Standard Error vs. Standard Deviation

Standard Error quantifies the variability between sample drawn from the same population, whereas the standard deviation quantifies the variability of values in a dataset. In this article we will discuss Standard Error vs. Standard Deviation.

Standard deviation and Standard error are the two important concepts of statistics. Standard deviation measures the variability within a single sample, whereas standard error estimates the variability across multiple samples of a population. In this article, we will briefly discuss Standard Error vs. Standard Deviation.
The differences between Standard Error and Standard Deviation are totally based on Descriptive and Inferential Statistics.
Now, let’s explore some more differences between them.

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Table of Content

• Standard Deviation vs. Standard Error
• What is Standard Deviation?
• What is Standard Error?
• Example of Standard Deviation and Standard Error
• Key differences and similarities between Standard Deviation and Standard Error

Standard Deviation vs. Standard Error

Parameter	Standard Deviation	Standard Error
Definition	Quantifies the variability of values in a dataset.	Quantifies variability between the sample drawn from the same population.
Statistics	Descriptive	Inferential
Distribution	Distribution of observation around the normal curve.	Distribution of an estimate around the normal curve.
Measures Variability	Within a sample	Between Samples
Sample Size	Increase in sample size gives a more specific measure of standard deviation.	An increase in the sample size decreases standard error.
Formula	Square root of the variance	Standard deviation / square root of sample size

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What is Standard Deviation?

Standard deviation is the square root of the averages of the square deviation from the mean. In simple terms, the standard deviation is defined as the square root of variance.

• Standard deviation measures how far a group of observations is from the mean.
• Standard deviation is very sensitive to outliers.
• The value of the standard deviation will be zero if all the dataset values are equal.

Formula

• Population

• Sample

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What is Standard Error?

As we take different samples from the mean, there will be different means corresponding to different samples for the same population ( this is called the sampling distribution of the mean). Now, the standard deviation is used to estimate the variance between different means of the sample. This is what you call the standard error of the estimate of the mean.

• The standard error is a measure of how precise our estimate is from the mean.
• It is mainly used in Hypothesis Testing and estimating intervals.
• It estimates the variability across multiple samples of a population.
• The sample size is inversely proportional to Standard Error, i.e., an increase in population size will decrease the standard error.

Formula

S.E. = (S.D) /sqrt (n), 

where,

S.D. – Standard Deviation

n – number of elements in the sample

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Example of Standard Deviation and Standard Error

The wholesale price of a commodity for seven consecutive days in a month is as follows:

Days	1	2	3	4	5	6	7
Commodity Price/Quintal	240	260	270	245	255	286	264

Calculate the Standard Deviation and standard error.

xi	(xi – mean)2
240	(240 – 260)2
260	(260 – 260)2
270	(270 – 260)2
245	(245 – 260)2
255	(255 – 260)2
286	(286 – 260)2
264	(264 – 260)2
Mean = 1820 / 7 = 260	Sum of Squared Deviation = 1442

Now, standard deviation = sqrt ( sum of squared difference / n) = sqrt ( 1442 /7) = sqrt (206) = 14.35

Hence, Standard Deviation = 14.35

Now, Standard Error = Standard Deviation / sqrt (n) = 14.35/ sqrt(7) = 14.35 / 2.646 = 5.42

Hence, Standard Error = 5.42.

Therefore, the standard deviation and standard error are 14.35 and 5.42 respectively.  

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The key difference between Standard Deviation and Standard Error

• Standard deviation assesses how far a data point is likely to fall from the mean. In contrast, Standard Error assesses how far a sample statistic falls from a population parameter.
• The standard error is an inferential statistic, whereas the standard deviation is a descriptive statistic.
• As the sample size increases, it gives a more specific measure of standard deviation, whereas an increase in the sample size decreases standard error.
• With reference to the normal curve, the standard deviation is the distribution of observations, whereas standard error is the distribution of estimates.
• Standard deviation is calculated by taking the square root of the variance, whereas standard error is calculated by dividing the standard deviation by the square root of the sample size.
• Standard deviation measures the variability within a single sample, whereas standard error estimates the variability across multiple population samples.