standarddeviationcalculator.net

Updated

How to calculate the relative standard deviation

The relative standard deviation, %RSD, is the standard deviation as a fraction of the mean, in percent. It answers "how large is the noise relative to the signal?", which an absolute standard deviation cannot, because 0.05 of points is negligible at a mean of 100 and enormous at a mean of 0.1.

Separate numbers with commas, spaces or new lines, or paste a spreadsheet column. Decimals and negatives are fine; write 10:3 for a value that occurs 3 times.

Try:
Calculation type

Not sure which? How to choose sample or population

Relative standard deviation

0.596845%

The standard deviation (0.06047) is 0.597% of the mean (10.13). Because it has no units, this figure can be compared across datasets measured on different scales.

Count (n)
6
Mean (x̄)
10.1317
Standard deviation (s)
0.0604704
Variance (s²)
0.00365667
Minimum
10.05
Q1 (25%)
10.09
Median
10.135
Q3 (75%)
10.1725
Maximum
10.21
Range
0.16
More statistics (5)
Standard error
0.0246869
Coefficient of variation
0.00596845
Sum (Σx)
60.79
Sum of squares, Σ(x − x̄)²
0.0182833
IQR (Q3 − Q1)
0.0825

Data distribution

10 10.1 10.2 10.3 mean 10.13 −1 SD +1 SD 10.12: 0.193 SD below the mean10.08: 0.854 SD below the mean10.21: 1.3 SD above the mean10.15: 0.303 SD above the mean10.05: 1.35 SD below the mean10.18: 0.799 SD above the mean Value

Shaded bands mark ±1, ±2 and ±3 SD from the mean. 4 of 6 values (67%) fall within ±1 SD.

Chart as text

Mean 10.1317, sample standard deviation s = 0.0604704, from 6 values between 10.05 and 10.21.

  • Within ±1 SD (10.07 to 10.19): 4 of 6 values (67%). About 68% for normal data.
  • Within ±2 SD: 6 (100%). About 95% for normal data.
  • Within ±3 SD: 6 (100%). About 99.7% for normal data.
Show the working, step by step
More statistics
The standard deviation (0.06047) is 0.597% of the mean (10.13). Because it has no units, this figure can be compared across datasets measured on different scales.

The formula

The relative standard deviation is the sample standard deviation s divided by the mean x̄, written as a percentage:

%RSD = s ÷ x̄ × 100%

It is the same quantity as the coefficient of variation, which the coefficient of variation calculator reports as a ratio, 0.055 on the example below, against 5.45% when written as a percentage.

Worked example

Six replicate measurements of the same quantity: 10.12, 10.08, 10.21, 10.15, 10.05, 10.18.

  1. Mean: 60.79 ÷ 6 = 10.132.
  2. Deviations from the mean: −0.012, −0.052, +0.078, +0.018, −0.082, +0.048.
  3. Squared deviations: 0.00014, 0.00267, 0.00614, 0.00034, 0.00667, 0.00234. Their sum is 0.01828.
  4. Sample variance: 0.01828 ÷ (6 − 1) = 0.00366, so s = 0.0605.
  5. Relative standard deviation: 0.0605 ÷ 10.132 × 100 = 0.597%.

The measurement holds to about six in a thousand of its value, which is the figure to quote when comparing this instrument against another: two measurements agree to better than one part in a thousand only if each is repeatable to that degree. The RSD calculator runs these five steps on any dataset.

Relative standard deviation guide: the worked example on this page, with its result and chart
Relative standard deviation guide: the worked example above, at a glance.

Common questions

What is a good relative standard deviation?

For a measurement that is supposed to be repeatable, RSD under 1% is usually tight, 1–5% is moderate, and above 5% the variation is dominating the signal. The number that "counts" depends on the field: quality labs quote %RSD on the method, not the value, and compare it with their own acceptance criterion, not a universal limit. If your RSD is larger than the difference you care about, the measurement noise swallows the signal, and you need more repeats, not a finer scale.

How do you calculate the relative standard deviation?

Divide the standard deviation by the mean and multiply by 100. On the six measurements above, s = 0.0605 and x̄ = 10.13, so %RSD = 0.0605 ÷ 10.13 × 100 = 0.597%. The sample standard deviation uses the n − 1 divisor, the same one the variance calculator uses; switch the calculator to population mode and both figures change slightly.

What is the difference between RSD and the coefficient of variation?

The same number with different names. The coefficient of variation, CV, is s ÷ x̄ as a ratio; RSD is the percentage form. On the example above the CV is 0.0605 ÷ 10.13 = 0.0060 and the RSD is 0.60%, the same relative spread written two ways.

When should I not use the relative standard deviation?

When the mean is near zero, where a tiny denominator blows the ratio up, when the data are not symmetric, or when the measurement cannot meaningfully have a zero point such as a temperature in Celsius. The standard deviation and variance pages cover the underlying measures; use RSD only when the unit matters and the mean is well away from zero.

Does RSD get smaller as I add more measurements?

RSD describes the data you have, so it changes with every added value. What shrinks with more repeats is the standard error of the mean, s ÷ √n, which is the uncertainty of the average itself. The standard error calculator gives that second figure, and the sample size calculator works out how many repeats you need to hit a target precision.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.

Advertisement

Advertisement