Misplaced Pages

#762237

98-407: The RMSD of a sample is the quadratic mean of the differences between the observed values and predicted ones. These deviations are called residuals when the calculations are performed over the data sample that was used for estimation (and are therefore always in reference to an estimate) and are called errors (or prediction errors) when computed out-of-sample (aka on the full set, referencing

196-472: A batch of material from production is of high enough quality to be released to the customer or should be scrapped or reworked due to poor quality. In this case, the batch is the population. Although the population of interest often consists of physical objects, sometimes it is necessary to sample over time, space, or some combination of these dimensions. For instance, an investigation of supermarket staffing could examine checkout line length at various times, or

294-406: A better understanding of human health, or one might study records from people born in 2008 in order to make predictions about people born in 2009. Time spent in making the sampled population and population of concern precise is often well spent because it raises many issues, ambiguities, and questions that would otherwise have been overlooked at this stage. In the most straightforward case, such as

392-399: A block-level city map for initial selections, and then a household-level map of the 100 selected blocks, rather than a household-level map of the whole city. Coefficient of Variation In probability theory and statistics , the coefficient of variation ( CV ), also known as normalized root-mean-square deviation (NRMSD) , percent RMS , and relative standard deviation ( RSD ),

490-416: A continuous range for the independent variable with sparse measurements across each value (e.g., scatter-plot) may be amenable to single CV calculation using a maximum-likelihood estimation approach. In the examples below, we will take the values given as randomly chosen from a larger population of values . In these examples, we will take the values given as the entire population of values . When only

588-435: A fairly accurate indicative result with a 95% confidence interval at a margin of error within 4-5%; ELD reminded the public that sample counts are separate from official results, and only the returning officer will declare the official results once vote counting is complete. Successful statistical practice is based on focused problem definition. In sampling, this includes defining the " population " from which our sample

686-744: A fixed range (e.g. like the Gini coefficient which is constrained to be between 0 and 1). It is, however, more mathematically tractable than the Gini coefficient. Archaeologists often use CV values to compare the degree of standardisation of ancient artefacts. Variation in CVs has been interpreted to indicate different cultural transmission contexts for the adoption of new technologies. Coefficients of variation have also been used to investigate pottery standardisation relating to changes in social organisation. Archaeologists also use several methods for comparing CV values, for example

784-415: A forthcoming election (in advance of the election). These imprecise populations are not amenable to sampling in any of the ways below and to which we could apply statistical theory. As a remedy, we seek a sampling frame which has the property that we can identify every single element and include any in our sample. The most straightforward type of frame is a list of elements of the population (preferably

882-506: A given country will on average produce five men and five women, but any given trial is likely to over represent one sex and underrepresent the other. Systematic and stratified techniques attempt to overcome this problem by "using information about the population" to choose a more "representative" sample. Also, simple random sampling can be cumbersome and tedious when sampling from a large target population. In some cases, investigators are interested in research questions specific to subgroups of

980-413: A given size, all subsets of a sampling frame have an equal probability of being selected. Each element of the frame thus has an equal probability of selection: the frame is not subdivided or partitioned. Furthermore, any given pair of elements has the same chance of selection as any other such pair (and similarly for triples, and so on). This minimizes bias and simplifies analysis of results. In particular,

1078-606: A given street, and interview the first person to answer the door. In any household with more than one occupant, this is a nonprobability sample, because some people are more likely to answer the door (e.g. an unemployed person who spends most of their time at home is more likely to answer than an employed housemate who might be at work when the interviewer calls) and it's not practical to calculate these probabilities. Nonprobability sampling methods include convenience sampling , quota sampling , and purposive sampling . In addition, nonresponse effects may turn any probability design into

SECTION 10

#1733086307763

1176-413: A linear transformation of the form a x + b {\displaystyle ax+b} with b ≠ 0 {\displaystyle b\neq 0} , whereas Kelvins can be converted to Rankines through a transformation of the form a x {\displaystyle ax} . Provided that negative and small positive values of the sample mean occur with negligible frequency,

1274-412: A meaningful zero ( ratio scale ) and hence allow relative comparison of two measurements (i.e., division of one measurement by the other). The coefficient of variation may not have any meaning for data on an interval scale . For example, most temperature scales (e.g., Celsius, Fahrenheit etc.) are interval scales with arbitrary zeros, so the computed coefficient of variation would be different depending on

1372-533: A more complex computational process is required. It has also been noted that CV values are not an ideal index of the certainty of a measurement when the number of replicates varies across samples − in this case standard error in percent is suggested to be superior. If measurements do not have a natural zero point then the CV is not a valid measurement and alternative measures such as the intraclass correlation coefficient are recommended. The coefficient of variation fulfills

1470-410: A nonprobability design if the characteristics of nonresponse are not well understood, since nonresponse effectively modifies each element's probability of being sampled. Within any of the types of frames identified above, a variety of sampling methods can be employed individually or in combination. Factors commonly influencing the choice between these designs include: In a simple random sample (SRS) of

1568-406: A percentage ("%RSD"). The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay . It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R , by economists and investors in economic models , and in psychology / neuroscience . The coefficient of variation (CV) is defined as

1666-469: A percentage, where lower values indicate less residual variance. This is also called Coefficient of Variation or Percent RMS . In many cases, especially for smaller samples, the sample range is likely to be affected by the size of sample which would hamper comparisons. Another possible method to make the RMSD a more useful comparison measure is to divide the RMSD by the interquartile range (IQR). When dividing

1764-445: A perfect fit to the data. In general, a lower RMSD is better than a higher one. However, comparisons across different types of data would be invalid because the measure is dependent on the scale of the numbers used. RMSD is the square root of the average of squared errors. The effect of each error on RMSD is proportional to the size of the squared error; thus larger errors have a disproportionately large effect on RMSD. Consequently, RMSD

1862-451: A population than for the overall population; in such cases, using a stratified sampling approach may be more convenient than aggregating data across groups (though this may potentially be at odds with the previously noted importance of utilizing criterion-relevant strata). Finally, since each stratum is treated as an independent population, different sampling approaches can be applied to different strata, potentially enabling researchers to use

1960-536: A probability proportionate to size sample. This is done by treating each count within the size variable as a single sampling unit. Samples are then identified by selecting at even intervals among these counts within the size variable. This method is sometimes called PPS-sequential or monetary unit sampling in the case of audits or forensic sampling. Example: Suppose we have six schools with populations of 150, 180, 200, 220, 260, and 490 students respectively (total 1500 students), and we want to use student population as

2058-616: A probability sample is the fact that each person's probability is known. When every element in the population does have the same probability of selection, this is known as an 'equal probability of selection' (EPS) design. Such designs are also referred to as 'self-weighting' because all sampled units are given the same weight. Probability sampling includes: simple random sampling , systematic sampling , stratified sampling , probability-proportional-to-size sampling, and cluster or multistage sampling . These various ways of probability sampling have two things in common: Nonprobability sampling

SECTION 20

#1733086307763

2156-439: A relative value rather than an absolute. Comparing the same data set, now in absolute units: Kelvin: [273.15, 283.15, 293.15, 303.15, 313.15] Rankine: [491.67, 509.67, 527.67, 545.67, 563.67] The sample standard deviations are still 15.81 and 28.46, respectively, because the standard deviation is not affected by a constant offset. The coefficients of variation, however, are now both equal to 5.39%. Mathematically speaking,

2254-468: A sample of data from a population is available, the population CV can be estimated using the ratio of the sample standard deviation s {\displaystyle s\,} to the sample mean x ¯ {\displaystyle {\bar {x}}} : But this estimator, when applied to a small or moderately sized sample, tends to be too low: it is a biased estimator . For normally distributed data, an unbiased estimator for

2352-544: A sample of size n is: Many datasets follow an approximately log-normal distribution. In such cases, a more accurate estimate, derived from the properties of the log-normal distribution , is defined as: where s ln {\displaystyle {s_{\ln }}\,} is the sample standard deviation of the data after a natural log transformation. (In the event that measurements are recorded using any other logarithmic base, b, their standard deviation s b {\displaystyle s_{b}\,}

2450-404: A single trip to visit several households in one block, rather than having to drive to a different block for each household. It also means that one does not need a sampling frame listing all elements in the target population. Instead, clusters can be chosen from a cluster-level frame, with an element-level frame created only for the selected clusters. In the example above, the sample only requires

2548-461: A study on endangered penguins might aim to understand their usage of various hunting grounds over time. For the time dimension, the focus may be on periods or discrete occasions. In other cases, the examined 'population' may be even less tangible. For example, Joseph Jagger studied the behaviour of roulette wheels at a casino in Monte Carlo , and used this to identify a biased wheel. In this case,

2646-468: A subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. The subset is meant to reflect the whole population and statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it

2744-437: A true value rather than an estimate). The RMSD serves to aggregate the magnitudes of the errors in predictions for various data points into a single measure of predictive power. RMSD is a measure of accuracy , to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent. RMSD is always non-negative, and a value of 0 (almost never achieved in practice) would indicate

2842-477: A using a skip which ensures jumping between the two sides (any odd-numbered skip). Another drawback of systematic sampling is that even in scenarios where it is more accurate than SRS, its theoretical properties make it difficult to quantify that accuracy. (In the two examples of systematic sampling that are given above, much of the potential sampling error is due to variation between neighbouring houses – but because this method never selects two neighbouring houses,

2940-433: Is s l n {\displaystyle s_{ln}\,} which is of most use in the context of log-normally distributed data. If necessary, this can be derived from an estimate of c v {\displaystyle c_{\rm {v}}\,} or GCV by inverting the corresponding formula. The coefficient of variation is useful because the standard deviation of data must always be understood in

3038-467: Is "everybody in the country, given access to this treatment" – a group that does not yet exist since the program is not yet available to all. The population from which the sample is drawn may not be the same as the population from which information is desired. Often there is a large but not complete overlap between these two groups due to frame issues etc. (see below). Sometimes they may be entirely separate – for instance, one might study rats in order to get

Root mean square deviation - Misplaced Pages Continue

3136-419: Is a standardized measure of dispersion of a probability distribution or frequency distribution . It is defined as the ratio of the standard deviation σ {\displaystyle \sigma } to the mean μ {\displaystyle \mu } (or its absolute value , | μ | {\displaystyle |\mu |} ) , and often expressed as

3234-435: Is a sample of a population with true mean value x 0 {\displaystyle x_{0}} , then the RMSD of the sample is The RMSD of predicted values y ^ t {\displaystyle {\hat {y}}_{t}} for times t of a regression's dependent variable y t , {\displaystyle y_{t},} with variables observed over T times,

3332-410: Is any sampling method where some elements of the population have no chance of selection (these are sometimes referred to as 'out of coverage'/'undercovered'), or where the probability of selection cannot be accurately determined. It involves the selection of elements based on assumptions regarding the population of interest, which forms the criteria for selection. Hence, because the selection of elements

3430-594: Is computed for T different predictions as the square root of the mean of the squares of the deviations: (For regressions on cross-sectional data , the subscript t is replaced by i and T is replaced by n .) In some disciplines, the RMSD is used to compare differences between two things that may vary, neither of which is accepted as the "standard". For example, when measuring the average difference between two time series x 1 , t {\displaystyle x_{1,t}} and x 2 , t {\displaystyle x_{2,t}} ,

3528-448: Is converted to base e using s ln = s b ln ⁡ ( b ) {\displaystyle s_{\ln }=s_{b}\ln(b)\,} , and the formula for c v ^ r a w {\displaystyle {\widehat {cv}}_{\rm {raw}}\,} remains the same. ) This estimate is sometimes referred to as the "geometric CV" (GCV) in order to distinguish it from

3626-421: Is drawn. A population can be defined as including all people or items with the characteristics one wishes to understand. Because there is very rarely enough time or money to gather information from everyone or everything in a population, the goal becomes finding a representative sample (or subset) of that population. Sometimes what defines a population is obvious. For example, a manufacturer needs to decide whether

3724-407: Is eliminated.) However, systematic sampling is especially vulnerable to periodicities in the list. If periodicity is present and the period is a multiple or factor of the interval used, the sample is especially likely to be un representative of the overall population, making the scheme less accurate than simple random sampling. For example, consider a street where the odd-numbered houses are all on

3822-407: Is even, sum only over odd values of i {\displaystyle i} . This is useful, for instance, in the construction of hypothesis tests or confidence intervals . Statistical inference for the coefficient of variation in normally distributed data is often based on McKay's chi-square approximation for the coefficient of variation. Methods for Liu (2012) reviews methods for

3920-449: Is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling , weights can be applied to the data to adjust for the sample design, particularly in stratified sampling . Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling

4018-404: Is nonrandom, nonprobability sampling does not allow the estimation of sampling errors. These conditions give rise to exclusion bias , placing limits on how much information a sample can provide about the population. Information about the relationship between sample and population is limited, making it difficult to extrapolate from the sample to the population. Example: We visit every household in

Root mean square deviation - Misplaced Pages Continue

4116-424: Is often available – for instance, a survey attempting to measure the number of guest-nights spent in hotels might use each hotel's number of rooms as an auxiliary variable. In some cases, an older measurement of the variable of interest can be used as an auxiliary variable when attempting to produce more current estimates. Sometimes it is more cost-effective to select respondents in groups ('clusters'). Sampling

4214-434: Is often clustered by geography, or by time periods. (Nearly all samples are in some sense 'clustered' in time – although this is rarely taken into account in the analysis.) For instance, if surveying households within a city, we might choose to select 100 city blocks and then interview every household within the selected blocks. Clustering can reduce travel and administrative costs. In the example above, an interviewer can make

4312-449: Is sensitive to outliers . The RMSD of an estimator θ ^ {\displaystyle {\hat {\theta }}} with respect to an estimated parameter θ {\displaystyle \theta } is defined as the square root of the mean squared error : For an unbiased estimator , the RMSD is the square root of the variance , known as the standard deviation . If X 1 , ..., X n

4410-429: Is sometimes introduced after the sampling phase in a process called "poststratification". This approach is typically implemented due to a lack of prior knowledge of an appropriate stratifying variable or when the experimenter lacks the necessary information to create a stratifying variable during the sampling phase. Although the method is susceptible to the pitfalls of post hoc approaches, it can provide several benefits in

4508-428: Is taken from each stratum so that the rare target class will be more represented in the sample. The model is then built on this biased sample . The effects of the input variables on the target are often estimated with more precision with the choice-based sample even when a smaller overall sample size is taken, compared to a random sample. The results usually must be adjusted to correct for the oversampling. In some cases

4606-414: Is the k moment about the mean, which are also dimensionless and scale invariant. The variance-to-mean ratio , σ 2 / μ {\displaystyle \sigma ^{2}/\mu } , is another similar ratio, but is not dimensionless, and hence not scale invariant. See Normalization (statistics) for further ratios. In signal processing , particularly image processing ,

4704-469: Is the quantile function . When normalizing by the mean value of the measurements, the term coefficient of variation of the RMSD, CV(RMSD) may be used to avoid ambiguity. This is analogous to the coefficient of variation with the RMSD taking the place of the standard deviation . Some researchers have recommended the use of the mean absolute error (MAE) instead of the root mean square deviation. MAE possesses advantages in interpretability over RMSD. MAE

4802-406: Is the average of the absolute values of the errors. MAE is fundamentally easier to understand than the square root of the average of squared errors. Furthermore, each error influences MAE in direct proportion to the absolute value of the error, which is not the case for RMSD. Sampling (statistics) In statistics , quality assurance , and survey methodology , sampling is the selection of

4900-578: Is used to assess flow uniformity in combustion systems, HVAC systems, ductwork, inlets to fans and filters, air handling units, etc. where performance of the equipment is influenced by the incoming flow distribution. CV measures are often used as quality controls for quantitative laboratory assays . While intra-assay and inter-assay CVs might be assumed to be calculated by simply averaging CV values across CV values for multiple samples within one assay or by averaging multiple inter-assay CV estimates, it has been suggested that these practices are incorrect and that

4998-468: Is widely used for gathering information about a population. Acceptance sampling is used to determine if a production lot of material meets the governing specifications . Random sampling by using lots is an old idea, mentioned several times in the Bible. In 1786, Pierre Simon Laplace estimated the population of France by using a sample, along with ratio estimator . He also computed probabilistic estimates of

SECTION 50

#1733086307763

5096-475: The CV , also referred to as Percent RMS , %RMS , %RMS Uniformity , or Velocity RMS , is a useful determination of flow uniformity for industrial processes. The term is used widely in the design of pollution control equipment, such as electrostatic precipitators (ESPs), selective catalytic reduction (SCR), scrubbers, and similar devices. The Institute of Clean Air Companies (ICAC) references RMS deviation of velocity in

5194-408: The cause system of which the observed population is an outcome. In such cases, sampling theory may treat the observed population as a sample from a larger 'superpopulation'. For example, a researcher might study the success rate of a new 'quit smoking' program on a test group of 100 patients, in order to predict the effects of the program if it were made available nationwide. Here the superpopulation

5292-459: The exponential distribution is often more important than the normal distribution . The standard deviation of an exponential distribution is equal to its mean, so its coefficient of variation is equal to 1. Distributions with CV < 1 (such as an Erlang distribution ) are considered low-variance, while those with CV > 1 (such as a hyper-exponential distribution ) are considered high-variance . Some formulas in these fields are expressed using

5390-2021: The probability distribution of the coefficient of variation for a sample of size n {\displaystyle n} of i.i.d. normal random variables has been shown by Hendricks and Robey to be d F c v = 2 π 1 / 2 Γ ( n − 1 2 ) exp ⁡ ( − n 2 ( σ μ ) 2 ⋅ c v 2 1 + c v 2 ) c v n − 2 ( 1 + c v 2 ) n / 2 ∑ ∑ ′ i = 0 n − 1 ⁡ ( n − 1 ) ! Γ ( n − i 2 ) ( n − 1 − i ) ! i ! ⋅ n i / 2 2 i / 2 ⋅ ( σ μ ) i ⋅ 1 ( 1 + c v 2 ) i / 2 d c v , {\displaystyle \mathrm {d} F_{c_{\rm {v}}}={\frac {2}{\pi ^{1/2}\Gamma {\left({\frac {n-1}{2}}\right)}}}\exp \left(-{\frac {n}{2\left({\frac {\sigma }{\mu }}\right)^{2}}}\cdot {\frac {{c_{\rm {v}}}^{2}}{1+{c_{\rm {v}}}^{2}}}\right){\frac {{c_{\rm {v}}}^{n-2}}{(1+{c_{\rm {v}}}^{2})^{n/2}}}\sideset {}{^{\prime }}\sum _{i=0}^{n-1}{\frac {(n-1)!\,\Gamma \left({\frac {n-i}{2}}\right)}{(n-1-i)!\,i!\,}}\cdot {\frac {n^{i/2}}{2^{i/2}\cdot \left({\frac {\sigma }{\mu }}\right)^{i}}}\cdot {\frac {1}{(1+{c_{\rm {v}}}^{2})^{i/2}}}\,\mathrm {d} c_{\rm {v}},} where

5488-446: The requirements for a measure of economic inequality . If x (with entries x i ) is a list of the values of an economic indicator (e.g. wealth), with x i being the wealth of agent i , then the following requirements are met: c v assumes its minimum value of zero for complete equality (all x i are equal). Its most notable drawback is that it is not bounded from above, so it cannot be normalized to be within

5586-524: The squared coefficient of variation , often abbreviated SCV. In modeling, a variation of the CV is the CV(RMSD). Essentially the CV(RMSD) replaces the standard deviation term with the Root Mean Square Deviation (RMSD) . While many natural processes indeed show a correlation between the average value and the amount of variation around it, accurate sensor devices need to be designed in such a way that

5684-435: The 'population' Jagger wanted to investigate was the overall behaviour of the wheel (i.e. the probability distribution of its results over infinitely many trials), while his 'sample' was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of properties of materials such as the electrical conductivity of copper . This situation often arises when seeking knowledge about

5782-521: The Kelvin scale can be used to compute a valid coefficient of variability. Measurements that are log-normally distributed exhibit stationary CV; in contrast, SD varies depending upon the expected value of measurements. A more robust possibility is the quartile coefficient of dispersion , half the interquartile range ( Q 3 − Q 1 ) / 2 {\displaystyle {(Q_{3}-Q_{1})/2}} divided by

5880-472: The RMSD with the IQR the normalized value gets less sensitive for extreme values in the target variable. with Q 1 = CDF − 1 ( 0.25 ) {\displaystyle Q_{1}={\text{CDF}}^{-1}(0.25)} and Q 3 = CDF − 1 ( 0.75 ) , {\displaystyle Q_{3}={\text{CDF}}^{-1}(0.75),} where CDF

5978-454: The US, the 1936 Literary Digest prediction of a Republican win in the presidential election went badly awry, due to severe bias [1] . More than two million people responded to the study with their names obtained through magazine subscription lists and telephone directories. It was not appreciated that these lists were heavily biased towards Republicans and the resulting sample, though very large,

SECTION 60

#1733086307763

6076-492: The approach best suited (or most cost-effective) for each identified subgroup within the population. There are, however, some potential drawbacks to using stratified sampling. First, identifying strata and implementing such an approach can increase the cost and complexity of sample selection, as well as leading to increased complexity of population estimates. Second, when examining multiple criteria, stratifying variables may be related to some, but not to others, further complicating

6174-449: The average of the quartiles (the midhinge ), ( Q 1 + Q 3 ) / 2 {\displaystyle {(Q_{1}+Q_{3})/2}} . In most cases, a CV is computed for a single independent variable (e.g., a single factory product) with numerous, repeated measures of a dependent variable (e.g., error in the production process). However, data that are linear or even logarithmically non-linear and include

6272-414: The basis for a PPS sample of size three. To do this, we could allocate the first school numbers 1 to 150, the second school 151 to 330 (= 150 + 180), the third school 331 to 530, and so on to the last school (1011 to 1500). We then generate a random start between 1 and 500 (equal to 1500/3) and count through the school populations by multiples of 500. If our random start

6370-443: The coefficient of variation is close to zero, i.e., yielding a constant absolute error over their working range. In actuarial science , the CV is known as unitized risk . In industrial solids processing, CV is particularly important to measure the degree of homogeneity of a powder mixture. Comparing the calculated CV to a specification will allow to define if a sufficient degree of mixing has been reached. In fluid dynamics ,

6468-461: The coefficient of variation is not entirely linear. That is, for a random variable X {\displaystyle X} , the coefficient of variation of a X + b {\displaystyle aX+b} is equal to the coefficient of variation of X {\displaystyle X} only when b = 0 {\displaystyle b=0} . In the above example, Celsius can only be converted to Fahrenheit through

6566-556: The construction of a confidence interval for the coefficient of variation. Notably, Lehmann (1986) derived the sampling distribution for the coefficient of variation using a non-central t-distribution to give an exact method for the construction of the CI. Standardized moments are similar ratios, μ k / σ k {\displaystyle {\mu _{k}}/{\sigma ^{k}}} where μ k {\displaystyle \mu _{k}}

6664-506: The context of the mean of the data. In contrast, the actual value of the CV is independent of the unit in which the measurement has been taken, so it is a dimensionless number . For comparison between data sets with different units or widely different means, one should use the coefficient of variation instead of the standard deviation. The coefficient of variation is also common in applied probability fields such as renewal theory , queueing theory , and reliability theory . In these fields,

6762-437: The criterion in question, instead of availability of the samples). Even if a stratified sampling approach does not lead to increased statistical efficiency, such a tactic will not result in less efficiency than would simple random sampling, provided that each stratum is proportional to the group's size in the population. Third, it is sometimes the case that data are more readily available for individual, pre-existing strata within

6860-420: The design of fabric filters (ICAC document F-7). The guiding principal is that many of these pollution control devices require "uniform flow" entering and through the control zone. This can be related to uniformity of velocity profile, temperature distribution, gas species (such as ammonia for an SCR, or activated carbon injection for mercury absorption), and other flow-related parameters. The Percent RMS also

6958-415: The design, and potentially reducing the utility of the strata. Finally, in some cases (such as designs with a large number of strata, or those with a specified minimum sample size per group), stratified sampling can potentially require a larger sample than would other methods (although in most cases, the required sample size would be no larger than would be required for simple random sampling). Stratification

7056-579: The entire population) with appropriate contact information. For example, in an opinion poll , possible sampling frames include an electoral register and a telephone directory . A probability sample is a sample in which every unit in the population has a chance (greater than zero) of being selected in the sample, and this probability can be accurately determined. The combination of these traits makes it possible to produce unbiased estimates of population totals, by weighting sampled units according to their probability of selection. Example: We want to estimate

7154-406: The error. These were not expressed as modern confidence intervals but as the sample size that would be needed to achieve a particular upper bound on the sampling error with probability 1000/1001. His estimates used Bayes' theorem with a uniform prior probability and assumed that his sample was random. Alexander Ivanovich Chuprov introduced sample surveys to Imperial Russia in the 1870s. In

7252-407: The first set is 15.81/20 = 79%. For the second set (which are the same temperatures) it is 28.46/68 = 42%. If, for example, the data sets are temperature readings from two different sensors (a Celsius sensor and a Fahrenheit sensor) and you want to know which sensor is better by picking the one with the least variance, then you will be misled if you use CV. The problem here is that you have divided by

7350-413: The first to the k th element in the list. A simple example would be to select every 10th name from the telephone directory (an 'every 10th' sample, also referred to as 'sampling with a skip of 10'). As long as the starting point is randomized , systematic sampling is a type of probability sampling . It is easy to implement and the stratification induced can make it efficient, if the variable by which

7448-445: The formula becomes Normalizing the RMSD facilitates the comparison between datasets or models with different scales. Though there is no consistent means of normalization in the literature, common choices are the mean or the range (defined as the maximum value minus the minimum value) of the measured data: This value is commonly referred to as the normalized root mean square deviation or error (NRMSD or NRMSE), and often expressed as

7546-428: The high end and too few from the low end (or vice versa), leading to an unrepresentative sample. Selecting (e.g.) every 10th street number along the street ensures that the sample is spread evenly along the length of the street, representing all of these districts. (If we always start at house #1 and end at #991, the sample is slightly biased towards the low end; by randomly selecting the start between #1 and #10, this bias

7644-403: The list is ordered is correlated with the variable of interest. 'Every 10th' sampling is especially useful for efficient sampling from databases . For example, suppose we wish to sample people from a long street that starts in a poor area (house No. 1) and ends in an expensive district (house No. 1000). A simple random selection of addresses from this street could easily end up with too many from

7742-562: The modified signed-likelihood ratio (MSLR) test for equality of CVs. Comparing coefficients of variation between parameters using relative units can result in differences that may not be real. If we compare the same set of temperatures in Celsius and Fahrenheit (both relative units, where kelvin and Rankine scale are their associated absolute values): Celsius: [0, 10, 20, 30, 40] Fahrenheit: [32, 50, 68, 86, 104] The sample standard deviations are 15.81 and 28.46, respectively. The CV of

7840-403: The north (expensive) side of the road, and the even-numbered houses are all on the south (cheap) side. Under the sampling scheme given above, it is impossible to get a representative sample; either the houses sampled will all be from the odd-numbered, expensive side, or they will all be from the even-numbered, cheap side, unless the researcher has previous knowledge of this bias and avoids it by

7938-478: The population. For example, researchers might be interested in examining whether cognitive ability as a predictor of job performance is equally applicable across racial groups. Simple random sampling cannot accommodate the needs of researchers in this situation, because it does not provide subsamples of the population, and other sampling strategies, such as stratified sampling, can be used instead. Systematic sampling (also known as interval sampling) relies on arranging

8036-458: The ratio of the standard deviation σ {\displaystyle \sigma } to the mean μ {\displaystyle \mu } , C V = σ μ . {\displaystyle CV={\frac {\sigma }{\mu }}.} It shows the extent of variability in relation to the mean of the population. The coefficient of variation should be computed only for data measured on scales that have

8134-405: The right situation. Implementation usually follows a simple random sample. In addition to allowing for stratification on an ancillary variable, poststratification can be used to implement weighting, which can improve the precision of a sample's estimates. Choice-based sampling is one of the stratified sampling strategies. In choice-based sampling, the data are stratified on the target and a sample

8232-403: The sample designer has access to an "auxiliary variable" or "size measure", believed to be correlated to the variable of interest, for each element in the population. These data can be used to improve accuracy in sample design. One option is to use the auxiliary variable as a basis for stratification, as discussed above. Another option is probability proportional to size ('PPS') sampling, in which

8330-418: The sample will not give us any information on that variation.) As described above, systematic sampling is an EPS method, because all elements have the same probability of selection (in the example given, one in ten). It is not 'simple random sampling' because different subsets of the same size have different selection probabilities – e.g. the set {4,14,24,...,994} has a one-in-ten probability of selection, but

8428-439: The sampling of a batch of material from production (acceptance sampling by lots), it would be most desirable to identify and measure every single item in the population and to include any one of them in our sample. However, in the more general case this is not usually possible or practical. There is no way to identify all rats in the set of all rats. Where voting is not compulsory, there is no way to identify which people will vote at

8526-425: The scale used. On the other hand, Kelvin temperature has a meaningful zero, the complete absence of thermal energy, and thus is a ratio scale. In plain language, it is meaningful to say that 20 Kelvin is twice as hot as 10 Kelvin, but only in this scale with a true absolute zero. While a standard deviation (SD) can be measured in Kelvin, Celsius, or Fahrenheit, the value computed is only applicable to that scale. Only

8624-446: The selection probability for each element is set to be proportional to its size measure, up to a maximum of 1. In a simple PPS design, these selection probabilities can then be used as the basis for Poisson sampling . However, this has the drawback of variable sample size, and different portions of the population may still be over- or under-represented due to chance variation in selections. Systematic sampling theory can be used to create

8722-448: The set {4,13,24,34,...} has zero probability of selection. Systematic sampling can also be adapted to a non-EPS approach; for an example, see discussion of PPS samples below. When the population embraces a number of distinct categories, the frame can be organized by these categories into separate "strata." Each stratum is then sampled as an independent sub-population, out of which individual elements can be randomly selected. The ratio of

8820-516: The simple estimate above. However, "geometric coefficient of variation" has also been defined by Kirkwood as: This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of c v {\displaystyle c_{\rm {v}}\,} itself. For many practical purposes (such as sample size determination and calculation of confidence intervals ) it

8918-517: The size of this random selection (or sample) to the size of the population is called a sampling fraction . There are several potential benefits to stratified sampling. First, dividing the population into distinct, independent strata can enable researchers to draw inferences about specific subgroups that may be lost in a more generalized random sample. Second, utilizing a stratified sampling method can lead to more efficient statistical estimates (provided that strata are selected based upon relevance to

9016-429: The study population according to some ordering scheme and then selecting elements at regular intervals through that ordered list. Systematic sampling involves a random start and then proceeds with the selection of every k th element from then onwards. In this case, k =(population size/sample size). It is important that the starting point is not automatically the first in the list, but is instead randomly chosen from within

9114-474: The symbol ∑ ∑ ′ {\textstyle \sideset {}{^{\prime }}\sum } indicates that the summation is over only even values of n − 1 − i {\displaystyle n-1-i} , i.e., if n {\displaystyle n} is odd, sum over even values of i {\displaystyle i} and if n {\displaystyle n}

9212-528: The total income of adults living in a given street. We visit each household in that street, identify all adults living there, and randomly select one adult from each household. (For example, we can allocate each person a random number, generated from a uniform distribution between 0 and 1, and select the person with the highest number in each household). We then interview the selected person and find their income. People living on their own are certain to be selected, so we simply add their income to our estimate of

9310-441: The total. But a person living in a household of two adults has only a one-in-two chance of selection. To reflect this, when we come to such a household, we would count the selected person's income twice towards the total. (The person who is selected from that household can be loosely viewed as also representing the person who isn't selected.) In the above example, not everybody has the same probability of selection; what makes it

9408-416: The variance between individual results within the sample is a good indicator of variance in the overall population, which makes it relatively easy to estimate the accuracy of results. Simple random sampling can be vulnerable to sampling error because the randomness of the selection may result in a sample that does not reflect the makeup of the population. For instance, a simple random sample of ten people from

9506-417: Was 137, we would select the schools which have been allocated numbers 137, 637, and 1137, i.e. the first, fourth, and sixth schools. The PPS approach can improve accuracy for a given sample size by concentrating sample on large elements that have the greatest impact on population estimates. PPS sampling is commonly used for surveys of businesses, where element size varies greatly and auxiliary information

9604-530: Was deeply flawed. Elections in Singapore have adopted this practice since the 2015 election , also known as the sample counts, whereas according to the Elections Department (ELD), their country's election commission, sample counts help reduce speculation and misinformation, while helping election officials to check against the election result for that electoral division. The reported sample counts yield

#762237