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What is the Standard Error of the Estimate? - Formula & Examples

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  • 0:00 Definition of Standard Error
  • 1:55 Calculating a Standard Error
  • 3:00 Example of Standard Error
  • 4:17 Interpreting Standard Error
  • 5:51 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

When you take measurements of some quantity in a population, it is good to know how well your measurements will approximate the entire population. In this lesson, learn how to calculate the standard error of your estimate and interpret your results.

Definition of Standard Error

As part of a high school project, let's say you decide to measure how tall each of the players on your school's basketball team are. You find that the average height of the players on the team is 72 inches. Is this a good estimate of the height of all basketball players? How would you know, and is there a way to quantify exactly HOW good of an estimate this measure is? In fact, there's a way to quantify this, but before you can answer these questions, we first need to think about the difference between a sample and a population.

In statistics, the word sample refers to the specific group of data collected. In this case, the sample would be the data you collected on the height of players on your school's team. A population is the entire group from which the sample was drawn. This could include all high school basketball players, all basketball players at any level, or any other group. There are many ways to define a population, and you always need to be very clear about what your population is. For this project, let's assume that you want to compare the heights of the basketball players on your school's team to the heights of all high school basketball players. Therefore, the population would be all high school basketball players.

Now, in order to determine how well your sample represents the population, do you need to go out and measure how tall every single high school basketball player is? No, of course you couldn't do this! Instead, you can calculate the standard error, which tells you how well your sample mean estimates the true population mean. A large standard error would mean that there is a lot of variability in the population, so different samples would give you different mean values. A small standard error would mean that the population is more uniform, so your sample mean is likely to be close to the population mean.

Calculating a Standard Error

To calculate the standard error, follow these steps:

  1. Record the number of measurements (n) and calculate the sample mean (μ). This is just the average of all the measurements.
  2. Calculate how much each measurement deviates from the mean (subtract the sample mean from the measurement).
  3. Square all the deviations calculated in step 2 and add these together:
    Σ(xi - μ)²
  4. Divide the sum from step 3 by one less than the total number of measurements (n - 1).
  5. Take the square root of the number you got in step 4. This is known as the standard deviation (σ).
  6. Finally, divide the standard deviation from step 5 by the square root of the number of measurements (n) to get the standard error of your estimate.

You'll often see these steps expressed as formulas like these, where σ is the standard deviation and SE is the standard error:


standard deviation and standard error


Example of Standard Error

There are a lot of steps required to find the standard error. Let's continue with our high school basketball player height example to make sure you understand how to perform these calculations.

Let's assume that this was the data you collected on basketball player height at your school:


table of player heights


The first step to find the standard error is to find the sample mean. You would do that by adding up all the heights and then dividing by the total number of measurements (n = 13). This would give you a sample mean of 72.

Next, calculate the difference between the sample mean and each measurement, square all these values, and then add them all up. This is easier to do if you make a table like the one you are looking at on your screen:


standard error data table


Then, divide the sum you just calculated by n - 1 and take the square root to get the standard deviation.


standard deviation calculation


Finally, to calculate the standard error of your estimate, divide the standard deviation by the square root of the number of measurements (remember: n = 13). So the standard error equals:


standard error calculation


So there you go.

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