What is the Difference Between Standard Deviation and Standard Error?

The difference between the standard deviation and the standard error is that the standard deviation states the variability within a single sample, on the other hand, the standard error states variability with various samples of a population. To know about them in more detail, you can read through this blog till the end.

What is Standard Deviation?

The standard deviation is a typical deviation from the “Mean”. It is a popular measure of variability as it returns to the original units of the data set. The standard deviation calculates the extent to which the values differ from the average. 

It is the most common means of dispersion, is based on all the possible values. The standard deviation is useful in certain advanced statistical problems. Also, this is independent of origin but not of scale.

Standard deviation is of two types:

• Population Standard Deviation
• Sample Standard Deviation

What is Standard Error?

In statistics, standard error means the standard deviation of the sample data distribution. The standard error is represented as SE. The smaller the standard error, the more representative the sample will be of the overall population. 

To find out the standard error, you need to divide the standard deviation by the square root of the sample size. Various statistical software packages calculate the standard errors automatically.

What is the Difference between Standard Deviation and Standard Error

The differences between the standard error and the standard deviation are substantial. The validity of the data displayed at a particular standard deviation stage is ascertained using standard deviation. On the other hand, standard errors are more useful in assessing sample accuracy through the examination of deviations from the means. You can look through the provided table of differences to learn more about the specific differences between the two. 

Particulars	Standard Deviation	Standard Error
Definition	Standard deviation is a typical deviation from the “Mean”. It is a popular measure of variability as it returns to the original units of the data set.	In statistics, standard error means the standard deviation of the sample data distribution. The standard error is represented as SE.
What it measures	Variability of data points around the sample mean.	Variability of the sample mean around the population mean.
Effect of sample size	Unaffected by sample size	Decreases with a larger sample size
Distribution	Distribution of observation around the normal curve.	Distribution of an estimate around the normal curve.
Formula	Square root of the variance	Standard deviation / square root of sample size

Standard Deviation and Standard Error Formulas 

The standard error can be calculated using the following formula: 

• Sample size squared root / standard deviation. 

To represent it in symbolic form, we can use the symbols like this:

Moreover, the square root of the variance’s result can be used to compute the standard deviation. To represent the formula for standard deviation in symbols, you an use the given formula:

Standard Deviation

Definition

Standard deviation is a typical deviation from the “Mean”. It is a popular measure of variability as it returns to the original units of the data set.

Example

Question: The weights (in kilograms) of 5 pet cats are recorded: 4, 6, 8, 5, and 7. Find the standard deviation of their weights.

Solution: Add all the weights and divide by the number of cats (5).

Mean weight = (4 + 6 + 8 + 5 + 7) / 5 = 6 kg

Now, by adding these values to the above-stated formula and solving the equation, we will get the result of 1.58 kilogrammes.

Sample Question

A bakery records the temperature (in degrees Celsius) of their ovens throughout the day: 200, 215, 190, 205, and 220. Find the standard deviation of the oven temperatures.

Standard Error

Definition

In statistics, standard error means the standard deviation of the sample data distribution. The standard error is represented as SE.

Example

Question: A software company is considering investing in a new cloud service. To estimate the potential monthly cost savings, they track the server costs for the past 35 days. The average daily cost saving is found to be $25, with a standard deviation of $3.20 across the sample. What is the standard error of the mean daily cost saving for this sample period?

Solution: Substituting the values in the formula: SE = standard deviation (SD) / √(sample size), and by solving them we get the result as $0.53.

Sample Question

During a taste test, 20 volunteers rated a new chocolate bar on a scale of 1 (dislike) to 10 (love). The average rating for the chocolate bar was 7.5 points, with a standard deviation of 1.2 points within the sample. What is the standard error of the mean rating for this taste test?

Application of Standard Deviation and Standard Error in Real Life

You can use standard deviation in various aspects of practical life. For example, standard deviation (SD) can help you understand about your students’ scores. It tells you how much, on average, each student’s score differs from the class average. 

Similarly, there can be another example for this, like, a streaming service collects ratings (1-5 stars) for a new movie and the average rating is 3.8 stars. The high standard deviation means viewers have mixed opinions.

Similarities Between Standard Deviation and Standard Error

Standard deviation and standard error are both important concepts in statistics that deal with the spread of data. Here we ave listed their similarities:

• Both standard deviation and standard error are used to quantify how spread out the data points are from an average value (mean). A higher value indicates greater variability or dispersion in the data.
• They are expressed in the same units as the data itself. For example, if your data is in meters, both standard deviation and standard error will be in meters.
• Both can be seen as a measure of uncertainty. Standard deviation reflects how certain we are about the average value representing the entire dataset (sample), while standard error reflects the uncertainty in estimating a population parameter (like the population mean) based on a sample.pen_spark

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