equals the mean squared dis-tance from the mean. Sample variance is obtained by dividing the sum of squares by n 2 1. Sample standard deviation is represented by the symbol s and equal the square root of the sample variance. Notice that the sample formulas divide by n 2 1 unlike the population formulas, which divide by N (see Equations 4.3 and 4.4). This is the adjustment that is necessary to correct for the bias in sample variability. The effect of the adjustment is to increase the value you will obtain. Dividing by a smaller number ( n 2 1 instead of n ) produces a larger result and makes sample variance an accurate and unbiased estimator of population variance. The following example demonstrates the calculation of variance and standard deviation for a sample. We have selected a sample of n 5 8 scores from a population. The scores are 4, 6, 5, 11, 7, 9, 7, 3. The frequency distribution histogram for this sample is shown in Figure 4.5. Before we begin any calculations, you should be able to look at the sample distribution and make a preliminary estimate of the outcome. Remember that standard deviation measures the standard distance from the mean. For this sample the mean is M 5 52 8 5 6.5. The scores closest to the mean are X 5 6 and X 5 7, both of which are exactly 0.50 points away. The score farthest from the mean is X 5 11, which is 4.50 points away. With the smallest dis-tance from the mean equal to 0.50 and the largest distance equal to 4.50, we should obtain a standard distance somewhere between 0.50 and 4.50, probably around 2.5. D E FINITIO N S E X A M P L E 4.6 Remember, sample vari-ability tends to under-estimate population variability unless some correction is made. SECTION 4.4 | Measuring Standard Deviation and Variance for a Sample 113 Copyright 2017 Cengage Learning.
eBook - PDF- Ned Freed, Stacey Jones, Timothy Bergquist(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
FIGURE 7.10 Standard Error vs. Margin of Error The margin of error is the Standard Error multiplied by the appropriate z-score. z Standard Error Margin of Error x x Standard Error and Margin of Error The National Center for Health Statistics reports that the average weight in a sample of US men aged 20 to 74 was 191 pounds. (In 1960, the average was 166.3 pounds.) ( Source: cdc.gov/nchs/) If the sample size is 1200 and the population standard deviation is 18.2 pounds, calculate the a. Standard Error of the sampling distribution of the sample mean here. b. margin of error for a 95% confidence interval estimate of the population mean weight. Solution: a. Standard Error 18.2 21200 .525 pounds. b. Margin of error 1.96 18.2 21200 1.03 pounds. DEMONSTRATION EXERCISE 7.5 25. You want to build a 90% confidence interval estimate of a population mean. You take a sample of size 100, and find a sample mean of 1300. If the population standard deviation is 25, calculate the a. Standard Error of the sampling distribution of the sample mean. b. margin of error for your interval. 26. Refer to Exercise 19. There you selected a simple ran- dom sample of 49 units from a large shipment and found that the sample average breaking strength was 814 pounds. Assuming that the standard deviation of the breaking strengths for the population of units in the shipment is known to be 35 pounds, calculate the a. Standard Error of the sampling distribution of the sample mean that could be used here to estimate the population mean breaking strength. b. margin of error in a 95% confidence interval esti- mate of the mean breaking strength for all the units in the shipment. 27. The average annual expense for groceries in a 2012 random sample of 600 US households is $8562. If the standard deviation of grocery expenses in the popu- lation of US households is $1230, compute the a.
eBook - PDFAn Introduction to Uncertainty in Measurement
Using the GUM (Guide to the Expression of Uncertainty in Measurement)
- L. Kirkup, R. B. Frenkel(Authors)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
4 Although the divisor in equation (5.8) is n − 1, the summation is over all n terms as indicated in the numerator of equation (5.8). 5 The smaller the sample size n , the larger the positive correlation between ¯ x and any x i ( i = 1 , 2 , . . . , n ). 6 When the values are very similar except for several least-significant digits, this simple formula may give serious round-off errors. It is better in such cases to take differences from the mean of the set. For example, if the set consists of the three similar values 1000.013, 1000.021 and 1000.002 with mean 1000.012, the differences from the mean are + 0 . 001, + 0 . 009 and − 0 . 010 (summing to zero, as a check). The variance of these three differences is the same as the variance of the three original values, but the simple formula can now safely be used on the differences. Alternatively, the common amount ‘1000’ can be subtracted from each value, leaving + 0 . 013 , + 0 . 021 and + 0 . 002 , and the simple formula can equally safely be used on these three values. 5 .1 Sampling from a population 57 Table 5.1. V alues of p H of river water pH 6.8 7.3 6.9 6.9 7.2 7.0 7.1 5.1.4 The standard deviation of a sample and the standard deviation of the population The standard deviation, s b , of a sample is the square root of equation (5.4): s b = ∑ n i = 1 ( x i − ¯ x ) 2 n . (5.12) We can now define the estimated standard deviation, s , of the population as the square root of the unbiased estimate, s 2 , of the population variance, σ 2 : s = ∑ n i = 1 ( x i − ¯ x ) 2 n − 1 . (5.13) This is only an approximately unbiased estimate of the population standard devia-tion. Although E ( s 2 ) = σ 2 as in equation (5.9), it does not follow that E ( s ) = σ . However, the standard deviation as defined in equation (5.13) is the accepted mea-sure of the amount of variation of some quantity in its population, as inferred from a sample.
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