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For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above and below the actual value.
The standard error (SE)[1] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution[2] or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).[1]
The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.
Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size.[1] In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.
In regression analysis, the term «standard error» refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).
Standard error of the sample mean[edit]
Exact value[edit]
Suppose a statistically independent sample of 
observations 
is taken from a statistical population with a standard deviation of 
. The mean value calculated from the sample, 
, will have an associated standard error on the mean, 
, given by:[1]
.
Practically this tells us that when trying to estimate the value of a population mean, due to the factor 
, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations.
Estimate[edit]
The standard deviation of the population being sampled is seldom known. Therefore, the standard error of the mean is usually estimated by replacing with the sample standard deviation 
instead:
- .
As this is only an estimator for the true «standard error», it is common to see other notations here such as:
- or alternately .
A common source of confusion occurs when failing to distinguish clearly between:
Accuracy of the estimator[edit]
When the sample size is small, using the standard deviation of the sample instead of the true standard deviation of the population will tend to systematically underestimate the population standard deviation, and therefore also the standard error. With n = 2, the underestimate is about 25%, but for n = 6, the underestimate is only 5%. Gurland and Tripathi (1971) provide a correction and equation for this effect.[3] Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20.[4] See unbiased estimation of standard deviation for further discussion.
Derivation[edit]
The standard error on the mean may be derived from the variance of a sum of independent random variables,[5] given the definition of variance and some simple properties thereof. If are independent samples from a population with mean and standard deviation , then we can define the total
which due to the Bienaymé formula, will have variance
where we’ve approximated the standard deviations, i.e., the uncertainties, of the measurements themselves with the best value for the standard deviation of the population. The mean of these measurements is simply given by
- .
The variance of the mean is then
The standard error is, by definition, the standard deviation of which is simply the square root of the variance:
- .
For correlated random variables the sample variance needs to be computed according to the Markov chain central limit theorem.
Independent and identically distributed random variables with random sample size[edit]
There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size 
is a random variable whose variation adds to the variation of 
such that,
- [6]
If has a Poisson distribution, then 
with estimator 
. Hence the estimator of 
becomes 
, leading the following formula for standard error:
(since the standard deviation is the square root of the variance)
Student approximation when σ value is unknown[edit]
In many practical applications, the true value of σ is unknown. As a result, we need to use a distribution that takes into account that spread of possible σ’s.
When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. The standard error is the standard deviation of the Student t-distribution. T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t-distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation «s» instead of σ, and we could use this value to calculate confidence intervals.
Note: The Student’s probability distribution is approximated well by the Gaussian distribution when the sample size is over 100. For such samples one can use the latter distribution, which is much simpler.
Assumptions and usage[edit]
An example of how 
is used is to make confidence intervals of the unknown population mean. If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where is equal to the sample mean, is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution:
- Upper 95% limit and
- Lower 95% limit
In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. In other words, it is the actual or estimated standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE.
Standard errors provide simple measures of uncertainty in a value and are often used because:
- in many cases, if the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated;
- when the probability distribution of the value is known, it can be used to calculate an exact confidence interval;
- when the probability distribution is unknown, Chebyshev’s or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and
- as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.
Standard error of mean versus standard deviation[edit]
In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.[7]
Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean.[8] If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.
Extensions[edit]
Finite population correction (FPC)[edit]
The formula given above for the standard error assumes that the population is infinite. Nonetheless, it is often used for finite populations when people are interested in measuring the process that created the existing finite population (this is called an analytic study). Though the above formula is not exactly correct when the population is finite, the difference between the finite- and infinite-population versions will be small when sampling fraction is small (e.g. a small proportion of a finite population is studied). In this case people often do not correct for the finite population, essentially treating it as an «approximately infinite» population.
If one is interested in measuring an existing finite population that will not change over time, then it is necessary to adjust for the population size (called an enumerative study). When the sampling fraction (often termed f) is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a »finite population correction» (a.k.a.: FPC):[9]
[10]
which, for large N:
to account for the added precision gained by sampling close to a larger percentage of the population. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N.
This happens in survey methodology when sampling without replacement. If sampling with replacement, then FPC does not come into play.
Correction for correlation in the sample[edit]
Expected error in the mean of A for a sample of n data points with sample bias coefficient ρ. The unbiased standard error plots as the ρ = 0 diagonal line with log-log slope −½.
If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor f:
where the sample bias coefficient ρ is the widely used Prais–Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover, this formula works for positive and negative ρ alike.[11] See also unbiased estimation of standard deviation for more discussion.
See also[edit]
- Illustration of the central limit theorem
- Margin of error
- Probable error
- Standard error of the weighted mean
- Sample mean and sample covariance
- Standard error of the median
- Variance
References[edit]
- ^ a b c d Altman, Douglas G; Bland, J Martin (2005-10-15). «Standard deviations and standard errors». BMJ: British Medical Journal. 331 (7521): 903. doi:10.1136/bmj.331.7521.903. ISSN 0959-8138. PMC 1255808. PMID 16223828.
- ^ Everitt, B. S. (2003). The Cambridge Dictionary of Statistics. CUP. ISBN 978-0-521-81099-9.
- ^ Gurland, J; Tripathi RC (1971). «A simple approximation for unbiased estimation of the standard deviation». American Statistician. 25 (4): 30–32. doi:10.2307/2682923. JSTOR 2682923.
- ^ Sokal; Rohlf (1981). Biometry: Principles and Practice of Statistics in Biological Research (2nd ed.). p. 53. ISBN 978-0-7167-1254-1.
- ^ Hutchinson, T. P. (1993). Essentials of Statistical Methods, in 41 pages. Adelaide: Rumsby. ISBN 978-0-646-12621-0.
- ^ Cornell, J R, and Benjamin, C A, Probability, Statistics, and Decisions for Civil Engineers, McGraw-Hill, NY, 1970, ISBN 0486796094, pp. 178–9.
- ^ Barde, M. (2012). «What to use to express the variability of data: Standard deviation or standard error of mean?». Perspect. Clin. Res. 3 (3): 113–116. doi:10.4103/2229-3485.100662. PMC 3487226. PMID 23125963.
- ^ Wassertheil-Smoller, Sylvia (1995). Biostatistics and Epidemiology : A Primer for Health Professionals (Second ed.). New York: Springer. pp. 40–43. ISBN 0-387-94388-9.
- ^ Isserlis, L. (1918). «On the value of a mean as calculated from a sample». Journal of the Royal Statistical Society. 81 (1): 75–81. doi:10.2307/2340569. JSTOR 2340569. (Equation 1)
- ^ Bondy, Warren; Zlot, William (1976). «The Standard Error of the Mean and the Difference Between Means for Finite Populations». The American Statistician. 30 (2): 96–97. doi:10.1080/00031305.1976.10479149. JSTOR 2683803. (Equation 2)
- ^ Bence, James R. (1995). «Analysis of Short Time Series: Correcting for Autocorrelation». Ecology. 76 (2): 628–639. doi:10.2307/1941218. JSTOR 1941218.
The standard error of the estimate is a way to measure the accuracy of the predictions made by a regression model.
Often denoted σest, it is calculated as:
σest = √Σ(y – ŷ)2/n
where:
- y: The observed value
- ŷ: The predicted value
- n: The total number of observations
The standard error of the estimate gives us an idea of how well a regression model fits a dataset. In particular:
- The smaller the value, the better the fit.
- The larger the value, the worse the fit.
For a regression model that has a small standard error of the estimate, the data points will be closely packed around the estimated regression line:
Conversely, for a regression model that has a large standard error of the estimate, the data points will be more loosely scattered around the regression line:
The following example shows how to calculate and interpret the standard error of the estimate for a regression model in Excel.
Example: Standard Error of the Estimate in Excel
Use the following steps to calculate the standard error of the estimate for a regression model in Excel.
Step 1: Enter the Data
First, enter the values for the dataset:
Step 2: Perform Linear Regression
Next, click the Data tab along the top ribbon. Then click the Data Analysis option within the Analyze group.
If you don’t see this option, you need to first load the Analysis ToolPak.
In the new window that appears, click Regression and then click OK.
In the new window that appears, fill in the following information:
Once you click OK, the regression output will appear:
We can use the coefficients from the regression table to construct the estimated regression equation:
ŷ = 13.367 + 1.693(x)
And we can see that the standard error of the estimate for this regression model turns out to be 6.006. In simple terms, this tells us that the average data point falls 6.006 units from the regression line.
We can use the estimated regression equation and the standard error of the estimate to construct a 95% confidence interval for the predicted value of a certain data point.
For example, suppose x is equal to 10. Using the estimated regression equation, we would predict that y would be equal to:
ŷ = 13.367 + 1.693*(10) = 30.297
And we can obtain the 95% confidence interval for this estimate by using the following formula:
- 95% C.I. = [ŷ – 1.96*σest, ŷ + 1.96*σest]
For our example, the 95% confidence interval would be calculated as:
- 95% C.I. = [ŷ – 1.96*σest, ŷ + 1.96*σest]
- 95% C.I. = [30.297 – 1.96*6.006, 30.297 + 1.96*6.006]
- 95% C.I. = [18.525, 42.069]
Additional Resources
How to Perform Simple Linear Regression in Excel
How to Perform Multiple Linear Regression in Excel
How to Create a Residual Plot in Excel
Correlation and Regression
Andrew F. Siegel, Michael R. Wagner, in Practical Business Statistics (Eighth Edition), 2022
The Standard Error of Estimate: How Large Are the Prediction Errors?
The standard error of estimate, denoted Se here (but often denoted S in computer printouts), tells you approximately how large the prediction errors (residuals) are for your data set in the same units as Y. How well can you predict Y? The answer is to within about Se above or below.16 Because you usually want your forecasts and predictions to be as accurate as possible, you would be glad to find a small value for Se. You can interpret Se as a standard deviation in the sense that if you have a normal distribution for the prediction errors, then you will expect about two-thirds of the data points to fall within a distance Se either above or below the regression line. Also, about 95% of the data values should fall within 2Se, and so forth. This is illustrated in Fig. 11.2.10 for the production cost example.
Fig. 11.2.10. The standard error of estimate, Se, indicates approximately how much error you make when you use the predicted value for Y (on the least-squares line) instead of the actual value of Y. You may expect about two-thirds of the data points to be within Se above or below the least-squares line for a data set with a normal linear relationship, such as this one.
The standard error of estimate may be found using the following formulas:
Standard Error of Estimate
Se=SY(1−r2)n−1n−2(forcomputation)=1n−2∑i=1n[Yi−(a+bXi)]2(forinterpretation)
The first formula shows how Se is computed by reducing SY according to the correlation and sample size. Indeed, Se will usually be smaller than SY because the line a + bX summarizes the relationship and therefore comes closer to the Y values than does the simpler summary, Y¯. The second formula shows how Se can be interpreted as the estimated standard deviation of the residuals: The squared prediction errors are averaged by dividing by n − 2 (the appropriate number of degrees of freedom when two numbers, a and b, have been estimated), and the square root undoes the earlier squaring, giving you an answer in the same measurement units as Y.
For the production cost data, the correlation was found to be r = 0.869193, the variability in the individual cost numbers is SY = $389.6131, and the sample size is n = 18. The standard error of estimate is therefore
Se=SY(1−r2)n−1n−2=389.6131(1−0.8691932)18−118−2=389.6131(0.0244503)1716=389.61310.259785=$198.58
This tells you that, for a typical week, the actual cost was different from the predicted cost (on the least-squares line) by about $198.58. Although the least-squares prediction line takes full advantage of the relationship between cost and number produced, the predictions are far from perfect.
Read full chapter
URL:
https://www.sciencedirect.com/science/article/pii/B9780128200254000117
Detection of a Trend in Population Estimates
William L. Thompson, … Charles Gowan, in Monitoring Vertebrate Populations, 1998
5.2 VARIANCE COMPONENTS
In this section, we discuss sources of variation that must be considered to make inferences from data when trying to detect trends. Three sources of variation must be considered: sampling variation, temporal variation in the population dynamics process, and spatial variation in the dynamics of the population across space. The latter two sources often are referred to as process variation, i.e., variation in the population dynamics process associated with environmental variation (such as rainfall, temperature, community succession, fires, or elevation). Methods to separate process variation from sampling variation will be presented.
Detection of a trend in a population’s size requires at least two abundance estimates. For example, if the population size of Mexican spotted owls in Mesa Verde National Park is determined as 50 pairs in 1990, and as only 10 pairs in 1995, we would be concerned that a significant negative trend in the population exists during this time period, and that action must be taken to alleviate the trend. However, if the 1995 estimate was 40 pairs, we might still be concerned, but would be less confident that immediate action is required. Two sources of variation must be assessed before we are confident of our inference from these estimates.
The first source of variation is the uncertainty we have in our population estimates. We want to be sure that the two estimates are different, i.e., the difference between the two estimates is greater than would be expected from chance alone because of the sampling errors associated with each estimate. Typically, we present our uncertainty in our estimate as its variance, and use this variance to generate a confidence interval for our estimate. Suppose that the 1990 estimate of Nˆ90 = 50 pairs has a sampling variance of Vaˆr(Nˆ90)=25. Then, under the assumption of the estimate being normally distributed with a large sample size (i.e., large degrees of freedom), we would compute a 95% confidence interval as 50 ± 1.96 25, or 40.2–59.8. If the 1995 estimate was Nˆ95 = 40 with a sampling variance of Vâr(Nˆ95) = 20, then the 95% confidence interval for this estimate is 40 ± 1.96 20, or 31.2–48.8. Based on the overlap of the two confidence intervals (Fig. 5.2), we would conclude that by chance alone, these two estimates are probably not different. We also could compute a simple test as
Figure 5.2. The 95% confidence intervals plotted with the 1990 and 1995 population estimates.
(5.3)z=Nˆ90−Nˆ95Vaˆr(Nˆ90)+Vaˆr(Nˆ95),
which for this example results in z = 1.491, with a probability of observing a z statistic this large or larger of P = 0.136. Although we might be alarmed, the chances are that 13.6 times out of 100 we would observe this large of a change just by random chance.
A variation of the previous test is commonly conducted for several reasons: (1) we often are interested in the ratio of two population estimates (rather than the difference) because a ratio represents the rate of change of the population, (2) the variance of Nˆ is usually linked to its estimate by Vaˆr(Nˆ)=NˆC (e.g., Skalski and Robson, 1992, pp. 28–29), and (3) ln(Nˆ) is more likely to be normally distributed than Nˆ. Fortuitously, a log transformation provides some correction to all three of the above reasons and results in a more efficient statistical procedure. Because
(5.4)Var[ln(Nˆ)]=Var(Nˆ)Nˆ2,
we construct the z test as
(5.5)z=ln(Nˆ90)−ln(Nˆ95)Vaˆr[ln(Nˆ90)]+Vaˆr[ln(Nˆ95)]
to provide a more efficient (i.e., more powerful) test.
Suppose we had made a much more intensive effort in sampling the owl population, so that the sampling variances were one-half of the values observed (which would generally take about 4 times the effort). Thus, Vâr(Nˆ90) = 12.5 and Vâr(Nˆ95) = 10, giving a z statistic of 2.108 with probability value of P = 0.035. Now, we would conclude that the owl population was lower in 1995 than in 1990, and that this difference is unlikely due to variation in our samples, i.e., that an actual reduction in population size has taken place.
This leads us to the second variance component associated with determining whether a trend in the population is important. We would expect the size of the owl population (and any other population, for that matter) to fluctuate through time. How can we determine if this reduction is important? The answer lies in determining what the variation in the owl population has been for some period of time in the past, and then if the observed reduction is outside the range expected from this past fluctuation. Consider the example in Fig. 5.3, where the true population size (no sampling variation) is plotted. The population fluctuates around a mean of 50, but values more extreme than the range 40 to 60 are common. Note that a decline from 76 to 29 pairs occurred from 1984 to 1985, and that declines from over 50 pairs to under 40 pairs are fairly common occurrences. Thus, based on our previous example, a decline from 50 to 40 is not at all unreasonable given the past population dynamics of this hypothetical population.
Figure 5.3. Actual number of pairs of owls that exist each year. In reality, we never know these values, and can only estimate them.
To determine the level of change in population size that should receive our attention and suggest management action, we need to know something about the temporal variation in the population. The only way to estimate this variance component is to observe the population across a number of years. The exact number of years will depend on the magnitude of the temporal variation. Thus, if the population does not change much from year to year, a few observations will show this consistency. On the other hand, if the population fluctuates a lot, as in Fig. 5.3, many years of observations are needed to estimate the temporal variance. For the example in Fig. 5.3, we could compute the temporal variance as the variance of the 15 years. We find a variance of 265.7, or a standard deviation of 16.3 (Example 5.1). With a SD of 16.3, we would expect roughly 95% of the population values to be in the range of ±2 SD of the mean population size. This inference is based on the population being stable, i.e., not having an upward or downward trend, and being roughly normally distributed. For a normal distribution, 95% of the values lie in the interval ±2 SD of the mean. Therefore, a change of 2 SD, or 32.6, is not a particularly big change given the temporal variation observed over the 15-year period. Such a change should occur with probability greater than 1/20, or 0.05.
A complicating problem with estimating the temporal variance of a population’s size is that we are seldom allowed to observe the true value of the population size. Rather, we are required to sample the population, and hence only obtain an estimate of the population size each year, with its associated sampling variance. Thus, we would need to include the 95% confidence bars on the annual estimates. As a result of this uncertainty from our sampling procedure, we would conclude that many of the year-to-year changes were not really changes because the estimates were not different. This complication leads to a further problem. If we compute the variance with the usual formula when estimates of population size replace the actual population size shown in Fig. 5.3, we obtain a variance estimate larger than the true temporal variance because our sampling uncertainty is included in the variance. For low levels of sampling effort each year, we would have a high sampling variance associated with each estimate, and as a result, we would have a high variance across years. The noise associated with our low sampling intensity would suggest that the population is fluctuating widely, when in fact the population could be constant (i.e., temporal variance is zero), and the estimated changes in the population are just due to sampling variance.
This mixture of sampling and temporal variation becomes particularly important in population viability analysis (PVA). The objective of a PVA is to estimate the probability of extinction for a population, given current size, and some idea of the variation in the population dynamics (i.e., temporal variation). If our estimate of temporal variation includes sampling variation, and the level of effort to obtain the estimates is relatively low, the high sampling variation causes our naive estimate of temporal variation to be much too large. When we apply our PVA analysis with this inflated estimate of temporal variance, we conclude that the population is much more likely to go extinct than it really is, and hence the importance of separating sampling variation from process variation.
Typically, we estimate variance components with analysis of variance (ANOVA) procedures. For the example considered here, we would have to have at least two estimates of population size for a series of years to obtain valid estimates of sampling and temporal variation. Further, typical ANOVA techniques assume that the sampling variation is constant, and so do not account for differences in levels of effort, or the fact that sampling variance is usually a function of population size. For our example, we have an estimate of sampling variance for each of our estimates, obtained from the population estimation methods considered in this manual. That is, capture–recapture, mark–resight, line transects, removal methods, and quadrat counts all produce estimates of sampling variation. Thus, we do not want to estimate sampling variation by obtaining replicate estimates, but want to use the available estimate. Therefore, we present a method of moments estimator developed in Burnham et al. (1987, Part 5). Skalski and Robson (1992, Chapter 2) also present a similar procedure, but do not develop the weighted estimator presented here.
Example 5.1 Population Size, Estimates, Standard Error of the Estimates, and Confidence Intervals for Owl Pairs in Fig. 5.3
| Standard | |||||
|---|---|---|---|---|---|
| Year | Population | Estimate | error | Lower 95% CI | Upper 95% CI |
| 1980 | 44 | 40.04 | 5.926 | 28.42 | 51.66 |
| 1981 | 48 | 50.51 | 11.004 | 28.94 | 72.08 |
| 1982 | 61 | 61.36 | 15.278 | 31.42 | 91.31 |
| 1983 | 48 | 47.6 | 11.062 | 25.92 | 69.28 |
| 1984 | 76 | 95.51 | 18.988 | 58.3 | 132.72 |
| 1985 | 29 | 33.81 | 8.803 | 16.56 | 51.06 |
| 1986 | 60 | 34.39 | 5.804 | 23.01 | 45.76 |
| 1987 | 59 | 38.52 | 11.168 | 16.63 | 60.41 |
| 1988 | 76 | 84.57 | 21.312 | 42.8 | 126.34 |
| 1989 | 42 | 30.04 | 6.918 | 16.48 | 43.6 |
| 1990 | 29 | 20.29 | 7.529 | 5.54 | 35.05 |
| 1991 | 68 | 68.42 | 17.969 | 33.2 | 103.64 |
| 1992 | 42 | 45.51 | 13.225 | 19.6 | 71.44 |
| 1993 | 27 | 27.01 | 6.137 | 14.98 | 39.04 |
| 1994 | 72 | 71.12 | 14.511 | 42.67 | 99.56 |
| 1995 | 54 | 51.45 | 8.054 | 35.66 | 67.24 |
The variance of the n = 16 populations is 265.628, whereas the variance of the 16 estimates is 450.376. Sampling variation causes the estimates to have a larger variance than the actual population. The difference of these two variances is an estimate of the sampling variation, i.e., 450.376 – 265.628 = 184.748. The square root of 184.748 is 13.592, and is the approximate mean of the 16 reported standard errors.
To obtain an unbiased estimate of the temporal variance, we must remove the sampling variation from the estimate of the total variance. Define σtotal2 as the total variance, estimated for n = 16 estimates of owl pairs (Nˆi, i = 1980, …, 1995) as
(5.6)σˆtotal2=Σi=19801995(Nˆi−N¯)2(n−1)=Σi=19801995Nˆi2(Σi−19801985Nˆi)2n(n−1),
where the symbol indicates the estimate of the parameter. Thus, Nˆi are the estimates of the actual populations, Ni, and σˆtotal2 is an estimate of the total variance σˆi2 For each estimate, Nˆi, we also have an associated sampling variance, σˆi2. Then, a simple estimator of the temporal variance, σ2time, is given by
(5.7)σˆtime2=σˆtotal2−Σi=19801995σˆi2n,
when we can assume that all of the sampling variances, σˆi2, are equal. The above equation corresponds to Eq. (2.6) of Skalski and Robson (1992). When the σˆi2 cannot all be assumed to be equal, a more complex calculation is required (Burnham et al., 1987, Section 4.3) because each estimate must be weighted by its sampling variance. We take as the weight of each estimate the reciprocal of the sum of temporal variance plus the sampling variance, 1/(σˆtime2+σˆi2). That is, Var(Nˆi)=σˆtime2+σˆi2, so wi=1/Var(Nˆi)=1/(σˆtime2+σˆi2). Then, the weighted total variance is computed as
(5.8)σˆtotal2=Σi=19801995wi(Nˆi−N¯)2(n−1)Σi=19801995wi
with the mean of the estimates now computed as a weighted mean,
(5.9)N¯=Σi=19801995wiNˆiΣi=19801995wi.
We now know that the theoretical variance N¯ is
(5.10)Var(N¯)=Var(Σi=19801995wiNˆiΣi=19801995wi)=1Σi=19801995wi
and the empirical variance estimator is Eq. (5.8). Setting these two equations equal,
(5.11)1Σi=19801995wi=Σi=19801995wi(Nˆi−N¯)2(n−1)Σi=19801995wi
or
(5.12)1=Σi=19801995wi(Nˆi−N¯)2(n−1).
Because we cannot solve for σˆtime2 directly, we have to use an iterative numerical approach to estimate σˆtime2 This procedure involves substituting values of σˆtime2 into Eq. (5.12) via the wi until the two sides are equal. When both sides are the same, we have our estimate of σˆtime2. Using this estimate of σˆtime2, we can now decide what level of change in Nˆi to Nˆi+1 is important and deserves attention. If the change from a series of estimates is greater than 2σˆtime2, we may want to take action.
Typically, we do not have the luxury of enough background data to estimate σˆtime2, so we end up trying to evaluate whether a series of estimated population sizes is in fact signaling a decline in the population when both sampling and process variance are present. Note that just because we see a decline of the estimates for 3–4 consecutive years, we cannot be sure that the population is actually in a serious decline without knowledge of the mean population size and the temporal variation prior to the decline. Usually, however, we do not have good knowledge of the population size prior to some observed decline, and make a decision to act based on biological perceptions. Keep in mind the kinds of trends displayed in Fig. 5.1. Is the suggested trend part of a cycle, or are we observing a real change in population size? In this discussion, we have only considered temporal variation. A similar procedure can be used to separate spatial variation from sampling variation.
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Multiple Regression
Andrew F. Siegel, Michael R. Wagner, in Practical Business Statistics (Eighth Edition), 2022
Typical Prediction Error: Standard Error of Estimate
Just as for simple regression, with only one X, the standard error of estimate indicates the approximate size of the prediction errors. For the magazine ads example, Se = $106,072. This tells you that actual page costs for these magazines are typically within about $106,072 from the predicted page costs, in the sense of a standard deviation. That is, if the error distribution is normal, then you would expect about two-thirds of the actual page costs to be within Se of the predicted page costs, about 95% to be within 2Se, and so forth.
The standard error of estimate, Se = $106,072, indicates the remaining variation in page costs after you have used the X variables (audience, percent male, and median income) in the regression equation to predict page costs for each magazine. Compare this to the ordinary univariate standard deviation, SY = $163,549, for the page costs, computed by ignoring all the other variables. This standard deviation, SY, indicates the remaining variation in page costs after you have used only Y¯ to predict the page costs for each magazine. Note that Se = $106,072 is smaller than SY = $163,549; your errors are typically smaller if you use the regression equation instead of just Y¯ to predict page costs. This suggests that the X variables are helpful in explaining page costs.
Think of the situation this way. If you knew nothing of the X variables, you would use the average page costs (Y¯=$187,628) as your best guess, and you would be wrong by about SY = $163,549. But if you knew the audience, percent male readership, and median reader income, you could use the regression equation to find a prediction for page costs that would be wrong by only Se = $106,072. This reduction in prediction error (from $163,549 to $106,072) is one of the helpful payoffs from running a regression analysis.
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Multiple Regression
Gary Smith, in Essential Statistics, Regression, and Econometrics, 2012
Confidence Intervals for the Coefficients
If the error term is normally distributed and satisfies the four assumptions detailed in the simple regression chapter, the estimators are normally distributed with expected values equal to the parameters they estimate:
a∼N[α, standard deviation of a]bi∼N[βi, standard deviation of bi]
To compute the standard errors (the estimated standard deviations) of these estimators, we need to use the standard error of estimate (SEE) to estimate the standard deviation of the error term:
(10.3)SEE=∑(Y−Y^)2n−(k+1)
Because n observations are used to estimate k + 1 parameters, we have n − (k + 1) degrees of freedom. After choosing a confidence level, such as 95 percent, we use the t distribution with n − (k + 1) degrees of freedom to determine the value t* that corresponds to this probability. The confidence interval for each coefficient is equal to the estimate plus or minus the requisite number of standard errors:
(10.4)a±t*(standard error ofa)bi±t*(standard error ofbi)
For our consumption function, statistical software calculates SEE = 59.193 and these standard errors:
standard error ofa=27.327standard error ofb1=0.019standard error ofb2=0.003
With 49 observations and 2 explanatory variables, we have 49 − (2 + 1) = 46 degrees of freedom. Table A.2 gives t* = 2.013 for a 95 percent confidence interval, so that 95 percent confidence intervals are
α:a±t*(standard error ofa)=−110.126±2.013(27.327)=−110.126±55.010β1:b1±t*(standard error ofb1)=0.798±2.013(0.019)=0.798±0.039β2:b2±t*(standard error ofb2)=0.026±2.013(0.003)=0.026±0.006
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Multiple Regression
Gary Smith, in Essential Statistics, Regression, and Econometrics (Second Edition), 2015
Confidence Intervals for the Coefficients
If the error term is normally distributed and satisfies the four assumptions detailed in the simple regression chapter, the estimators are normally distributed with expected values equal to the parameters they estimate:
a∼N[α,standarddeviationofa]bi∼N[βistandarddeviationofbi]
To compute the standard errors (the estimated standard deviations) of these estimators, we need to use the standard error of estimate (SEE) to estimate the standard deviation of the error term:
(10.5)SEE=∑(y−yˆ)2n−(k+1)
Because n observations are used to estimate k + 1 parameters, we have n − (k + 1) degrees of freedom. After choosing a confidence level, such as 95 percent, we use the t distribution with n − (k + 1) degrees of freedom to determine the value t∗ that corresponds to this probability. The confidence interval for each coefficient is equal to the estimate plus or minus the requisite number of standard errors:
(10.6)a±t∗(standarderrorofa)bi±t∗(standarderrorofbi)
For our consumption function, statistical software calculates SEE = 59.193 and these standard errors:
standarderrorofa=27.327standarderrorofb1=0.019standarderrorofb2=0.003
With 49 observations and two explanatory variables, we have 49 − (2 + 1) = 46 degrees of freedom. Table A.2 gives t∗ = 2.013 for a 95 percent confidence interval, so that 95 percent confidence intervals are:
α:a±t∗(standarderrorofa)=−110.126±2.013(27.327)=−110.126±55.010β1:b1±t∗(standarderrorofb1)=0.798±2.013(0.019)=0.798±0.039β2:b2±t∗(standarderrorofb2)=0.026±2.013(0.003)=0.026±0.006
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Simple Regression
Gary Smith, in Essential Statistics, Regression, and Econometrics (Second Edition), 2015
Abstract
The simple regression model assumes a linear relationship, Y = α + βX + ε, between a dependent variable Y and an explanatory variable X, with the error term ε encompassing omitted factors. The least squares estimates a and b minimize the sum of squared errors when the fitted line is used to predict the observed values of Y. The standard error of estimate (SEE) is our estimate of the standard deviation of the error term. The standard errors of the estimates a and b can be used to construct confidence intervals for α and β and test null hypotheses, most often that the value of β is zero (Y and X are not linearly related). The coefficient of determination R2 compares the model’s sum of the squared prediction errors to the sum of the squared deviations of Y about its mean, and can be interpreted as the fraction of the variation in the dependent variable that is explained by the regression model. The correlation coefficient is equal to the square root of R2.
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Bootstrap Method
K. Singh, M. Xie, in International Encyclopedia of Education (Third Edition), 2010
Approximating Standard Error of a Sample Estimate
Let us suppose, information is sought about a population parameter θ. Suppose θˆ is a sample estimator of θ based on a random sample of size n, that is, θˆ is a function of the data (X1, X2, …,Xn). In order to estimate standard error of θˆ, as the sample varies over the class of all possible samples, one has the following simple bootstrap approach.
Computeθ1*,θ2*,…,θN*, using the same computing formula as the one used for θˆ, but now base it on N different bootstrap samples (each of size n). A crude recommendation for the size N could be N = n2 (in our judgment), unless n2 is too large. In that case, it could be reduced to an acceptable size, say nlogen. One defines
SEB(θˆ)=[(1/N)∑i=1N(θi*−θˆ)2]1/2
following the philosophy of bootstrap: replace the population by the empirical population.
An older resampling technique used for this purpose is Jackknife, though bootstrap is more widely applicable. The famous example where Jackknife fails while bootstrap is still useful is that of θˆ = the sample median.
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Pearson, Karl
M. Eileen Magnello, in Encyclopedia of Social Measurement, 2005
The Biometric School
Although Pearson’s success in attracting such large audiences in his Gresham lectures may have played a role in encouraging him to further develop his work in biometry, he resigned from the Gresham Lectureship due to his doctor’s recommendation. Following the success of his Gresham lectures, Pearson began to teach statistics to students at UCL in October 1894. Not only did Galton’s work on his law of ancestral heredity enable Pearson to devise the mathematical properties of the product– moment correlation coefficient (which measures the relationship between two continuous variables) and simple regression (used for the linear prediction between two continuous variables) but also Galton’s ideas led to Pearson’s introduction of multiple correlation and part correlation coefficients, multiple regression and the standard error of estimate (for regression), and the coefficient of variation. By then, Galton had determined graphically the idea of correlation and regression for the normal distribution only. Because Galton’s procedure for measuring correlation involved measuring the slope of the regression line (which was a measure of regression instead), Pearson kept Galton’s “r” to symbolize correlation. Pearson later used the letter b (from the equation for a straight line) to symbolize regression. After Weldon had seen a copy of Pearson’s 1896 paper on correlation, he suggested to Pearson that he should extend the range for correlation from 0 to +1 (as used by Galton) so that it would include all values from −1 to +1.
Pearson achieved a mathematical resolution of multiple correlation and multiple regression, adumbrated in Galton’s law of ancestral heredity in 1885, in his seminal paper Regression, Heredity, and Panmixia in 1896, when he introduced matrix algebra into statistical theory. (Arthur Cayley, who taught at Cambridge when Pearson was a student, created matrix algebra by his discovery of the theory of invariants during the mid-19th century.) Pearson’s theory of multiple regression became important to his work on Mendel in 1904 when he advocated a synthesis of Mendelism and biometry. In the same paper, Pearson also introduced the following statistical methods: eta (η) as a measure for a curvilinear relationship, the standard error of estimate, multiple regression, and multiple and part correlation. He also devised the coefficient of variation as a measure of the ratio of a standard deviation to the corresponding mean expressed as a percentage.
By the end of the 19th century, he began to consider the relationship between two discrete variables, and from 1896 to 1911 Pearson devised more than 18 methods of correlation. In 1900, he devised the tetrachoric correlation and the phi coefficient for dichotomous variables. The tetrachoric correlation requires that both X and Y represent continuous, normally distributed, and linearly related variables, whereas the phi coefficient was designed for so-called point distributions, which implies that the two classes have two point values or merely represent some qualitative attribute. Nine years later, he devised the biserial correlation, where one variable is continuous and the other is discontinuous. With his son Egon, he devised the polychoric correlation in 1922 (which is very similar to canonical correlation today). Although not all of Pearson’s correlational methods have survived him, a number of these methods are still the principal tools used by psychometricians for test construction. Following the publication of his first three statistical papers in Philosophical Transactions of the Royal Society, Pearson was elected a fellow of the Royal Society in 1896. He was awarded the Darwin Medal from the Royal Society in 1898.
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The standard error of estimate is used to determine how well a straight line can describe values of a data set. When you have a collection of data from some measurement, experiment, survey or other source, you can create a line of regression to estimate additional data. With the standard error of estimate, you get a score that describes how good the regression line is.
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1
Create a five column data table. Any statistical work is generally made easier by having your data in a concise format. A simple table serves this purpose very well. To calculate the standard error of estimate, you will be using five different measurements or calculations. Therefore, creating a five-column table is helpful. Label the five columns as follows:[1]
-
2
Enter the data values for your measured data. After collecting your data, you will have pairs of data values. For these statistical calculations, the independent variable is labeled
and the dependent, or resulting, variable is . Enter these values into the first two columns of your data table.[2]
- The order of the data and the pairing is important for these calculations. You need to be careful to keep your paired data points together in order.
- For the sample calculations shown above, the data pairs are as follows:
- (1,2)
- (2,4)
- (3,5)
- (4,4)
- (5,5)
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3
Calculate a regression line. Using your data results, you will be able to calculate a regression line. This is also called a line of best fit or the least squares line. The calculation is tedious but can be done by hand. Alternatively, you can use a handheld graphing calculator or some online programs that will quickly calculate a best fit line using your data.[3]
- For this article, it is assumed that you will have the regression line equation available or that it has been predicted by some prior means.
- For the sample data set in the image above, the regression line is .
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4
Calculate predicted values from the regression line. Using the equation of that line, you can calculate predicted y-values for each x-value in your study, or for other theoretical x-values that you did not measure.[4]
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1
Calculate the error of each predicted value. In the fourth column of your data table, you will calculate and record the error of each predicted value. Specifically, subtract the predicted value (
) from the actual observed value ().[5]
- For the data in the sample set, these calculations are as follows:
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2
Calculate the squares of the errors. Take each value in the fourth column and square it by multiplying it by itself. Fill in these results in the final column of your data table.
- For the sample data set, these calculations are as follows:
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3
Find the sum of the squared errors (SSE). The statistical value known as the sum of squared errors (SSE) is a useful step in finding standard deviation, variance and other measurements. To find the SSE from your data table, add the values in the fifth column of your data table.[6]
- For this sample data set, this calculation is as follows:
- For this sample data set, this calculation is as follows:
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4
Finalize your calculations. The Standard Error of the Estimate is the square root of the average of the SSE. It is generally represented with the Greek letter
. Therefore, the first calculation is to divide the SSE score by the number of measured data points. Then, find the square root of that result.[7]
- If the measured data represents an entire population, then you will find the average by dividing by N, the number of data points. However, if you are working with a smaller sample set of the population, then substitute N-2 in the denominator.
- For the sample data set in this article, we can assume that it is a sample set and not a population, just because there are only 5 data values. Therefore, calculate the Standard Error of the Estimate as follows:
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5
Interpret your result. The Standard Error of the Estimate is a statistical figure that tells you how well your measured data relates to a theoretical straight line, the line of regression. A score of 0 would mean a perfect match, that every measured data point fell directly on the line. Widely scattered data will have a much higher score.[8]
- With this small sample set, the standard error score of 0.894 is quite low and represents well organized data results.
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Article SummaryX
To calculate the standard error of estimate, create a five-column data table. In the first two columns, enter the values for your measured data, and enter the values from the regression line in the third column. In the fourth column, calculate the predicted values from the regression line using the equation from that line. These are the errors. Fill in the fifth column by multiplying each error by itself. Add together all of the values in column 5, then take the square root of that number to get the standard error of estimate. To learn how to organize the data pairs, keep reading!
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To estimate the error in a measurement, we need to know the expected or standard value and compare how far our measured values deviate from the expected value. The absolute error, relative error, and percentage error are different ways to estimate the errors in our measurements.
Error estimation can also use the mean value of all the measurements if there is no expected value or standard value.
The mean value
To calculate the mean, we need to add all measured values of x and divide them by the number of values we took. The formula to calculate the mean is:
Let’s say we have five measurements, with the values 3.4, 3.3, 3.342, 3.56, and 3.28. If we add all these values and divide by the number of measurements (five), we get 3.3764.
As our measurements only have two decimal places, we can round this up to 3.38.
Estimation of errors
Here, we are going to distinguish between estimating the absolute error, the relative error, and the percentage error.
Estimating the absolute error
To estimate the absolute error, we need to calculate the difference between the measured value x0 and the expected value or standard xref:
Imagine you calculate the length of a piece of wood. You know it measures 2.0m with a very high precision of ± 0.00001m. The precision of its length is so high that it is taken as 2.0m. If your instrument reads 2.003m, your absolute error is | 2.003m-2.0m | or 0.003m.
Estimating the relative error
To estimate the relative error, we need to calculate the difference between the measured value x0 and the standard value xref and divide it by the total magnitude of the standard value xref:
Using the figures from the previous example, the relative error in the measurements is | 2.003m-2.0m | / | 2.0m | or 0.0015. As you can see, the relative error is very small and has no units.
Estimating the percentage error
To estimate the percentage error, we need to calculate the relative error and multiply it by one hundred. The percentage error is expressed as ‘error value’%. This error tells us the deviation percentage caused by the error.
Using the figures from the previous example, the percentage error is 0.15%.
What is the line of best fit?
The line of best fit is used when plotting data where one variable depends on another one. By its nature, a variable changes value, and we can measure the changes by plotting them on a graph against another variable such as time. The relationship between two variables will often be linear. The line of best fit is the line that is closest to all the plotted values.
Some values might be far away from the line of best fit. These are called outliers. However, the line of best fit is not a useful method for all data, so we need to know how and when to use it.
Obtaining the line of best fit
To obtain the line of best fit, we need to plot the points as in the example below:
Fig. 1 — Data plotted from several measurements showing variation on the y-axis
Here, many of our points are dispersed. However, despite this data dispersion, they appear to follow a linear progression. The line that is closest to all those points is the line of best fit.
When to use the line of best fit
To be able to use the line of best fit, the data need to follow some patterns:
- The relationship between the measurements and the data must be linear.
- The dispersion of the values can be large, but the trend must be clear.
- The line must pass close to all values.
Data outliers
Sometimes in a plot, there are values outside the normal range. These are called outliers. If the outliers are fewer in number than the data points following the line, the outliers can be ignored. However, outliers are often linked to errors in the measurements. In the image below, the red point is an outlier.
Fig. 2 — Data plotted from several measurements showing variation on the y-axis in green and an outlier in pink
Drawing the line of best fit
To draw the line of best fit, we need to draw a line passing through the points of our measurements. If the line intersects with the y-axis before the x-axis, the value of y will be our minimum value when we measure.
The inclination or slope of the line is the direct relationship between x and y, and the larger the slope, the more vertical it will be. A large slope means that the data changes very fast as x increases. A gentle slope indicates a very slow change of the data.
Figure 3 — The line of best fit is shown in pink, with the slope being shown in light green
Calculating uncertainty in a plot
In a plot or a graph with error bars, there can be many lines passing between the bars. We can calculate the uncertainty of the data using the error bars and the lines passing between them. See the following example of three lines passing between values with error bars:
Fig. 4 — Plot showing uncertainty bars and three lines passing between them. The blue and purple lines begin at the extreme values of the uncertainty bars
How to calculate the uncertainty in a plot
To calculate the uncertainty in a plot, we need to know the uncertainty values in the plot.
- Calculate two lines of best fit.
- The first line (the green one in the image above) goes from the highest value of the first error bar to the lowest value of the last error bar.
- The second line (red) goes from the lowest value of the first error bar to the highest value of the last error bar.
- Calculate the slope m of the lines using the formula below.
- For the first line, y2 is the value of the point minus its uncertainty, while y1 is the value of the point plus its uncertainty. The values x2 and x1 are the values on the x-axis.
- For the second line, y2 is the value of the point plus its uncertainty, while y1 is the value of the point minus its uncertainty. The values x2 and x1 are the values on the x-axis.
- You add both results and divide them by two:
Let’s look at an example of this, using temperature vs time data.
Calculate the uncertainty of the data in the plot below.
Figure 6. Plot showing uncertainty bars and three lines passing between them. The red and green lines begin at the extreme values of the uncertainty bars. Source: Manuel R. Camacho, StudySmarter.
The plot is used to approximate the uncertainty and calculate it from the plot.
| Time (s) | 20 | 40 | 60 | 80 |
| Temperature in Celsius | 84.5 ± 1 | 87 ± 0.9 | 90.1 ± 0.7 | 94.9 ± 1 |
To calculate the uncertainty, you need to draw the line with the highest slope (in red) and the line with the lowest slope (in green).
In order to do this, you need to consider the steeper and the less steep slopes of a line that passes between the points, taking into account the error bars. This method will give you just an approximate result depending on the lines you choose.
You calculate the slope of the red line as below, taking the points from t=80 and t=60.
You now calculate the slope of the green line, taking the points from t=80 and t=20.
Now you subtract the slope of the green one (m2) from the slope of the red one (m1) and divide by 2.
As our temperature measurements take only two significant digits after the decimal point, we round the result to 0.06 Celsius.
Estimation of Errors — Key takeaways
- You can estimate the errors of a measured value by comparing it to a standard value or reference value.
- The error can be estimated as an absolute error, a percentage error, or a relative error.
- The absolute error measures the total difference between the value you expect from a measurement (X0) and the obtained value (Xref), equal to the absolute value difference of both Abs = | Xo-Xref |.
- The relative and percentage errors measure the fraction of the difference between the expected value and the measured value. In this case, the error is equal to the absolute error divided by the expected value rel = Abs / Xo for the relative error, and divided by the expected value and expressed as a percentage for the percentage error per = (Abs / Xo) * 100. You must add the percentage symbol for percentage errors.
- You can approximate the relationship between your measured values using a linear function. This approximation can be made simply by drawing a line, which must be the line that passes closest to all values (the line of best fit).
What Is the Standard Error?
The standard error (SE) of a statistic is the approximate standard deviation of a statistical sample population.
The standard error is a statistical term that measures the accuracy with which a sample distribution represents a population by using standard deviation. In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error of the mean.
Key Takeaways
- The standard error (SE) is the approximate standard deviation of a statistical sample population.
- The standard error describes the variation between the calculated mean of the population and one which is considered known, or accepted as accurate.
- The more data points involved in the calculations of the mean, the smaller the standard error tends to be.
Standard Error
Understanding Standard Error
The term «standard error» is used to refer to the standard deviation of various sample statistics, such as the mean or median. For example, the «standard error of the mean» refers to the standard deviation of the distribution of sample means taken from a population. The smaller the standard error, the more representative the sample will be of the overall population.
The relationship between the standard error and the standard deviation is such that, for a given sample size, the standard error equals the standard deviation divided by the square root of the sample size. The standard error is also inversely proportional to the sample size; the larger the sample size, the smaller the standard error because the statistic will approach the actual value.
The standard error is considered part of inferential statistics. It represents the standard deviation of the mean within a dataset. This serves as a measure of variation for random variables, providing a measurement for the spread. The smaller the spread, the more accurate the dataset.
Standard error and standard deviation are measures of variability, while central tendency measures include mean, median, etc.
Formula and Calculation of Standard Error
The standard error of an estimate can be calculated as the standard deviation divided by the square root of the sample size:
SE = σ / √n
where
- σ = the population standard deviation
- √n = the square root of the sample size
If the population standard deviation is not known, you can substitute the sample standard deviation, s, in the numerator to approximate the standard error.
Requirements for Standard Error
When a population is sampled, the mean, or average, is generally calculated. The standard error can include the variation between the calculated mean of the population and one which is considered known, or accepted as accurate. This helps compensate for any incidental inaccuracies related to the gathering of the sample.
In cases where multiple samples are collected, the mean of each sample may vary slightly from the others, creating a spread among the variables. This spread is most often measured as the standard error, accounting for the differences between the means across the datasets.
The more data points involved in the calculations of the mean, the smaller the standard error tends to be. When the standard error is small, the data is said to be more representative of the true mean. In cases where the standard error is large, the data may have some notable irregularities.
The standard deviation is a representation of the spread of each of the data points. The standard deviation is used to help determine the validity of the data based on the number of data points displayed at each level of standard deviation. Standard errors function more as a way to determine the accuracy of the sample or the accuracy of multiple samples by analyzing deviation within the means.
Standard Error vs. Standard Deviation
The standard error normalizes the standard deviation relative to the sample size used in an analysis. Standard deviation measures the amount of variance or dispersion of the data spread around the mean. The standard error can be thought of as the dispersion of the sample mean estimations around the true population mean. As the sample size becomes larger, the standard error will become smaller, indicating that the estimated sample mean value better approximates the population mean.
Example of Standard Error
Say that an analyst has looked at a random sample of 50 companies in the S&P 500 to understand the association between a stock’s P/E ratio and subsequent 12-month performance in the market. Assume that the resulting estimate is -0.20, indicating that for every 1.0 point in the P/E ratio, stocks return 0.2% poorer relative performance. In the sample of 50, the standard deviation was found to be 1.0.
The standard error is thus:
SE = 1.0/√50 = 1/7.07 = 0.141
Therefore, we would report the estimate as -0.20% ± 0.14, giving us a confidence interval of (-0.34 — -0.06). The true mean value of the association of the P/E on returns of the S&P 500 would therefore fall within that range with a high degree of probability.
Say now that we increase the sample of stocks to 100 and find that the estimate changes slightly from -0.20 to -0.25, and the standard deviation falls to 0.90. The new standard error would thus be:
SE = 0.90/√100 = 0.90/10 = 0.09.
The resulting confidence interval becomes -0.25 ± 0.09 = (-0.34 — -0.16), which is a tighter range of values.
What Is Meant by Standard Error?
Standard error is intuitively the standard deviation of the sampling distribution. In other words, it depicts how much disparity there is likely to be in a point estimate obtained from a sample relative to the true population mean.
What Is a Good Standard Error?
Standard error measures the amount of discrepancy that can be expected in a sample estimate compared to the true value in the population. Therefore, the smaller the standard error the better. In fact, a standard error of zero (or close to it) would indicate that the estimated value is exactly the true value.
How Do You Find the Standard Error?
The standard error takes the standard deviation and divides it by the square root of the sample size. Many statistical software packages automatically compute standard errors.
The Bottom Line
The standard error (SE) measures the dispersion of estimated values obtained from a sample around the true value to be found in the population. Statistical analysis and inference often involves drawing samples and running statistical tests to determine associations and correlations between variables. The standard error thus tells us with what degree of confidence we can expect the estimated value to approximate the population value.