How do you explain the concept of standard error in simple terms?

Lets take a simple example we are estimating average height of student in school, We measure height of sample students and calculate average height from that sample, But we know that our sample might not perfectly represent entire population of school, now this is where standard error comes in picture It tell us how much our sample average might differ from true average height of all students in school. If standard error is small, it means that we can trust our sample average as it is closer to true average, But if standard error is large it mean our sample average could be further off from true average The standard error is like margin of error of our sample average Smaller the standard error the more reliable our estimate is

Imagine you're training an AI model to predict the average height of people in Pakistan. You take a random sample of, say, 100 individuals and calculate the average height. This average height is like a "prediction" made by your AI model. Now, if you were to repeat this process many times, taking new random samples of 100 individuals each time, you'd get slightly different average heights each time. This variability in the average heights is like the "error" or "uncertainty" in your AI model's predictions. In summary, standard error is a statistical measure that quantifies the variability in sample statistics due to sampling error, much like the uncertainty in AI model predictions.

Imagine que você está tentando descobrir a altura média dos estudantes de uma escola, mas em vez de medir todos os estudantes, você mede apenas um grupo menor (uma amostra). A média das alturas desse grupo é uma estimativa da média real de todos os estudantes. No entanto, cada vez que você escolhe um grupo diferente de estudantes para medir, a média das alturas pode ser um pouco diferente. O erro padr?o nos diz o quanto essas médias podem variar se você repetisse o processo de medi??o várias vezes com diferentes grupos de estudantes. Em outras palavras, o erro padr?o é uma medida da varia??o esperada na média da amostra em rela??o à verdadeira média da popula??o.

In statistics, understanding variability is Important. Standard error helps us understand the variability associated with sample means, considering the spread of the data and the limitations of sample size. The sample means will likely not be identical. Standard error tells us how much we can expect these sample means to fluctuate around the population mean. Standard error is a crucial component in constructing confidence intervals. A confidence interval represents a range of values that likely holds the true population mean with a certain level of confidence (e.g., 95% confidence interval). A smaller standard error translates to a narrower confidence interval, indicating a more precise estimate of the population mean.

SE measures the uncertainty associated with the sampling scheme chosen. Standard errors of the mean close to zero indicates how close the sample mean is from the population mean, and vice versa.

you are newly married and you start observing action of your partner to understand . After some time you start guessing what your partner is expecting from your behaviour/action. Some time you will be near to expectations or sometime not able to meet expectations, either overestimate or underestimate. Now to measure how much you have understood your partner start giving numerical value to your understanding. You will get a lower value , average value and upper value. Probably you may be interested in knowing how much you have understood your partner 95%of time. For each percentage your lower value and upper value will be different. Deviation from average value to upper value or lower value is called standard error.

Building intuition about the standard error (SE) is helped by realizing that SE is an estimate of a particular standard deviation. Namely, one can take many samples to estimate the mean of a population. Such multiple sampling gives rise to a new distribution, one that shows where the estimates of the population mean fall. The new distribution has its own variance and thereby standard deviation. SE estimates this standard deviation.

The standard error tells us the variability of sample means. The standard deviation of the sampling distribution of the mean is the standard error (S.E.). It reflects how much the mean varies from sample to sample. The sampling distribution is only a theoretical distribution as in practice we take only one sample and not a repeated sample. We estimate the standard error from one sample.

The standard error (SE) is directly proportional to the standard deviation (SD). SD represents the spread or variability of individual data points in the sample. A larger SD indicates that the data points are more spread out from the sample mean, while a smaller SD indicates that they are closer to the sample mean. This variability directly impacts the standard error; greater variability in the data leads to a larger standard error, indicating less precision in the estimate of the population mean. The increase in sample size (n) results in a decrease of random sampling variability. This formula makes use of the 'Law of Large Numbers', which states that the sample mean converges to the population mean as sample size increases.

Given the formula that SD/ sqrt(n); the SE essentially reflects the mean residual between the population and each observation in the sample; where the sqrt comes from the eucledian solution for the absolute deviance (i.e., the residual). Assuming that the sample is adequately representative for the corresponding population; the SE corresponds to the standard deviation of the observations that we have in our sample (which ultimately creates a distribution that is formally referred to as the “sampling distribution”), where the mean of this distribution approximates the population mean.

The standard error looks at how precise the sample mean is as an estimate of the population mean. It reflects the expected variability of the sample mean if many samples were drawn from the same population. A smaller standard error indicates a more precise estimate which suggests that the sample mean is close to the true population mean, whereas a larger standard error suggests less precision. This is a good measure to assess the reliability of statistical conclusions taken from sample data. The mathematical equation for standard error is the standard deviation of the population divided by the square root of the sample size.

Standard error is the standard deviation of different samples within a given sample population. For instance, we have to work with 30 mice; our sample size/population is 30 of course. If we are checking weights of the 30 mice, we will have an average (mean) weight of the entire sample +/- the standard deviation. However, if we group the same mice into 5 groups of 6 mice each, and we take averages (means) of each group, it will be the average weight (of each group)+/- standard error of the mean.

Standard error of the mean assesses the precision of the sample mean as an estimate for the population mean. The sampling distribution of the mean converges to the population mean and has a variance of sigma squared divided by n (the sample size). So, the standard deviation of the sampling distribution becomes sigma divided by the square root of n. This estimate of uncertainty is also referred to as the standard error. The larger the standard error, the less precise is the mean estimate.

Standard Error measures the dispersion of means in the study population. It estimates how varied the actual population mean μ may be from the sample mean. Hence it is a population parameter. It is calculated as :SD/√n=SE, where n is the number of individuals in the sample and SD is the standard deviation of the sample. In simple words, standard error indicates how different the population mean is likely to be from the sample mean. Or you can say it’s a statistical measure of representing how accurately a set of data likely represents the actual population.

Once you understand PopulationMean, SampleMean, SampleStandardDeviation, and the difference between StandardError and ConfidenceIntervals; a good way to think about Standard Error is - How big could the gap between my sample mean and the real mean possibly be? Hence, how much should we think about alternatives that cover the spread from the population to sample mean in case of nearly worst case scenario?

Let’s try to understand standard error using a scenario. Think of standard error as a measure of how confident we are that the average score from a group of students on a test is close to the true average score of all students. If we only ask a few students, we might not be very sure, so the standard error is bigger. But if we ask lots of students, we can be more confident, and the standard error gets smaller. It’s like trying to guess the average height in a class by measuring just a few friends versus measuring everyone. The more friends you measure, the better your guess will be!

The standard error is the standard deviation of the measure of the sample mean of the population. Normally, the estimator of the population mean is the sample mean. But, if we draw multiple samples from the same population, it may provide a distinct values. The standard error shows us how the mean changes with different tests, estimating the same quantity. Thus, if there is notable difference in distribution of random samples, then the standard error will have a higher value. But, if the random samples follow somewhat similar distribution, then the value of the standard error will be minimum.

Explaining how to interpret the standard error in practical terms is crucial. The standard error provides a measure of how much error there is likely to be in a sample mean compared to the true population mean. In research and reporting, a smaller standard error suggests a more reliable estimate of the mean. It's important to convey that while the standard error gives an idea of the precision of the sample mean's estimate, it does not reflect the accuracy of the estimate — it only tells us about the consistency if the sampling were to be repeated under the same conditions.

Suppose we’re studying the average height of students in a grade level. We take a sample of 50 students and measure their heights. The sample mean height is 165 cm, and the standard deviation of their heights is 10 cm. For this data SE is approximately 1.41 (Using the formula in part 3) Interpreting the Standard Error: The standard error of 1.41 cm means that the sample mean height (165 cm) is expected to vary by about 1.41 cm from the true population mean height. This gives an idea of the precision of the sample mean as an estimate of the true population mean. If we were to take multiple samples of 50 students, the average of all these sample means would be within 1.41 cm of the true population mean height most of the time.

Em termos práticos o Erro Padr?o (EP) é a varia??o das médias amostrais, ou seja, se o EP for grande, há uma grande chance de haver algum distúrbio nos dados coletados.

Standard error measures how accurately a sample's average represents the population average. Take five tree heights: 50, 55, 53, 47, 52 feet. The average is 51.4 feet. Subtract each height from 51.4, giving deviations. We square these deviations (-1.4, 3.6, 1.6, -4.4, 0.6) to eliminate negative values and focus on the magnitude of each difference. Sum these squares, divide by the number of samples minus one to get the variance (9.3), and calculate the standard error by dividing the square root of the variance by the square root of the sample size. This results in 1.36 feet, indicating our sample mean might be off by about 1.36 feet from the true average.

From my experience esp. with business data, one often encounters the problem that data are far from being normally distributed. The usage of the mean, the standard deviation, and therefore also the standard error assumes, that data are more or less normally distributed. That should be checked. There are tests for that, see e.g. the article on normality tests in wikipedia. Often, one assumes normality because one believes thatthe law of large numbers applies. But for business data, this is often not the case.

Statistics results are only as good as the data that went into them, garbage in garbage out. So real accuracy depends on the quality of data as well as the sample size. It’s not just the sample size matter if the quality of data is not credible and not biased. With field of stats, though you can’t measure bias, one can try to minimize it by designing good samples and studies.

En resumen, el error estándar es una medida estadística fundamental que se utiliza para evaluar la precisión de una estimación y construir intervalos de confianza. Su cálculo se basa en la desviación estándar de la población o de la muestra, y su interpretación es crucial para entender la confiabilidad de los resultados estadísticos.

While standard error (SE) is valuable, it's not the only measure you should consider. For example, if your sample shows an L10 life of a bearing to be 1000 hours, SE tells you how much this estimate might vary. But SE alone doesn't give the full picture. Confidence Intervals A 90% confidence interval for instance, provides a range, such as 917.75 to 1082.25 hours, within which the true L10 life is likely to fall. This range adds context, helping you understand the potential accuracy of your sample mean. Statistics is about painting a complete picture. SE is just one part; confidence intervals provide the depth, showing the reliability of your estimate and offering a fuller understanding of your data.

We need to carefully consider if our data would be best represented by SE or not. Rightly so, sample mean variability around the population mean is measured by the SE. Its benefit is that it offers a precision metric that aids in evaluating the sample mean's dependability as a population mean estimate. Its drawback is that it may be susceptible to outliers, which could distort how accurately it represents the population mean. Hence, we would be required to carefully assess any outlier, what it could mean for the data set of interest.

The standard error is a measure of how much sample statistics (like the sample mean) are expected to vary from the true population parameter. In simpler terms, it tells us how much the results from different samples of the same population are likely to differ from each other. Here’s an easy analogy: imagine you’re trying to guess the average height of people in a city by measuring the height of a few randomly chosen groups. Each group might give you a slightly different average height. The standard error helps you understand how spread out these different group averages are likely to be.

Bravo, Stephen Senn, for your valued contributions! Much could be said about the misuse of the term "intuitive" in the teaching of statistics. However, in terms of adding in the extra detail to explain the truth about statistics, there is a critical need for more statistically trained instructors to be teaching statistics to non-specialists. Without this need being addressed, instructors are going to continue perpetuating traditional statistical misconceptions and receive praise for their over-simplification of statistics and 'intuitive' approach at the expense of the safety of stakeholders, including patients. The explosion in data science, so called, is squeezing statistics out of institutions. Statistical integrity matters.

Es importante destacar que el error estándar no disminuye directamente proporcional con el tama?o de la muestra. Para reducir el error estándar a la mitad, es necesario cuadruplicar el tama?o de la muestra. Esto se debe a que el error estándar se calcula como la raíz cuadrada de N, lo que significa que el tama?o de la muestra debe aumentar significativamente para lograr una disminución significativa del error estándar.

If data is far from being normally distributed (see above), one can use robust statistics which provides measures that can be used instead of SE, SD and mean.