To calculate standard deviation from %RSD and the mean, use SD = (RSD × mean) / 100. To calculate the mean from %RSD and the standard deviation, use mean = (SD × 100) / RSD. For example, an RSD of 2.5% at a mean of 80 mg gives an SD of 2.5 × 80 / 100 = 2.0 mg.
Both formulas come from the standard RSD definition by simple algebra. The rest of this guide shows each rearrangement step by step, works through laboratory examples, and covers the unit and percentage mistakes that cause most wrong answers.
The Two Rearranged Formulas
Start from the definition used on the RSD Calculator home page:
Here s is the standard deviation, x̄ is the arithmetic mean and RSD is expressed as a percentage. Solving that one equation for each unknown gives two working formulas.
Pick the formula based on what you already have.
| You know | You want | Formula |
|---|---|---|
| %RSD and mean | Standard deviation | SD = (RSD × mean) / 100 |
| %RSD and standard deviation | Mean | Mean = (SD × 100) / RSD |
| SD and mean | %RSD | RSD = (SD / mean) × 100 |
| CV (decimal) and mean | Standard deviation | SD = CV × mean |
The last row matters because some reports give the coefficient of variation as a decimal (0.025) rather than a percentage (2.5%). A decimal CV has already been divided by 100, so the factor of 100 disappears from the formula.
Rearranging for Standard Deviation, Step by Step
The goal is to isolate s on one side of the equation. Each step applies the same operation to both sides, so the equation stays balanced.
- Write the definition: RSD = (s / x̄) × 100
- Divide both sides by 100: RSD / 100 = s / x̄
- Multiply both sides by x̄: (RSD / 100) × x̄ = s
- Reorder: s = (RSD × x̄) / 100
Step 2 is where the percentage turns back into a plain ratio. An RSD of 2.5% becomes 0.025, which is the fraction of the mean that one standard deviation represents.
A quick sanity check helps. If the RSD doubles while the mean stays fixed, the SD must double too. The formula agrees, because s is directly proportional to RSD.
Rearranging for the Mean, Step by Step
This time the goal is to isolate x̄. Because the mean sits in the denominator, one extra step is needed.
- Write the definition: RSD = (s / x̄) × 100
- Multiply both sides by x̄: RSD × x̄ = s × 100
- Divide both sides by RSD: x̄ = (s × 100) / RSD
The mean is inversely proportional to RSD when the standard deviation is fixed. A larger RSD from the same SD means the mean must be smaller. That relationship is the reason RSD grows quickly as the mean approaches zero, which is covered in the edge-case section below.
Worked Examples: Standard Deviation From %RSD and Mean
Example 1: Method precision stated as %RSD
An analytical method report states a repeatability of 1.8% RSD for a control sample with a mean of 250 mg/L. What is the standard deviation in concentration units?
One standard deviation is 4.5 mg/L, so a band of plus or minus one SD around the mean runs from 245.5 to 254.5 mg/L. Having the SD in concentration units makes it easier to compare with a specification written in mg/L.
Example 2: Fill volume on a certificate
A supplier’s certificate reports a mean fill volume of 500.0 mL with an RSD of 0.35%.
If your own filling check has an SD of 3.1 mL at the same mean, your process is noticeably more variable than the supplier’s figure. Comparing SDs this way is only fair when both results use the same kind of standard deviation, which is discussed below.
Example 3: Getting the variance
Some statistical tests and uncertainty budgets need the variance rather than the SD. Recover the SD first, then square it. From Example 1:
Variance carries squared units. Do not try to square the RSD and multiply by the mean in one step, because that mixes percentages and units in a way that is easy to get wrong.
Worked Examples: Mean From Standard Deviation and %RSD
Example 4: Recovering an unreported mean
A paper reports that replicate measurements of a trace element had a standard deviation of 0.42 µg and an RSD of 3.5%, but the mean is only shown in a figure. The mean is:
Check it by working forward: 0.42 / 12 × 100 = 3.5%. Running the original formula on your answer is the fastest way to catch an algebra slip.
Example 5: The lowest level that meets an RSD target
Suppose an instrument shows a roughly constant standard deviation of 0.06 mg/L at low concentrations, and you need an RSD of 2% or less. The mean that gives exactly 2% is:
Below about 3.0 mg/L, the same SD becomes a larger percentage of a smaller mean, so the RSD rises above 2%. This is a useful planning estimate, but real SDs usually change with concentration. Confirm the estimate with replicate measurements near the level you care about.
Which Standard Deviation Did You Recover?
The rearranged formula returns whichever standard deviation was used to calculate the RSD. If the source used the sample SD (dividing by n − 1), you recover a sample SD. If it used the population SD (dividing by n), you recover a population SD.
Most laboratory RSD values are based on the sample SD. When you need to convert between the two and you know the number of values, n, use:
For s = 4.5 mg/L from n = 6 replicates, σ = 4.5 × √(5/6) = 4.11 mg/L. The difference shrinks as n grows. The sample vs population standard deviation guide explains when each divisor is appropriate.
Keep the Units Consistent
RSD has no units, but the mean and the standard deviation always share the units of the original measurements. The rearranged formulas only work when those units match.
| Quantity | Units | Example |
|---|---|---|
| Mean (x̄) | Same as the data | 2.4 mg/L |
| Standard deviation (s) | Same as the data | 0.036 mg/L |
| Variance (s²) | Data units squared | 0.001296 (mg/L)² |
| RSD | Percent, no units | 1.5% |
A common trap is a mean in mg/L and an SD reported in µg/L. Take a mean of 2.4 mg/L and an SD of 36 µg/L. Converting the SD first gives 0.036 mg/L and an RSD of 0.036 / 2.4 × 100 = 1.5%. Skipping the conversion gives 36 / 2.4 × 100 = 1,500%, a result that is wrong by a factor of 1,000.
The same problem appears in reverse. If you recover an SD from an RSD and a mean in g, the SD is in g. Convert it before comparing it with a limit written in mg.
Percentage Conversion Mistakes
Most wrong answers from these formulas come from handling the percent sign. The table lists the usual errors with the same inputs (RSD = 1.8%, mean = 250).
| Mistake | What happens | Result | Correct |
|---|---|---|---|
| Forgetting to divide by 100 | s = 1.8 × 250 | 450 | 4.5 |
| Dividing by 100 twice (using 0.018 in the /100 formula) | s = 0.018 × 250 / 100 | 0.045 | 4.5 |
| Treating a decimal CV as a percent | Mean = 0.42 × 100 / 0.035 | 1,200 | 12 |
| Rounding RSD before back-calculating | Uses 1.2% instead of 1.2449% | Small bias | See below |
The rule is simple. Use the percentage value (1.8) with the formulas that contain 100, or use the decimal value (0.018) with the formulas that do not. Never combine a decimal with a formula that also divides by 100.
How rounding changes a recovered SD
Reported RSD values are often rounded to one or two decimal places. When you back-calculate, that rounding carries straight into the SD because the relationship is proportional.
Suppose the unrounded RSD was 1.2449% at a mean of 842.6. The SD is 842.6 × 1.2449 / 100 = 10.49. If you only have the rounded value 1.2%, you get 842.6 × 1.2 / 100 = 10.11, which is about 3.6% lower. For quick checks that difference rarely matters. For uncertainty budgets or pooled calculations, ask for unrounded values or recompute from the raw data.
Edge Cases: Zero RSD, Near-Zero Means and Negative Means
Zero RSD
An RSD of 0% means every value in the data set was identical, so the standard deviation is zero. The SD formula still works (0 × mean / 100 = 0). The mean formula does not, because it divides by RSD and division by zero is undefined. You cannot recover a mean from an SD of 0 and an RSD of 0%.
In practice, a reported RSD of exactly 0% usually means the instrument resolution was too coarse to show variation, or the result was rounded. Treat it as “below the reporting resolution” rather than perfect precision.
Near-zero means
When the mean is close to zero, a modest SD produces a very large RSD. An SD of 0.05 with a mean of 0.02 gives an RSD of 250%. The rearranged mean formula confirms this: 0.05 × 100 / 250 = 0.02.
Be careful with very large RSD values. An RSD of several hundred percent usually signals a mean near zero rather than unusually poor precision. Recovered means and SDs in that region are extremely sensitive to rounding and to small changes in the data. The home page section on limitations of RSD explains why relative measures break down near zero.
Negative means
RSD is intended for positive, ratio-scale data such as concentrations, masses and peak areas. NIST’s coefficient of variation reference notes that the measure should typically only be used for ratio-scale data, and defines the relative standard deviation with the absolute value of the mean, 100 × s / |x̄|, so that the percentage is always positive. If an RSD was calculated that way, the mean you recover with the rearranged formula is the magnitude of the mean. The formula cannot tell you whether the original mean was positive or negative. When a data set can be negative, report the SD and the mean directly instead of relying on RSD.
Using the Rearranged Formulas in a Spreadsheet
The formulas translate directly into cell references. With RSD (as a plain number such as 1.8) in B2 and the mean in C2:
- Standard deviation:
=B2*C2/100 - Mean, with the SD in D2 and RSD in B2:
=D2*100/B2
If B2 is formatted as a percentage and displays 1.8%, the stored value is 0.018. In that case use =B2*C2 without the division by 100. This is the spreadsheet version of the decimal-versus-percent mistake above. The Excel RSD guide covers percentage formatting and error handling in detail.
The same one-line rearrangements work in code. In R or Python, s = rsd * xbar / 100 and xbar = s * 100 / rsd give the same results as the worked examples, provided rsd holds the percentage value (1.8) rather than the fraction (0.018). Avoid naming the variables sd or mean in R, where those names belong to built-in functions.
Check Your Answer
Always run the original formula on your result. If you recovered an SD of 4.5 mg/L from a mean of 250 mg/L, then 4.5 / 250 × 100 should return 1.8%.
The RSD Calculator has a known SD and mean input mode for exactly this check. Enter the standard deviation and mean you recovered and confirm that the %RSD matches the value you started from. If you have the raw measurements, enter them instead and compare all three statistics at once.
Quick Reference
| Given | Find | Formula | Example |
|---|---|---|---|
| RSD = 2.5%, mean = 80 mg | SD | (2.5 × 80) / 100 | 2.0 mg |
| RSD = 3.4%, mean = 15.2 g | SD | (3.4 × 15.2) / 100 | 0.517 g |
| SD = 2.6, RSD = 0.85% | Mean | (2.6 × 100) / 0.85 | 305.9 |
| SD = 0.42 µg, RSD = 3.5% | Mean | (0.42 × 100) / 3.5 | 12 µg |
| CV = 0.018, mean = 250 | SD | 0.018 × 250 | 4.5 |
For a refresher on what the ratio itself represents, see What Is Relative Standard Deviation?. To practise these rearrangements alongside other RSD calculations, the RSD practice problems include a full set of reverse-formula questions with answers.
Frequently Asked Questions
Can I calculate standard deviation from RSD alone?
No. RSD is a ratio, so it only tells you the standard deviation relative to the mean. You need the mean as well: SD = (RSD × mean) / 100. The same RSD of 2% corresponds to an SD of 0.2 at a mean of 10 and an SD of 20 at a mean of 1,000.
How do I get the variance from %RSD and the mean?
First recover the standard deviation with SD = (RSD × mean) / 100, then square it. For RSD = 1.8% and mean = 250 mg/L, SD = 4.5 mg/L and the variance is 4.5² = 20.25 (mg/L)². Variance carries squared units.
What if the RSD is given as a decimal instead of a percentage?
A decimal value such as 0.018 is a coefficient of variation, not a percentage. Drop the 100 from the formulas: SD = CV × mean and mean = SD / CV. Using 0.018 in a formula that divides by 100 gives an answer 100 times too small.
Can I recover the original data values from the RSD?
No. Many different data sets share the same mean, standard deviation and RSD. The rearranged formulas give you summary statistics only, never the individual measurements.
Does the rearranged formula give the sample or the population standard deviation?
It gives whichever standard deviation was used to calculate the RSD in the first place. If the source RSD was based on the sample SD (n − 1), the SD you recover is the sample SD. Check the source before comparing it with other results.
