Relative standard deviation is defined as a percentage standard deviation that calculates how much the data entries in a set are distributed around the mean value. It tells whether the regular standard deviation is a small or high number when compared to the data set’s mean. In other words, it indicates the percentage distribution of the data. If a data set has a greater relative standard deviation, it clearly indicates that the numbers are significantly far off from the meanwhile, a lower value means that the figures are closer than the average.

It is also called the coefficient of variation. Its formula is equal to the ratio of the standard deviation of the data set to the mean multiplied by 100. Its unit of measurement is a percentage (%).

What is Relative Standard Deviation?

Relative Standard Deviation (RSD) is a statistical measure that describes the precision of a set of measurements relative to the mean of the data set. It is expressed as a percentage and provides an indication of how spread out the values are around the mean. RSD is particularly useful in comparing the variability of datasets that have different units or different magnitudes.

The image added below shows the formula for the Relative Standard Deviation.

Relative Standard Deviation Formula

Relative Standard Deviation Formula

Relative Standard Deviation Formula is added below:

R = (σ / x̄) × 100

where,

• R is Relative Standard Deviation
• σ is Standard Deviation
• x̄ is Mean of Data Set

Examples Relative Standard Deviation Formula

Example 1: Calculate the relative standard deviation of the data set: 2, 5, 7, 3, 1. 

Solution:

We have,

x̄ = (2 + 5 + 7 + 3 + 1)/5 = 3.6 

σ = √((2 – 3.6)² + (5 – 3.6)² + (7 – 3.6)² + (3 – 3.6)² + (1 – 3.6)²)/(5 – 1)

= √(23.2/4)

= 2.4

Using the formula we get,

R = (σ / x̄) × 100

= (2.4/3.6) × 100

= 66.9%

Example 2: Calculate the relative standard deviation of the data set: 4, 7, 1, 3, 6.

Solution:

We have,

x̄ = (4 + 7 + 1 + 3 + 6)/5 = 4.2

σ = √((4 – 4.2)² + (7 – 4.2)² + (1 – 4.2)² + (3 – 4.2)² + (6 – 4.2)²)/(5 – 1)

= √(22.8/4)

= 2.38

Using the formula we get,

R = (σ / x̄) × 100

= (2.38/4.2) × 100

= 56.84%

Example 3: Calculate the relative standard deviation of the data set: 5, 9, 3, 6, 4.

Solution:

We have,

x̄ = (5 + 9 + 3 + 6 + 4)/5 = 5.4

σ = √((5 – 5.4)² + (9 – 5.4)² + (3 – 5.4)² + (6 – 5.4)² + (4 – 5.4)²)/(5 – 1)

= √(21.2/4)

= 2.30

Using the formula we get,

R = (σ / x̄) × 100

= (2.30/5.4) × 100

= 42.63%

Example 4: Calculate the standard deviation of the data set if the relative deviation is 45% and the mean is 6.

Solution:

We have,

x̄ = 6

R = 45%

Using the formula we get,

R = (σ / x̄) × 100

⇒ σ = Rx̄/100

⇒ σ = (45 × 6)/100

⇒ σ = (270)/100

⇒ σ = 27

Example 5: Calculate the standard deviation of the data set if the relative deviation is 67% and the mean is 3.4.

Solution:

We have,

x̄ = 3.4

R = 67%

Using the formula we get,

R = (σ / x̄) × 100

⇒ σ = Rx̄/100

⇒ σ = (67 × 3.4)/100

⇒ σ = (227.8)/100

⇒ σ = 22.78

Example 6: Calculate the mean of the data set if the relative deviation is 47% and the standard deviation is 10.

Solution:

We have,

σ = 10

R = 47%

Using the formula we get,

R = (σ / x̄) × 100

⇒ x̄ = (σ / R) × 100

⇒ x̄ = (10/47) × 100

⇒ x̄ = 21.2

Example 7: Calculate the mean of the data set if the relative deviation is 78% and the standard deviation is 1.5.

Solution:

We have,

σ = 1.5

R = 78%

Using the formula we get,

R = (σ / x̄) × 100

⇒ x̄ = (σ / R) × 100

⇒ x̄ = (1.5/78) × 100

⇒ x̄ = 1.92

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Practice Problems on Relative Standard Deviation Formula

Q1. Dataset: 15, 18, 21, 24, 27 .Calculate the RSD.

Q2. A group of students received the following test scores: 72, 85, 90, 78, and 88. Calculate the RSD of the test scores.

Q3. Dataset: 100, 105, 95, 110, 90 .Find the RSD to two decimal places.

Q4. The heights (in cm) of six plants are measured as follows: 25, 30, 28, 32, 27, and 29. Determine the RSD of the plant heights.

Q5. Dataset: 5.2, 5.4, 5.1, 5.3, 5.5 .Determine the RSD as a percentage.

Q6. Dataset: 1000, 1200, 950, 1100, 1050.Calculate the RSD and express it as a decimal.

Q7. Dataset: 0.01, 0.015, 0.012, 0.018, 0.014.Find the RSD to three significant figures.

Q8. Following are the marks obtained in by 4 students in mathematics examination: 60, 68, 65, 85. Calculate the relative standard deviation ?

Q9. 5 measurements were collected as a sample 51,55,49,58 and 52. Calculate the relative standard deviation.

Q10. A survey asked participants to rate a product on a scale of 1 to 10, with the following ratings: 7, 8, 6, 9, and 7. Calculate the RSD of the survey rating.

Summary

Relative standard deviation (RSD) is a special type of standard deviation (SD). The formula for relative standard deviation helps you understand if the standard deviation is small or large compared to the mean of a data set. The more precise the data, the smaller the is the RSD. The RSD is generally written with the mean and a plus/minus symbol: 3.5 ± 2.86% For example, if the standard deviation is 0.1 and the mean is 3.5, then the RSD for this set of numbers is 100 x 0.1 / |3.5| = 2.86%. The standard deviation is 2.86% of the mean value of 3.5, which is very small. In other words, the data points are close to the mean. On the other hand, if the relative standard deviation is high (say 56%), it means that the data is more dispersed.

FAQs on Relative Standard Deviation Formula

How to Calculate Relative Standard Deviation?

Often the relative standard deviation (RSD) is more useful: it is expressed as a percentage and is obtained by multiplying the standard deviation by 100 and dividing this product by the mean.

What is an Example of a RSD Calculation?

RSD tells you whether a “normal” standard deviation is small or large compared to the mean of a dataset. For example, say your experiment finds that the standard deviation is 0.1 and the mean is 4.4. The RSD for this set of numbers is: 100 x 0.1 / |4.4| = 2.3%.

What is the Mean Relative Deviation Formula?

To compare the mean deviation of several samples of several variables with different units, it is recommended to use the relative version of the mean deviation, which is defined by dividing the mean deviation by the absolute value of the mean

What is the RSD in Statistics?

Relative standard deviation (RSD or %RSD) is the absolute value of the coefficient of variation. It is often expressed as a percentage. A similar term is sometimes used: relative variance. It is the square of the coefficient of variation.