Standard Error Formula

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Standard Error

In statistics, the standard error is the standard deviation of the sample distribution. The sample mean of a data is generally varied from the actual population mean. It is represented as SE. It is used to measure the amount of accuracy by which the given sample represents its population. Statistics is a vast topic in which we learn about data, sample and population, mean, median, mode, dependent and independent variables, standard deviation, variance, etc. Here you will learn the standard error formula along with SE of the mean and estimation.

Standard Error Meaning

The standard error is one of the mathematical tools used in statistics to estimate the variability. It is abbreviated as SE. The standard error of a statistic or an estimate of a parameter is the standard deviation of its sampling distribution. We can define it as an estimate of that standard deviation.

Standard Error Formula

The accuracy of a sample that describes a population is identified through the SE formula. The sample mean which deviates from the given population and that deviation is given as;

Where S is the standard deviation and n is the number of observations.

Standard Error of the Mean (SEM)

The standard error of the mean also called the standard deviation of mean, is represented as the standard deviation of the measure of the sample mean of the population. It is abbreviated as SEM. For example, normally, the estimator of the population mean is the sample mean. But, if we draw another sample from the same population, it may provide a distinct value.

Thus, there would be a population of the sampled means having its distinct variance and mean. It may be defined as the standard deviation of such sample means of all the possible samples taken from the same given population. SEM defines an estimate of standard deviation which has been computed from the sample. It is calculated as the ratio of the standard deviation to the root of sample size, such as:

Where ‘s’ is the standard deviation and n is the number of observations.

The standard error of the mean shows us how the mean changes with different tests, estimating the same quantity. Thus if the outcome of random variations is notable, then the standard error of the mean will have a higher value. But, if there is no change observed in the data points after repeated experiments, then the value of the standard error of the mean will be zero.

Standard Error of Estimate (SEE)

The standard error of the estimate is the estimation of the accuracy of any predictions.  It is denoted as SEE. The regression line depreciates the sum of squared deviations of prediction. It is also known as the sum of squares error. SEE is the square root of the average squared deviation. The deviation of some estimates from intended values is given by standard error of estimate formula.

Where xi stands for data values, x bar is the mean value and n is the sample size.

How to calculate Standard Error

Step 1: Note the number of measurements (n) and determine the sample mean (μ). It is the average of all the measurements.

Step 2: Determine how much each measurement varies from the mean.

Step 3: Square all the deviations determined in step 2 and add altogether: Σ(xi – μ)²

Step 4: Divide the sum from step 3 by one less than the total number of measurements (n-1).

Step 5: Take the square root of the obtained number, which is the standard deviation (σ).

Step 6: Finally, divide the standard deviation obtained by the square root of the number of measurements (n) to get the standard error of your estimate.
Go through the example given below to understand the method of calculating standard error.

Standard Error Example

Calculate the standard error of the given data:

y: 5, 10, 12, 15, 20

Solution: First we have to find the mean of the given data;

Mean = (5+10+12+15+20)/5 = 62/5 = 10.5

Now, the standard deviation can be calculated as;

S = Summation of difference between each value of given data and the mean value/Number of values.

Hence,

After solving the above equation, we get;

S = 5.35

Therefore, SE can be estimated with the formula;

SE = S/√n

SE = 5.35/√5 = 2.39

Standard Error vs Standard Deviation

The below table shows how we can calculate the standard deviation (SD) using population parameters and standard error (SE) using sample parameters.

Importance of Standard Error

Standard errors produce simplistic measures of uncertainty in a value. They are often used because, in many cases, if the standard error of some individual quantities is known, then we can easily calculate the standard error of some function of the quantities. Also, when the probability distribution of the value is known, we can use it to calculate an exact confidence interval. However, the standard error is an essential indicator of how precise an estimate of the sample statistic’s population parameter is.

Frequently Asked Questions – FAQs

How do you calculate standard error?

The standard error is calculated by dividing the standard deviation by the sample size’s square root. It gives the precision of a sample mean by including the sample-to-sample variability of the sample means.

What does the standard error mean?

The standard error of a statistic or an estimate of a parameter is the standard deviation of its sampling distribution.

Is standard error the same as SEM?

The standard error (SE) can be defined more precisely like the standard error of the mean (SEM) and is a property of our estimate of the mean.

What is a good standard error?

SE is an implication of the expected precision of the sample mean as compared with the population mean. The bigger the value of standard error, the more the spread and likelihood that any sample means are not close to the population’s mean. A small standard error is thus a good attribute.

What is a big standard error?

The bigger the standard error, the more the spread means there will be less accurate statistics.

What is the Standard Error of a Sample ?

What is the standard error?

The standard error(SE) is very similar to standard deviation. Both are measures of spread. The higher the number, the more spread out your data is.

In statistics, you’ll come across terms like “the standard error of the mean” or “the standard error of the median.” The SE tells you how far your sample statistic (like the sample mean) deviates from the actual population mean. The larger your sample size, the smaller the SE. In other words, the larger your sample size, the closer your sample mean is to the actual population mean.

What is the SE Calculation?

How you find the standard error depends on what stat you need. For example, the calculation is different for the mean or proportion. When you are asked to find the sample error, you’re probably finding the standard error. That uses the following formula: s/√n. You might be asked to find standard errors for other stats like the mean or proportion.

What is the Standard Error Formula?

The following tables show how to find the standard deviation (first table) and SE (second table). That assumes you know the right population parameters. If you don’t know the population parameters, you can find the standard error:

  • Sample mean.
  • Sample proportion.
  • Difference between means.
  • Difference between proportions.

Key for above tables:
P = Proportion of successes. Population.
p = Proportion of successes. Sample.
n = Number of observations. Sample.
n2 = Number of observations. Sample 1.
n2 = Number of observations. Sample 2.
σ21 = Variance. Sample 1.
σ22 = Variance. Sample 2.

Standard Error Examples

With random sampling of binomial values (in-favor vs. not-in-favor; heads vs. tails):

  1. Sampling from populations with percent-in-favor close to 50% have wider sampling distributions than populations with percentages closer to 0% or 100%.
  2. Larger sample sizes have narrower sampling distributions.

The various sampling distributions have different locations on the horizontal axis and they have different widths. It would be useful to convert them all to one standard scale. We’ll need a common unit. And the rescaling to that unit must account for the effects of the population percent-in-favor value (number 1above) and sample size (number 2 above).

The unit to be used is called Standard Error. It’s labeled “Standard” because it serves as a standard unit. And it’s labelled “Error” because we don’t expect our sample statistic values to be exactly equal to the population parameter; there will be some amount of error. The Standard Error formula, which I’ll explain a piece at a time, is as follows:

The variable p is the proportion rather than percentage: .5 rather than 50% (and 0 rather than 0%; .01 rather than 1%; .1 rather than 10%; and 1 rather than 100%).
The p * (1 – p) term in the numerator is called the proportion variance.

The variance p * (1 – p) reflects this dynamic:

  • 0.0 * (1 – 0) = 0.00
  • .01 * (1 – .01) = .01
  • .1 *(1 – .1) = .09
  • .3 *(1 – .3) = .21
  • .5 *(1 – .5) = .25
  • .7 *(1 – .7) = .21
  • .9 *(1 – .9) = .09
  • .99 *(1 – .99) = .01
  • 1.0 *(1 – 1) = 0.00

So, as p moves from .5 towards 0 or 1, variance decreases, and since variance is in the numerator, Standard Error decreases. Decreases in Standard Error correspond to narrowing of the sampling distribution. This reflects lower uncertainty. Lower variance, lower uncertainty. Variance is itself a statistic and is very important in statistical analysis. We’ll be seeing it in formulas from now on. Now let’s consider sample size, which is represented in the denominator of the formula by n.

Larger sample sizes have narrower sampling distributions. Since n is in the denominator of the Standard Error formula, as n increases Standard Error decreases. Again, decreases in Standard Error correspond to narrowing of the sampling distribution. Again, this reflects lower uncertainty. Larger sample size, lower uncertainty.

Now we can use the Standard Error scale to determine 95% intervals. First, an important fact: The boundary lines of the 95% interval on the Standard Error scale are always -2 and +2 (they’re actually -1.95996… and +1.95996…, but I’m rounding to -2 and +2 for the present purposes). Let’s clarify all this by looking at several example calculations and illustrations.

Let’s start with random sampling of 100 from a population that is 50% in favor of the new public health policy (Figure 1.2, below).

Plugging in the numbers gives

Standard Error is .05 and two Standard Errors is .1 in proportions and 10% in percentages. Since we want to center the interval on the percentage p of 50%, we’ll add and subtract 10% from 50%. This yields a calculated 95% interval of 50% + 10% (50% minus 10% to 50% plus 10%) or 40%-to-60%. That’s also what Figure 1.2 shows!

Putting everything we just computed into a formula for calculating 95% intervals we get

Next let’s consider the 95% interval of random sampling of 100 from a population that is 30% in favor of the new public health policy (Figure 2.7, reproduced below).

Standard Error is .045 and two Standard Errors is .09 in proportions and 9% in percentages. We want to center the interval on 30%, so we’ll add and subtract 9% from 30%. This yields a 95% interval of 30% ± 9% (30% minus 9% to 30% plus 9%) or 21%-to-39%. That’s also what Figure 2.7 shows!

Last let’s consider the 95% interval of random sampling of 1000 from a population that is 50% in favor of the new public health policy (Figure 2.3, below).

Standard Error is .015 and two Standard Errors is .03 in proportions and 3% in percentages. We want to center the interval on 50%, so we’ll add and subtract 3% from 50%. This yields a 95% interval of 50% + 3% or 47%-to-53%. That’s also what Figure 2.3 shows!

The formula works! The reason the formula works is because the sampling distributions are “bell shaped”. More than that, they approximate the very special bell shape called the Normal distribution.

Let’s go one step further and standardize an entire sampling distribution to get what’s called the Standard Normal distribution. The Standard Normal Distribution is a normal distribution that uses Standard Error as its unit (rather than percentages or proportions). To illustrate, let’s standardize Figure 1.1 (below).

Figure 3.1 is a standardized version of Figure 1.1.

Notice that Standard Error is the unit used on the horizontal axis of Figure 3.1. This is done by rescaling the horizontal axis unit of Figure 1.1 to the Standard Error unit of Figure 3.1 using the below formula.

This formula gives us how many Standard Errors a proportion, p, is from .5. First, we convert the percentages to proportions. Next, we recenter the axis: whereas Figure 1.1 is centered on the proportion value .5 (50%), Figure 3.1 is centered on zero Standard Errors; the numerator p-.5 centers the horizonal axis of Figure 3.1 onto zero. Finally, these differences are divided by the Standard Error to rescale the horizontal axis. Voila, Figure 1.1 has been standardized to the Standard Error scale of Figure 3.1.

Figure 3.2 shows its 95% interval below Figure 1.2.

The boundary lines of the 95% interval on the Standard Error scale are -2 and 2 (rounded). Plugging .4 (40%) and .6 (60%) from Figure 1.2 into the above formula gives us -2 and 2 Standard Errors as the 95% boundary lines in the Standard Error unit. As emphasized above: The boundary lines of the 95% interval on the Standard Error scale are always -2 and +2 (rounded). If we standardized Figures 2.3 and 2.7,
we’ll again find the 95% interval boundary lines to be -2 and 2. (You can use the formula and do the arithmetic if you want to confirm this.)

We can convert our units (e.g., percent-in-favor, percent-heads) into the Standard Error unit and vice versa by multiplying and dividing by Standard Error. That comes in very handy. All of the sampling distributions we’ve looked at so far can be standardized in this way. In practice, we don’t convert entire sampling distributions to the standardized distribution; we use Standard Error in formulas as multipliers and divisors to calculate individual values, like we do to calculate the boundary lines for 95% intervals and to convert proportions to the Standard Error scale.

Formula

SE = standard error of the sample

σ  = sample standard deviation

n  = number of samples

Steps to Calculate Standard Error

Standard error is calculated by dividing the standard deviation of the sample by the square root of the sample size.

  1. Calculate the mean of the total population.
  2. Calculate each measurements deviation from the mean.
  3. Square each deviation from the mean.
  4. Add the squared deviations from Step 3.
  5. Divide the sum of the squared deviations by one less than the sample size (n-1).
  6. Calculate the square root of the value obtained from Step 5. This result gives you the standard deviation.
  7. Divide the standard deviation by the square root of the sample size (n). This results gives you the standard error.
  8. Subtracting the standard error from the mean / adding the standard error to the mean will give the mean ± 1 standard error.

Example:

The values in your sample are 52, 60, 55, and 65.

  • Calculate the mean of these values by adding them together and dividing by 4. (52 + 60 + 55 + 65)/4 = 58 (Step 1).
  • Next, calculate the sum of the squared deviations of each sample value from the mean (Steps 2-4).
  • Using the values in this example, the squared deviations are (58 – 52)^2= 36, (58 – 60)^2= 4, (58 – 55)^2=9 and (58 – 65)^2=49. Therefore, the sum of the squared deviations is 98 (36 + 4 + 9 + 49).
  • Next, divide the sum of the squared deviations by the sample size minus one and take the square root (Steps 5-6). The standard deviation in this example is the square root of [98 / (4 – 1)], which is about 5.72.
  • Lastly, divide the standard deviation, 5.72, by the square root of the sample size, 4 (Step 7). The resulting value is 2.86 which gives the standard error of the values in this example.

Frequently asked questions

1. When to use Standard Error?

We use standard error to indicate the uncertainty around the estimate of the mean measurement. It tells us how well our sample data represents the whole population. This is useful when we want to calculate a confidence interval.

2. What is the Difference between Standard Error and Standard Deviation?

Standard error and standard deviation are both measures of variability, but 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 tells us how concentrated the data is around the mean. It describes variability within a single sample. On the other hand, standard error tells us how the mean itself is distributed.

It estimates the variability across multiple samples of a population. The formula for standard error calculates the standard deviation divided by the square root of the sample size.

What is Standard Error Formula?

In statistics, the term “standard error” of a statistic refers to the estimate of the standard deviation of the sample mean from the true population mean. To put it simply, just as standard deviation measures each individual’s dispersion value from the sample mean, the standard error of mean measures the dispersion of all the sample means around the population mean.

The formula for standard error can be derived by dividing the sample standard deviation by the square root of the sample size. Although population standard deviation should be used in the computation, it is seldom available, and as such a sample, the standard deviation is used as a proxy for population standard deviation. Mathematically, it is represented as,

Standard Error = s / √n

Where,

  • s: √Σni(xi-x̄)2 / n-1
  • xi: ith Random Variable
  • : Sample Mean
  • n: Sample Size

Examples of Standard Error Formula (With Excel Template)

Standard Error Formula – Example #1

Let us take the example of a survey where 100 respondents were asked to provide their feedback on the recently concluded college fest. They were asked to rate the fest on a scale of 1 to 5, with 5 being the best. Now, a random sampling method was used to build a sample of 5 responses out of the 100 responses. The selected responses are – 3, 2, 5, 3 and 4. Calculate the standard error of the statistic based on the selected responses.

Solution:

Sample Mean ( x̄ ) is calculated using the formula given below

x̄ = Σnixi/n

  • Sample Mean ( x̄ ) = (3 + 2 + 5 + 3 + 4) / 5
  • Sample Mean ( x̄ ) = 3.4

Calculated Deviation as

Similarly Calculated as below

Standard Deviation (s)  is calculated using the formula given below

s = √Σni(xi-x̄)2 / n-1

  • Standard Deviation = √ [{(3 – 3.4)2  + (2 – 3.4)2 + (5 – 3.4)2 + (3 – 3.4)2 + (4 – 3.4)2} / (5 – 1)]
  • Standard Deviation = 1.14

Standard Error is calculated using the formula given below

Standard Error = s / √n 

  • Standard Error = 1.14 / √5
  • Standard Error = 0.51

Therefore, the standard error of the sample mean is 0.51.

Standard Error Formula – Example #2

Let us take the example of a survey conducted at an office in New York where around 1,000 employees were asked how much they liked the work that they were doing in their current profile. They were to rate on a scale of 1 to 10, with 10 being the best. Then a sample of 10 responses was selected, and the responses are – 4, 5, 8, 10, 9, 5, 9, 8, 9 and 7. Calculate the standard error of the statistic based on the selected responses.

Solution:

Sample Mean ( x̄ ) is calculated using the formula given below

x̄ = Σnixi/n

  • Sample Mean ( x̄ ) = (4 + 5 + 8 + 10 + 9 + 5 + 9 + 8 + 9 + 7) / 10
  • Sample Mean ( x̄ )= 7.2

Calculated Deviation as

Similarly Calculated as below

Standard Deviation (s)  is calculated using the formula given below

s = √Σni(xi-x̄)2 / n-1

  • Standard Deviation  = √ [{(4 – 7.2)2  + (5 – 7.2)2 + (8 – 7.2)2 + (10 – 7.2)2 + (9 – 7.2)2 + (5 – 7.2)2  + (9 – 7.2)2 + (8 – 7.2)2 + (9 – 7.2)2 + (7 – 7.2)2} / (10 – 1)]
  •   Standard Deviation = 2.44

Standard Error is calculated using the formula given below

Standard Error = s / √n 

  • Standard Error = 2.44 / √10
  • Standard Error = 0.77

Therefore, the standard error of the sample mean is 0.77.

Explanation

The formula for standard error can be derived by using the following steps:

Step 1: Firstly, collect the sample variables from the population-based on a certain sampling method. The sample variables are denoted by x such that xi refers to the ith variable of the sample.

Step 2: Next, determine the sample size, which is the total number of variables in the sample.   It is denoted by n.

Step 3: Next, compute the sample mean, which can be derived by dividing the summation of all the variables in the sample (step 1) by the sample size (step 2). It is denoted by, and mathematically it is represented as,

x̄ = Σnixi/n

Step 4: Next, compute the sample standard deviation (s), which involves a complex calculation that uses each sample variable (step 1), sample mean (step 3) and sample size (step 2) as shown below.

s = √Σni(xi-x̄)2 / n-1

Step 5: Finally, the formula for standard error can be derived by dividing the sample standard deviation (step 4) by the square root of the sample size (step 2), as shown below.

Standard Error = s / √n

Relevance and Use of Standard Error Formula

It is very important to understand the concept of standard error as it predominantly used by statisticians as it allows them to measure the precision of their sampling method. Statisticians usually use the sample from a large pool of data as it is difficult to process such a huge data set, and as such, sampling makes the task a lot easier. So, standard error helps estimate how far the sample mean from the true population means.

In the case of finite population standard deviation, an increase in sample size will eventually reduce the standard error of the sample mean to zero as the population’s estimation will improve. Additionally, the sample standard deviation will also become approximately equal to the population standard deviation with the increase in sample size.

In the normally distributed sampling distribution, the sample mean, quantiles of the normal distribution and standard error can be used in the calculation of the population mean’s confidence intervals.**

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.

Example

To understand the standard error formula better, it may help to go through an example. Say we have a population of 80 people and we are interested in their height. We measure their height and calculate the standard deviation as 30.6 cm. We now need to plug these values into our equation:

If you are uncomfortable with inputting equations into calculators, you can break down the formula into manageable chunks. Here are the steps you can take.

  1. Firstly, calculate the square root of the number of samples (n). In this case, n is 80. The square root of 80 is 8.94.
  2. Next, divide the standard deviation (30.6) by the square root of 80 (8.94). Doing this gives a value of 3.42.
  3. Therefore, the standard error in our population for height is 3.42 cm.

Example: Using the standard error formulaTo estimate the standard error for math SAT scores, you follow two steps.

First, find the square root of your sample size (n).

Next, divide the sample standard deviation by the number you found in step one.

The standard error of math SAT scores is 12.8.

How should you report the standard error?

You can report the standard error alongside the mean or in a confidence interval to communicate the uncertainty around the mean. Example: Reporting the mean and standard errorThe mean math SAT score of a random sample of test takers is 550 ± 12.8 (SE).

The best way to report the standard error is in a confidence interval because readers won’t have to do any additional math to come up with a meaningful interval.

A confidence interval is a range of values where an unknown population parameter is expected to lie most of the time, if you were to repeat your study with new random samples.

With a 95% confidence level, 95% of all sample means will be expected to lie within a confidence interval of ± 1.96 standard errors of the sample mean.

Based on random sampling, the true population parameter is also estimated to lie within this range with 95% confidence. Example: Constructing a 95% confidence intervalYou construct a 95% confidence interval (CI) to estimate the population mean math SAT score.

For a normally distributed characteristic, like SAT scores, 95% of all sample means fall within roughly 4 standard errors of the sample mean.

With random sampling, a 95% CI [525 575] tells you that there is a 0.95 probability that the population mean math SAT score is between 525 and 575.

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