Standard error mean and standard deviation

Standard Error of the Mean vs. Standard Deviation: An Overview

Standard deviation (SD) measures the amount of variability, or dispersion, from the individual data values to the mean. SD is a frequently-cited statistic in many applications from math and statistics to finance and investing.

Standard error of the mean (SEM) measures how far the sample mean (average) of the data is likely to be from the true population mean. The SEM is always smaller than the SD.

Standard deviation and standard error are both used in statistical studies, including those in finance, medicine, biology, engineering, and psychology. In these studies, the SD and the estimated SEM are used to present the characteristics of sample data and explain statistical analysis results.

However, even some researchers occasionally confuse the SD and the SEM. Such researchers should remember that the calculations for SD and SEM include different statistical inferences, each of them with its own meaning. SD is the dispersion of individual data values. In other words, SD indicates how accurately the mean represents sample data.

However, the meaning of SEM includes statistical inference based on the sampling distribution. SEM is the SD of the theoretical distribution of the sample means (the sampling distribution).

Calculating SD and SEM

standard deviation 

σ

=

∑

i

=

1

n

(

x

i

−

x

ˉ

)

2

n

−

1

variance

=

σ

2

standard error 

(

σ

x

ˉ

)

=

σ

n

where:

x

ˉ

=

the sample’s mean

n

=

the sample size

begin{aligned} &text{standard deviation } sigma = sqrt{ frac{ sum_{i=1}^n{left(x_i — bar{x}right)^2} }{n-1} } \ &text{variance} = {sigma ^2 } \ &text{standard error }left( sigma_{bar x} right) = frac{{sigma }}{sqrt{n}} \ &textbf{where:}\ &bar{x}=text{the sample’s mean}\ &n=text{the sample size}\ end{aligned}

​standard deviation σ=n−1∑i=1n​(xi​−xˉ)2​​variance=σ2standard error (σxˉ​)=n​σ​where:xˉ=the sample’s meann=the sample size​

Standard Deviation

The formula for the SD requires a few steps:

1. First, take the square of the difference between each data point and the sample mean, finding the sum of those values.
2. Next, divide that sum by the sample size minus one, which is the variance.
3. Finally, take the square root of the variance to get the SD.

Standard Error of the Mean

SEM is calculated simply by taking the standard deviation and dividing it by the square root of the sample size.

Standard error gives the accuracy of a sample mean by measuring the sample-to-sample variability of the sample means. The SEM describes how precise the mean of the sample is as an estimate of the true mean of the population.

As the size of the sample data grows larger, the SEM decreases vs. the SD. As the sample size increases, the sample mean estimates the true mean of the population with greater precision.

Increasing the sample size does not make the SD necessarily larger or smaller; it just becomes a more accurate estimate of the population SD.

Standard Error and Standard Deviation in Finance

In finance, the SEM daily return of an asset measures the accuracy of the sample mean as an estimate of the long-run (persistent) mean daily return of the asset.

On the other hand, the SD of the return measures deviations of individual returns from the mean. Thus, SD is a measure of volatility and can be used as a risk measure for an investment.

Assets with greater day-to-day price movements have a higher SD than assets with lesser day-to-day movements. Assuming a normal distribution, around 68% of daily price changes are within one SD of the mean, with around 95% of daily price changes within two SDs of the mean.

The Bottom Line

Investors and analysts measure standard deviation as a way to estimate the potential volatility of a stock or other investment. It helps determine the level of risk to the investor that is involved. When reading an analyst’s report, the level of riskiness of an investment may be labeled «standard deviation.»

Standard error of the mean is an indication of the likely accuracy of a number. The larger the sample size, the more accurate the number should be.

From Wikipedia, the free encyclopedia

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 S_E.

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]

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]

Perhaps you’ve come across the terms “standard deviation” and “standard error” and are wondering what the difference is. What are they used for, and what do they actually mean for data analysts? Well, you’ve come to the right place. Keep reading for a beginner-friendly explanation.

When analyzing and interpreting data, you’re trying to find patterns and insights that can tell you something useful. For example, you might use data to better understand the spending habits of people who live in a certain city. In this case, it most likely wouldn’t be possible to collect the data you need from every single person living in that city—rather, you’d use a sample of data and then apply your findings to the general population. As part of your analysis, it’s important to understand how accurately or closely the sample data represents the whole population. In other words, how applicable are your findings?

This is where statistics like standard deviation and standard error come in. In this post, we’ll explain exactly what standard deviation and standard error mean, as well as the key differences between them. First, though, we’ll set the scene by briefly recapping the difference between descriptive and inferential statistics (as standard deviation is a descriptive statistic, while standard error an inferential statistic). Sound confusing? Don’t worry! All will become clear by the end of this post.

If you’re already familiar with descriptive vs inferential statistics, just use the clickable menu to skip ahead.

1. Quick recap: What is the difference between descriptive and inferential statistics?
2. What is standard deviation?
3. How to calculate standard deviation
4. What is standard error?
5. How to calculate standard error
6. Standard error vs standard deviation: What is the difference?
7. Standard error vs standard deviation: When should you use them?
8. Key takeaways and further reading

Are you ready to explore the difference between standard error and standard deviation? Let’s dive in.

1. What is the difference between descriptive and inferential statistics?

The first main difference between standard deviation and standard error is that standard deviation is a descriptive statistic while standard error is an inferential statistic. So what’s the difference?

Descriptive statistics are used to describe the characteristics or features of a dataset. This includes things like distribution(the frequency of different data points within a data sample—for example, how many people in the chosen population have brown hair, blonde hair, black hair, etc), measures of central tendency (the mean, median, and mode values), and variability (how the data is distributed—for example, looking at the minimum and maximum values within a dataset).

While descriptive statistics simply summarize your data, with inferential statistics, you’re making generalizations about a population (e.g. residents of New York City) based on a representative sample of data from that population. Inferential statistics are often expressed as a probability.

You can learn more about the difference between descriptive and inferential statistics in this guide, but for now, we’ll focus on the topic at hand: Standard deviation vs standard error.

So without further ado: What is standard deviation?

As already mentioned, standard deviation is a descriptive statistic, which means it helps you to describe or summarize your dataset. In simple terms, standard deviation tells you, on average, how far each value within your dataset lies from the mean. A high standard deviation means that the values within a dataset are generally positioned far away from the mean, while a low standard deviation indicates that the values tend to be clustered close to the mean. So, in a nutshell, it measures how much “spread” or variability there is within your dataset.

An example of standard deviation

Let’s illustrate this further with the help of an example. Suppose two shops X and Y have four employees each. In shop X, two employees earn $14 per hour and the other two earn $16 per hour. In shop Y, one employee earns $11 per hour, one earns $10 per hour, the third earns $19, and the fourth receives $20 per hour. The average hourly wage for each shop is $15, but you can see that some employees earn much closer to this average value than others.

For shop X, the employees’ wages are close to the average value of $15, with little variation (just one dollar difference either side), while for shop Y, the values are spread quite far apart from each other, and from the average. In this simple example, we can see this at a glance without doing any heavy calculations. But, in a more comprehensive and complex dataset, you’d calculate the standard deviation to tell you how far each individual value sits from the mean value.

We’ll look at how to calculate standard deviation in section three. For now, we’ll introduce two key concepts: Normal distribution and the empirical rule.

Normal distribution in standard deviation

Standard deviation can be interpreted by using normal distribution. In graph form, normal distribution is a bell-shaped curve which is used to display the distribution of independent and similar data values. In any normal distribution, data is symmetrical and distributed in fixed intervals around the mean. In terms of standard deviation, a graph (or curve) with a high, narrow peak and a small spread indicates low standard deviation, while a flatter, broader curve indicates high standard deviation.

What is the empirical rule?

If your dataset follows a normal distribution, you can interpret it using the empirical rule. The empirical rule states that almost all observed data will fall within three standard deviations of the mean:

• Around 68% of values fall within the first standard deviation of the mean
• Around 95% of values fall within the first two standard deviations of the mean
• Around 99.7% of values fall within the first three standard deviations of the mean

The empirical rule gives a quick overview of data and determines extreme values that don’t follow a pattern of normal distribution.

Now we know what standard deviation tells us, let’s take a look at how to calculate it.

3. How to calculate standard deviation

Now, you must be wondering about the formula used to calculate standard deviation. There are actually two formulas which can be used to calculate standard deviation depending on the nature of the data—are you calculating the standard deviation for population data or for sample data?

• Population data is when you have data for the entire group (or population) that you want to analyze. For example, if you’re collecting data on employees in your company and have data for all 100 employees, you are working with population data.
• Sample data is when you collect data from just a sample of the population you want to gather insights for. For example, if you wanted to collect data on residents of New York City, you’d likely get a sample rather than gathering data for every single person who lives in New York.

With that in mind, you can calculate standard deviation as follows:

How to calculate standard deviation for population data

To calculate standard deviation for population data, the formula is:

Standard deviation vs standard error: Population data[/caption]

Where:

• refers to population standard deviation
• ∑ refers to sum of values
• xi refers to each value
• refers to population mean
• N refers to number of values in the sample

How to calculate standard deviation for sample data

To calculate standard deviation for sample data, you can use the following formula:

Where:

• s refers to sample standard deviation
• ∑ refers to sum of values
• xi refers to each value
• x̅  refers to sample mean
• N refers to number of values in the sample

At this stage, simply having the mathematical formula may not be all that helpful. Let’s take a look at the actual steps involved in calculating the standard deviation.

How to calculate the standard deviation (step by step)

Here we’ll break down the formula for standard deviation, step by step.

1. Find the mean: Add up all the scores (or values) in your dataset and divide them by the total number of scores or data points.
2. Calculate the deviation from the mean for each individual score or value: Subtract the mean value (from step one) from each individual value or score you have in your dataset. You’ll end up with a set of deviation values.
3. Square each deviation from the mean: Multiply each deviation value you got in step two by itself. E.g. if the deviation value is 4, multiply it by 4.
4. Find the sum of squares: Add up all of the squared deviations as calculated in step three. This will give you a single value known as the sum of squares.
5. Find the variance: Divide the sum of the squares by n − 1 for sample data, or by N for population data. N denotes the total number of scores or values within your dataset, so if you collected data on thirty employees, N is thirty. This will give you a variance value.
6. Find the square root of the variance: Calculate the square root of the variance (as calculated in step five). This gives you the standard deviation (SD).

Let’s further illustrate the step-by-step procedure of calculating standard deviation through an interesting example.

How to calculate standard deviation with an example (in Excel or Google Sheets)

Let’s imagine a group of fifteen employees took part in an assessment, and their employer wants to know how much variation there is in the test scores. Did all employees perform at a similar level, or was there a high standard deviation? The test scores are as follows:

Now let’s calculate the standard deviation for our dataset, following the step-by-step process laid out previously. We’ll use formulas in Google Sheets / Excel, but you can also calculate these values manually.

1. Find the mean: Add all test scores together and divide the total score by the number of scores (1280 / 15 = 85.3). Your mean value is 85.3. In Google Sheets, we used the formula =SUM(A2:A16)/15
2. Calculate the deviation from the mean for each score and then square this value: Subtract the mean value (85.3) from each test score, and then square it. In Google Sheets, we used the formula =(A2-85.3)^2 (and so on). 3
3. Find the sum of the squares: Add up all of the squared deviations (in column B) to find the sum of the squares. In Google Sheets, we used the formula =SUM(B2:B16) to get the value 1139.35.
4. Find the variance: Divide the sum of the squares by N (as we’re using population data). So: 1139.35 / 15 = 75.96.
5. Find the square root of the variance to get the standard deviation: You can calculate the square root in Excel or Google Sheets using the following formula: =B18^0.5. In our example, the square root of 75.96 is 8.7.

So, for the employee test scores, the standard deviation is 8.7. This is low variance, indicating that all employees performed at a similar level.

4. What is standard error?

Standard error (or standard error of the mean) is an inferential statistic that tells you, in simple terms, how accurately your sample data represents the whole population. For example, if you conduct a survey of people living in New York, you’re collecting a sample of data that represents a segment of the entire population of New York. Different samples of the same population will give you different results, so it’s important to understand how applicable your findings are. So, when you take the mean results from your sample data and compare it with the overall population mean on a distribution, the standard error tells you what the variance is between the two means. In other words, how much would the sample mean vary if you were to repeat the same study with a different sample of people from the New York City population?

Just like standard deviation, standard error is a measure of variability. However, the difference is that standard deviationdescribes variability within a single sample, while standard error describes variability across multiple samples of a population. We’ll explore those differences in more detail in section six. For now, let’s continue to explore standard error.

Standard error can either be high or low. In the case of high standard error, your sample data does not accurately represent the population data; the sample means are widely spread around the population mean. In the case of low standard error, your sample is a more accurate representation of the population data, with the sample means closely distributed around the population mean.

What is the relationship between standard error (SE) and the sample size?

Sample size is inversely proportional to standard error, and so the standard error can be minimized by using a large sample size. As you can see from this graph, the larger the sample size, the lower the standard error.

5. How to calculate standard error

The computational method for calculating standard error is very similar to that of standard deviation, with a slight difference in formula. The exact formula you use will depend on whether or not the population standard deviation is known. It’s also important to note that the following formulas can only be applied to data samples containing more than 20 values.

So, if the population standard deviation is known, you can use this formula to calculate standard error:

Where:

• SE refers to standard error of all possible samples from a single population
• σ refers to population standard deviation
• n refers to the number of values in the sample

If the population standard deviation is not known, use this formula: 

Where:

• SE refers to standard error of all possible samples from a single population
• s refers to sample standard deviation which is a point estimate of population standard deviation
• n refers to the number of values in the sample

How to calculate standard error (step by step)

Let’s break that process down step by step.

1. Find the square root of your sample size (n)
2. Find the standard deviation for your data sample (following the steps laid out in section three of this guide)
3. Divide the sample standard deviation (as found in step 2) by the square root of your sample size (as calculated in step 1)

Let’s solve a problem step-by-step to show you how to calculate the standard error of mean by hand.

How to calculate standard error with an example

Suppose a large number of students from multiple schools participated in a design competition. From the whole population of students, evaluators chose a sample of 300 students for a second round. The mean of their competition scores is 650, while the sample standard deviation of  scores is 220. Now let’s calculate the standard error.

1. Find the square root of the sample size. In our example,n = 300, and you can calculate the square root in Excel or Google Sheets using the following formula: =300^0.5. So n= 17.32
2. Find the standard deviation for your data sample. You can do this following the steps laid out in section three, but for now we’ll take it as known that the sample standard deviation S = 220.
3. Divide the sample standard deviation by the square root of the sample size. So, in our example, 220 / 17.32 = 12.7. So, the standard error is 12.7.

When reporting the standard error, you would write (for our example): The mean test score is 650 ± 12.7 (SE).

6. Standard error vs standard deviation: What’s the difference?

Now we know what standard deviation and standard error are, let’s examine the differences between them. The key differences are:

• Standard deviation describes variability within a single sample, while standard error describes variability across multiple samples of a population.
• Standard deviation is a descriptive statistic that can be calculated from sample data, while standard error is an inferential statistic that can only be estimated.
• Standard deviation measures how much observations vary from one another, while standard error looks at how accurate the mean of a sample of data is compared to the true population mean.
• The formula for standard deviation calculates the square root of the variance, while the formula for standard error calculates the standard deviation divided by the square root of the sample size.

7. Standard error vs standard deviation: When should you use which?

With those differences in mind, when should you use standard deviation and when should you use standard error?

Standard deviation is useful when you need to compare and describe different data values that are widely scattered within a single dataset. Because standard deviation measures how close each observation is to the mean, it can tell you how precise the measurements are. So, if you have a dataset forecasting air pollution for a certain city, a standard deviation of 0.89 (i.e. a low standard deviation) shows you that the data is precise.

Standard error is useful if you want to test a hypothesis, as it allows you to gauge how accurate and precise your sample data is in relation to drawing conclusions about the actual overall population. For example, if you want to investigate the spending habits of everyone over 50 in New York City, using a sample of 500 people, standard error can tell you how “powerful” or applicable your findings are.

8. Key takeaways and further reading

In this guide, we’ve explained how to calculate standard error and standard deviation, and outlined the key differences between the two. In summary, standard deviation tells you how far each value lies from the mean within a single dataset, while standard error tells you how accurately your sample data represents the whole population.

Statistical concepts such as these form the very basis of data analytics, so it’s important to get your head around them if you’re considering a career in data analytics or data science. If you’d like to try your hand at analyzing real data, we can recommend this free introductory data analytics short course. And, for more useful guides, check out the following:

• What’s the difference between covariance and correlation?
• What is exploratory data analysis?
• What is multivariate data analysis?

Стандартное отклонение и стандартная ошибка: в чем разница?

Стандартное отклонение (SD), измеряет количество изменчивости или дисперсии, из отдельных значений данных, к среднему значению, в то время как стандартная ошибка среднего (SEM) мер, как далеко образец среднее (среднее) данных, вероятно, будет от истинного среднего значения населения. SEM всегда меньше SD.

SEM против SD

Стандартное отклонение и стандартная ошибка используются во всех типах статистических исследований, включая исследования в области финансов, медицины, биологии, инженерии, психологии и т. Д. В этих исследованиях стандартное отклонение (SD) и расчетная стандартная ошибка среднего (SEM) ) используются для представления характеристик данных выборки и объяснения результатов статистического анализа. Однако некоторые исследователи иногда путают SD и SEM. Таким исследователям следует помнить, что расчеты SD и SEM включают разные статистические выводы, каждый из которых имеет свое значение. SD – это разброс отдельных значений данных.

Другими словами, SD указывает, насколько точно среднее значение представляет данные выборки. Однако значение SEM включает статистический вывод, основанный на распределении выборки. SEM – это стандартное отклонение теоретического распределения выборочных средних (выборочное распределение).

Расчет стандартного отклонения

Формула SD требует нескольких шагов:

1. Во-первых, возьмите квадрат разницы между каждой точкой данных и средним значением выборки, найдя сумму этих значений.
2. Затем разделите эту сумму на размер выборки минус один, который представляет собой дисперсию.
3. Наконец, извлеките квадратный корень из дисперсии, чтобы получить стандартное отклонение.

Стандартная ошибка среднего

SEM рассчитывается путем деления стандартного отклонения на квадратный корень из размера выборки.

Стандартная ошибка дает точность выборочного среднего путем измерения изменчивости выборочного среднего от образца к образцу. SEM описывает, насколько точное среднее значение выборки является оценкой истинного среднего значения совокупности. По мере увеличения размера выборки данных SEM уменьшается по сравнению с SD; следовательно, по мере увеличения размера выборки среднее значение выборки оценивает истинное среднее значение генеральной совокупности с большей точностью. Напротив, увеличение размера выборки не обязательно делает SD больше или меньше, это просто становится более точной оценкой SD населения.

Стандартная ошибка и стандартное отклонение в финансах

В финансах стандартная ошибка средней дневной доходности актива измеряет точность выборочного среднего как оценки долгосрочной (постоянной) средней дневной доходности актива.

С другой стороны, стандартное отклонение доходности измеряет отклонения индивидуальных доходов от среднего значения. Таким образом, SD является мерой волатильности и может использоваться в качестве меры риска для инвестиций. Активы с более высокими ежедневными движениями цен имеют более высокое SD, чем активы с меньшими ежедневными движениями. Предполагая нормальное распределение, около 68% дневных изменений цен находятся в пределах одного стандартного отклонения от среднего, при этом около 95% дневных изменений цен находятся в пределах двух стандартных значений среднего.