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Relative Standard Deviation Formula

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  • Last Updated : 25 Jun, 2022
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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 (%).

Relative Standard Deviation Formula

R = (σ / x̄) × 100

Where,

  • R is the relative standard deviation,
  • σ is the standard deviation,
  • x̄ is the mean of data set.

Sample problems

Problem 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%

Problem 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%

Problem 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%

Problem 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

Problem 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

Problem 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

Problem 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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