О калькулятор стандартных ошибок (Высокая точность)
Калькулятор стандартной ошибки используется для расчета стандартной ошибки среднего значения набора чисел (Шаг за шагом).
стандартная ошибка среднего
Стандартная ошибка среднего — это стандартное отклонение выборочной средней оценки среднего значения генеральной совокупности. Обычно он рассчитывается путем деления стандартного отклонения оценки выборки (стандартное отклонение выборки) на квадратный корень из размера выборки (при условии статистической независимости медианных значений выборки):
в:
SEM = стандартная ошибка среднего
s = стандартное отклонение выборки (см. формулу ниже)
n = размер выборки (количество наблюдений)
СледующееСтандартное отклонение выборкиформула:
в:
s = стандартное отклонение выборки
x 1 ，…，x N= образец набора данных
x̄ = среднее значение выборочного набора данных
N = размер выборочного набора данных
Стандартная ошибка данных Решение
ШАГ 0: Сводка предварительного расчета
ШАГ 1. Преобразование входов в базовый блок
Стандартное отклонение данных: 4 —> Конверсия не требуется
Размер образца: 10 —> Конверсия не требуется
ШАГ 2: Оцените формулу
ШАГ 3: Преобразуйте результат в единицу вывода
1.26491106406735 —> Конверсия не требуется
7 Ошибки Калькуляторы
Стандартная ошибка данных формула
Стандартная ошибка данных = Стандартное отклонение данных/sqrt(Размер образца)
σμ = σ/sqrt(N)
Что такое стандартная ошибка и ее важность?
В статистике и анализе данных большое значение имеет стандартная ошибка. Термин «стандартная ошибка» используется для обозначения стандартного отклонения различных выборочных статистических данных, таких как среднее значение или медиана. Например, «стандартная ошибка среднего» относится к стандартному отклонению распределения выборочных средних, взятых из совокупности. Чем меньше стандартная ошибка, тем более репрезентативной будет выборка для генеральной совокупности. Соотношение между стандартной ошибкой и стандартным отклонением таково, что для данного размера выборки стандартная ошибка равна стандартному отклонению, деленному на квадратный корень размера выборки. Стандартная ошибка также обратно пропорциональна размеру выборки; чем больше размер выборки, тем меньше стандартная ошибка, потому что статистика будет приближаться к фактическому значению.
Инструкции:
Этот стандартный калькулятор ошибок позволяет вычистить стандартную ошибку, показывая все шаги.Пожалуйста, предоставьте стандартное отклонение населения (Σ) и размер выборки (N):
Подробнее о стандартной ошибке
Стандартная ошибка \(se\) соответствует стандартному отклонению образца среднего.Напомним, что если мы сделаем образец \(X_1\), \(X_2\), …, \(X_n\), в котором каждый \(X_i\) — случайная переменная с стандартным отклонением населения \(\sigma\), то образец среднее значение
\[ \bar X = \sum_{i=1}^n X_i\]
также является случайной величиной, со стандартным отклонением, равным \(\frac{\sigma}{\sqrt n}\).Это стандартное отклонение отбора дискретизации распределения образца среднего называется
стандартная ошибка
\(s_e\).
Created by Wojciech Sas, PhD
Reviewed by
Bogna Szyk and Jack Bowater
Last updated:
Jun 05, 2023
Welcome to the standard error calculator, also known as the standard error of the mean (SEM) calculator. This is a great tool that helps you estimate the standard error of the mean of any dataset in the blink of an eye.
If you’ve studied statistics at all, you’ve probably already heard of the mean, median, and mode, but do you know how to calculate the standard error? If not – this is the right place to begin!
In the article below, you can also find the equation of the standard error of the mean, as well as a comparison of standard error vs. standard deviation. There is no time to waste, let’s go!
💡 Check also our mean absolute deviation calculator, which estimates the error using a slightly different approach.
What is a standard error?
To learn how to find a standard error, we first need to ask ourselves standard error of what?
In statistics, we can estimate the standard error of any parameter – a mean, a proportion, a difference of means, and many many more. Typically, if someone wants to know how to calculate the standard error, it’s the standard error of the mean or SEM for short.
Great! So then, why do we want to know how to find the standard error?
Let’s say we have a task to find the average height of adults within a country. Such measures usually form a normal distribution for large populations.
Ideally, we should measure everybody one by one, and eventually, we would get a precise, well-defined number. In practice, however, it’s impossible from a time, money, and technical point of view, so we need to estimate such a value.
One approach is to take a relatively small group of people (a sample) and find their average height. It’s almost certain that it won’t be precisely the same as the one for the whole country. Still, we should be able to say that there is a high probability that the real result is within a range of values that we evaluated using the standard error. That’s an example of a situation where this SEM calculator comes in handy!
💡 Omni’s error propagation calculator will help you evaluate the total error when dealing with multiple variables.
Standard error formula
The equation of the standard error of the mean in its most compact form is:
∑i(xi−x‾)2N(N−1),\sqrt{\frac{\sum_i(x_i — \overline{x})^2}{N (N — 1)}},
where:
- xix_i is the ith measure;
- x‾\overline{x} (x-bar) stands for the mean value of our dataset; and
- NN is the number of data points.
To get familiar with the standard error formula, it’s a good practice to follow these steps:
-
Evaluate the mean value (x‾\overline{x}). It’s usually the arithmetic average.
-
Find the differences xi−x‾x_i — \overline{x} for every point.
-
Square the differences for each of the points separately, (xi−x‾)2(x_i — \overline{x})^2.
-
Add up all of the squared differences ∑(xi−x‾)2\sum(x_i — \overline{x})^2.
-
Divide the sum by the product N(N−1)N(N — 1).
-
Finally, work out the square root of this ratio.
As you can see, estimating the standard error of the mean for multiple points by hand can be really time-consuming. In such cases, using our standard error calculator is always a good idea, and you’ll also avoid any mistakes!
Standard error vs. standard deviation
In statistics, the standard deviation tells us about the variability of the respective measures from the mean. So what is the difference between standard deviation and standard error, then?
Simply speaking, the standard deviation is a parameter of a population (or a sample), while the standard error is an estimation of a particular value. In general, we can compute the standard error of any statistical value, but in most cases, we want to find the standard error of the mean.
To compare standard error vs. standard deviation, let’s take a look at their formulas:
-
standard deviation=σ2=σ=∑(xi−μ)2N\text{standard deviation} = \sqrt{\sigma^2} = \sigma = \frac{\sum(x_i — \mu)^2}{N} for a population, where σ2\sigma^2 is the variance of the set; or
-
standard deviation=s2=s=∑(xi−x‾)2N−1\text{standard deviation} = \sqrt{s^2} = s = \frac{\sum(x_i — \overline{x})^2}{N-1} for a sample, where s2s^2 is the estimate of the variance.
-
standard error of the mean=sN=∑(xi−x‾)2N(N−1).\text{standard error of the mean} = \frac{s}{\sqrt{N}} = \frac{\sum(x_i — \overline{x})^2}{N(N-1)}.
μ\mu and x‾\overline{x} stand for the mean and sample mean, respectively.
In other words, we can say that SEM tries to estimate the mean value of the whole population within a certain margin of error.
🔎 Read more about standard deviation in Omni’s standard deviation calculator.
Take a look at an example:
We want to estimate the average height of students in a school (a population). We take a group of 12 random pupils (a sample) whose heights (in cm) are 177, 182, 175, 194, 181, 177, 169, 180, 182, 186, 179, and 172.
The average height of this sample is x̄ = 179.5. Using our SEM calculator, you can find that the standard error of the mean equals SEM = 1.88. It tells us that the (real) mean height of students in this school is most likely between 177.62 and 181.38 (which is 179.5 ± 1.88).
At the same time, this sample’s standard deviation is s = 6.52. This means that we can expect that the majority of students’ heights (roughly 70%) to lie within the range [172.98, 186.02] (that is 179.5 ± 6.52).
How to find a standard error of the mean?
Let’s say we have a set of ten different values related to the weight of balls taken randomly from a production line. The numbers are: [5.5, 5.8, 6.1, 5.4, 5.5, 5.4, 5.9, 5.6, 5.9, 5.5]. The question is: what is the standard error of the mean for these measures? Let’s do it step by step:
-
Work out the mean value of this set. x̄ = (5.5+5.8+6.1+5.4+5.5+5.4+5.9+5.6+5.9+5.5) / 10 = 56.6/10 = 5.66.
-
Calculate the difference between every number and the mean (xᵢ — x̄): [-0.16, 0.14, 0.44, -0.26, -0.16, -0.26, 0.24, -0.6, 0.24, -0.16].
-
Square them: [0.0256, 0.0196, 0.1936, 0.0676, 0.0256, 0.0676, 0.0576, 0.0036, 0.0576, 0.0256].
-
Sum them up: 0.0256 + 0.0196 + 0.1936 + 0.0676 + 0.0256 + 0.0676 + 0.0576 + 0.0036 + 0.0576 + 0.0256 = 0.544.
-
Make a fraction from this value and 90 (that is N(N-1)=10×9): 0.544 / 90 = 0.0060.
-
Compute the square root of the latter, resulting in the standard error of the mean: SEM = √0.0060 = 0.078.
-
We can also write that our estimated mean in the form x̄ = 5.660 ± 0.078, taking two significant figures into account.
You can always check the result with our standard error calculator!
🙋 If you find this tool useful, our margin of error calculator will be a valuable extension on the topic.
By the way, have you heard about the skewness or mean absolute deviation? These are other parameters you can find while working with statistical data. Each of them carries additional information about your numbers!
Data (Enter up to 30 numbers)
| Observations (N): | 0 |
| Mean (x̄): | 0 |
| Standard error (SEM): | 0 |
5 number summary5★ rating averageCoefficient of variation… 41 more
Standard error calculator helps you to calculate the sample mean dispersion from the given raw data set or from the sample mean for statistical data analysis.
Give a complete read to this useful & important content to know about the basic terms, formulas and calculations related to the standard error.
What is the Standard Error Formula?
The standard error equation is as follows:
$$ S.E = \frac{s}{\sqrt{n}} $$
S.E = s/√n
Where,
s is the standard deviation of the numbers.
n is the number of samples.
When You Have Raw Data:
When you have raw data points, first you need to find the standard deviation and sample mean of the data. The formulas for standard deviation & population mean are:
S.D = √⅀(Xi -µ)2/N-1
Where,
Xi is each value in the data set.
µ is the mean of all values in the data set.
N is the total number of values in the data set.
Now, the formula for sample mean is:
µ =X1 + X2 + X3 + X4 +…….+ XN / N
These are the formulas which are also used by this online standard error calculator for the estimated results of your problem. Just read on, we have the complete step by step manual example for both the calculations.
Difference Between SEM & SD:
The SD and SEM both are used in statistical studies, in Finance, biology, engineering, psychology, medicine etc. The standard deviation (SD) & standard error of the mean (SEM) are used to represent the characteristics of the sample data and explain statistical analysis results.
Remember that, SD & SEM both are different, each have its own meaning. Standard deviation (SD) is the measure of dispersion of the individual data values. In simple words, SD determines how the sample data represents the mean accurately.
While, the SEM includes the statistical results of a particular value or sampling distribution. The SEM is the SD of theoretical distribution of sample means. Here you get a central tendency calculator that helps you to calculate mean, median. mode and range of the given date set.
You can try the online standard error of the mean calculator that allows you to find out the standard error from the sample mean & standard deviation.
Applications of Standard Error:
It is the most important and widely used measure in Statistics to determine the reliability of the sample data or mean. The major applications of standard error include, test of significance or hypothesis testing for large & small sample size (Z & t statistic) to measure the reliability of sample, to determine the confidence interval for sample.
It’s a statistical measure calculated from the sampling distribution where the large size samples minimize the SE of the statistic & vice versa. For the accurate & quick results, people may use this standard error calculator to determine the results with complete step-by-step calculations.
How to Calculate Standard Error Manually (Step-by-Step):
The formula for standard error is discussed earlier. Now, we have an example with complete step-by-step calculations.
Example:
Let’s have raw data 12, 23, 45, 33, 65, 54. Find the standard error of the given data?
Solution:
The formula to calculate SE is:
S.E = s/√n
Step1:
First of all, we have to calculate the mean of the data. The formula is:
µ =X1 + X2 + X3 + X4 +…….+ XN / N
So,
µ =12 + 23 + 45 + 33 + 65 + 54/ 6
µ =232/ 6
µ =38.66
Step 2:
Then, determine the standard deviation of the data.
S.D = √⅀(Xi -µ)2/N-1
Here,
µ = 38.66
So,
S.D = √⅀(Xi -µ)2/N-1
S.D = √ {(12-38.66)2 + (23-38.66)2 + (45-38.66)2 + (33-38.66)2 + (65-38.66)2 + (54-38.66)2}/6-1
S.D = √ {(-26.66)2 + (-15.66)2 + (6.34)2 + (-5.66)2 + (26.34)2 + (15.34)2}/5
S.D = √ {710.75 + 245.23 + 40.19 + 32.03 + 693.79 + 235.31}/5
S.D = √1957.3/5
S.D = √391.46
S.D = 19.7
Step 3:
Now,
S.E = s/√n
S.E = 19.7/√6
S.E = 19.7/2.44
S.E = 8.07
You can use the online standard error calculator to verify your answers with complete step-by-step calculations.
How to Find Standard Error With Standard Error Calculator:
Calculating standard error becomes very easy with this online sem calculator. Give a detailed calculation of the standard error.
Swipe on!
Calculations from the Raw Data:
To calculate the standard error from the raw data, just stick to these following points:
Inputs:
- First of all, enter the data points in the designated field separated with commas.
- Then, hit the calculate button.
Outputs:
The calculator will show:
- Standard Error of the numbers.
- Total numbers.
- Sum of the numbers.
- Mean of the numbers.
- Standard deviation between the numbers.
- Complete Step-by-Step calculation.
Calculations from the Summary Data:
Inputs:
- Firstly, enter the standard deviation between the numbers.
- Very next, plug-in the total number of samples (n) in the designated field.
- Lastly, click on the calculate button.
Outputs:
The tool shows:
- Standard Error of the numbers.
- Complete Step-by-Step calculation.
Simply, use the online standard error calculator for the complete step-by-step calculations from both raw data & from summary of the data.
Frequently Ask Questions (FAQ’s):
What is Standard Error?
It is a statistical term that measures the accuracy of the sample by using the standard deviation. The SE of a statistic is the standard deviation of the statistical sample population. In Statistics, the sample mean diverges from the actual mean and this deviation is the standard error of the mean.
What is a good value for standard error?
As the SE is an indication of the accuracy of sample mean as compared with the population mean. The smaller it is, the less spreading of data and more likely it is. So, the smaller value of standard error is a good thing.
How do I calculate standard error in Excel?
For the calculations in the excel, you can simply use the following function.
=STDEV (Sampling Range) / SQRT(COUNT(sampling range))
How do you read standard error (SE) bars?
SE bars can tell how the data is spread around the mean value. Smaller the SD bar lowers the suspension, larger SD bar larger suspension of data around the mean.
What does a SE of 2 mean?
As we know from the empirical rule, the 95% values fall in the range of 2 standard errors and approximately 99.7% of means will be in the range of 3 standard errors of the mean.
Why is the standard error important?
It is very important because it tells the fluctuation of the sample in statistics. The standard error helps in the construction of confidence intervals & significance testing, so it is really helpful.
End-Note:
Thankfully, you come to know about the standard error, its applications including; hypothesis testing, confidence interval of the sample and many others in the Statistics. Simply, use this online standard error calculator that helps you to determine the reliability of the sample data. Typically, students & education experts use this online tool to solve their education related problems.
References:
From the source Wikipedia: Standard error of the mean, Student approximation when σ value is unknown and all other statistical approaches.