random error

(noun)

an error which is a combination of results both higher and lower than the desired measurement; precision error

Related Terms

  • systematic error
  • Accuracy
  • Precision

Examples of random error in the following topics:

  • Chance Error

    • Random, or chance, errors are errors that are a combination of results both higher and lower than the desired measurement.
    • Uncertainties are measures of random errors.
    • Random error is due to factors which we cannot (or do not) control.
    • In this case, there is more systematic error than random error.
    • In this case, there is more random error than systematic error.
  • Chance Error and Bias

    • In sampling, there are two main types of error: systematic errors (or biases) and random errors (or chance errors).
    • Of course, this is not possible, and the error that is associated with the unpredictable variation in the sample is called random, or chance, error.
    • Random error always exists.
    • The size of the random error, however, can generally be controlled by taking a large enough random sample from the population.
    • If the observations are collected from a random sample, statistical theory provides probabilistic estimates of the likely size of the error for a particular statistic or estimator.
  • Bias

    • While conducting measurements in experiments, there are generally two different types of errors: random (or chance) errors and systematic (or biased) errors.
    • To better understand the outcome of experimental data, an estimate of the size of the systematic errors compared to the random errors should be considered.
    • If it is within the margin of error for the random errors, then it is most likely that the systematic errors are smaller than the random errors.
    • In this case, there is more random error than systematic error.
    • In this case, there is more systematic error than random error.
  • Caveat Emptor and the Gallup Poll

    • The margin of error statistic expresses the amount of random sampling error in a survey's results.
    • For a simple random sample from a large population, the maximum margin of error is a simple re-expression of the sample size $n$.
    • As an example of the above, a random sample of size 400 will give a margin of error, at a 95% confidence level, of $\frac{0.98}{20}$ or 0.049 (just under 5%).
    • A random sample of size 1,600 will give a margin of error of $\frac{0.98}{40}$, or 0.0245 (just under 2.5%).
    • A random sample of size 10,000 will give a margin of error at the 95% confidence level of $\frac{0.98}{100}$, or 0.0098 - just under 1%.
  • Which Standard Deviation (SE)?

    • Although they are often used interchangeably, the standard deviation and the standard error are slightly different.
    • The standard error is the standard deviation of the sampling distribution of a statistic.
    • Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time.
    • However, the mean and standard deviation are descriptive statistics, whereas the mean and standard error describes bounds on a random sampling process.
    • Standard error should decrease with larger sample sizes, as the estimate of the population mean improves.
  • Expected Value and Standard Error

    • Expected value and standard error can provide useful information about the data recorded in an experiment.
    • In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on.
    • The weights used in computing this average are probabilities in the case of a discrete random variable, or values of a probability density function in the case of a continuous random variable.
    • The standard error is the standard deviation of the sampling distribution of a statistic.
    • Solve for the standard error of a sum and the expected value of a random variable
  • Standard Error

    • This is due to the fact that the standard error of the mean is a biased estimator of the population standard error.
    • However, while the mean and standard deviation are descriptive statistics, the mean and standard error describe bounds on a random sampling process.
    • The relative standard error (RSE) is simply the standard error divided by the mean and expressed as a percentage.
    • If one survey has a standard error of $10,000 and the other has a standard error of $5,000, then the relative standard errors are 20% and 10% respectively.
    • Paraphrase standard error, standard error of the mean, standard error correction and relative standard error.
  • Computing R.M.S. Error

    • The differences between values occur because of randomness or because the estimator doesn't account for information that could produce a more accurate estimate.
    • Root-mean-square error serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power.
    • RMS error is the square root of mean squared error (MSE), which is a risk function corresponding to the expected value of the squared error loss or quadratic loss.
    • RMS error is simply the square root of the resulting MSE quantity.
    • $\displaystyle \sqrt { \frac { { \left( \text{error}\ 1 \right) }^{ 2 }+{ \left(\text{error}\ 2 \right) }^{ 2 }+\cdots +{ \left( \text{error n} \right) }^{ 2 } }{ n } }$.
  • Examining the standard error formula

    • The formula for the standard error of the difference in two means is similar to the formula for other standard errors.
    • Recall that the standard error of a single mean, $\bar{x}_1$, can be approximated by
    • The standard error of the difference of two sample means can be constructed from the standard errors of the separate sample means:
    • 5.14: The standard error squared represents the variance of the estimate.
    • If X and Y are two random variables with variances $\sigma^2_{x_1}$ and $\sigma^2_y$, then the variance of X−Y is $\sigma^2_x+\sigma^2_y$.
  • Standard error of the mean

    • From the random sample represented in run10Samp, we guessed the average time it takes to run 10 miles is 95.61 minutes.
    • Suppose we take another random sample of 100 individuals and take its mean: 95.30 minutes.
    • A reliable method to ensure sample observations are independent is to conduct a simple random sample consisting of less than 10% of the population.
    • Because the sample is simple random and consists of less than 10% of the population, the observations are independent.
    • 4.3: (a) Consider two random samples: one of size 10 and one of size 1000.
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