How to Calculate Percentile from Mean and Standard Deviation -

Summary (What you’ll learn)

The exact formula to calculate percentile from mean and standard deviation under the normal distribution
How to convert a raw score to a z-score and then to a percentile
Step-by-step examples, plus Excel/Google Sheets formulas
When this approach is valid and when it’s not
A simple, embeddable infographic + mini-calculator you can use on your page

Prefer a quick tool? Try our percentile from mean and standard deviation calculator.

If you’ve ever asked how to calculate percentile from mean and standard deviation, you’re usually dealing with test scores, grading curves, or benchmarked KPIs. Under the assumption of a normal distribution, you can translate any raw value into a percentile rank with just the mean (μ) and standard deviation (σ). This guide explains the method clearly, shows examples, and gives you a copy-paste infographic you can embed. We’ll also cover related workflows like z-score to percentile, finding a score from a target percentile, and caveats when the data isn’t normal.

Key concepts (semantic overview)

Mean (μ): the center of your distribution
Standard Deviation (σ): average spread around the mean
Z-score (z): standardized distance from the mean
Percentile (%): proportion of observations at or below a value

The formula (normal model)

1) Raw score → z-score

z = x−μσz \;=\; \frac{x – \mu}{\sigma}z=σx−μ

2) z-score → percentile

Percentile = Φ(z)×100%\text{Percentile} \;=\; \Phi(z)\times 100\%Percentile=Φ(z)×100%

Here, Φ(z)\Phi(z)Φ(z) is the cumulative distribution function (CDF) of the standard normal. In plain English: it’s the area under the bell curve to the left of zzz.

Shortcut in spreadsheets:

Excel/Google Sheets: =NORM.DIST(x, mean, sd, TRUE)*100

If you already have z: =NORM.S.DIST(z, TRUE)*100

Z-Score Explained

Worked examples

Example 1 — What percentile is a score of 82?

Mean μ=75\mu = 75μ=75
Standard deviation σ=10\sigma = 10σ=10
Value x=82x = 82x=82

Step 1: z=(82−75)/10=0.7z = (82 – 75) / 10 = 0.7z=(82−75)/10=0.7
Step 2: Percentile =Φ(0.7)×100≈75.8%= \Phi(0.7)\times 100 \approx 75.8\%=Φ(0.7)×100≈75.8%

Interpretation: A score of 82 is around the 76th percentile.

Convert Percentile to Score Calculator

Example 2 — What score corresponds to the 90th percentile?

Sometimes you want the reverse: given a percentile, find the score.

Formula: x = μ+zp⋅σx \;=\; \mu + z_p \cdot \sigmax=μ+zp⋅σ

where zpz_pzp is the z-value for percentile ppp. For the 90th percentile, z0.90≈1.2816z_{0.90} \approx 1.2816z0.90≈1.2816.

Mean μ=500\mu = 500μ=500
SD σ=100\sigma = 100σ=100
90th percentile score x=500+1.2816×100≈628.16x = 500 + 1.2816 \times 100 \approx 628.16x=500+1.2816×100≈628.16

Excel/Google Sheets: =NORM.INV(0.9, 500, 100)

Practical notes and assumptions

The method assumes approximate normality. Many educational test scores and measurement errors are near-normal, but not all datasets are.
For skewed or bounded data (e.g., times, percentages with floor/ceil), consider transformations or use empirical percentiles from the actual data.
With small samples, the sample mean/SD make the percentile an estimate, not an exact value.
If you need a quick, user-friendly result without the math.

Use our percentile from mean and standard deviation calculator.

Step-by-step guide (checklist)

Gather inputs: mean (μ), standard deviation (σ), and your score (x).
Standardize: compute z=(x−μ)/σz = (x – \mu)/\sigmaz=(x−μ)/σ.
Convert to percentile: use a z-table or software to compute Φ(z)×100%\Phi(z)\times 100\%Φ(z)×100%.
Interpret: state the result clearly (“This is the 84th percentile”).
Caveat: confirm the data is roughly normal or present the result as an approximation.

How to do it in Excel/Google Sheets

Raw → Percentile:
=NORM.DIST(x, mean, sd, TRUE)*100
Z → Percentile:
=NORM.S.DIST((x-mean)/sd, TRUE)*100
Percentile → Raw:
=NORM.INV(p, mean, sd) (where p is between 0 and 1)

Percentile from Mean & Standard Deviation

Assumes an approximately normal distribution

Mean (μ)

Standard Deviation (σ)

Score (x)

Result: Enter values and click “Calculate Percentile”.

Bell curve (normal) Marker = your score

Formulas

z = (x − μ) / σ

Percentile (%) = Φ(z) × 100

Need a full-featured tool? Try our percentile from mean and standard deviation calculator.

Interpreting your results (and common questions)

Is “calculate percentile from mean and standard deviation” always valid?

Only if the underlying distribution is approximately normal. If your data are skewed or have outliers, prefer empirical percentiles (rank the data and compute positions) or use a non-parametric percentile estimator.

What’s the relationship between z-score and percentile?

z-score is how many standard deviations a value is from the mean.
Percentile is the proportion of observations at or below a value.
The standard normal CDF, Φ(z)\Phi(z)Φ(z), converts one to the other.

What does the 50th percentile mean?

It’s the median under symmetry (normal distribution), corresponding to z=0z=0z=0 ⇒ Φ(0)=0.5\Phi(0)=0.5Φ(0)=0.5.

How do I get the score for the 95th percentile?

Use x=μ+z0.95σx = \mu + z_{0.95}\sigmax=μ+z0.95σ where z0.95≈1.6449z_{0.95}\approx1.6449z0.95≈1.6449.
Excel/Sheets: =NORM.INV(0.95, mean, sd).

Can I avoid the math?

Yes—use our <a href=”https://gradewisecalculator.com/percentile-from-mean-standard-deviation-calculator/” target=”_blank” rel=”noopener”>percentile from mean and standard deviation calculator</a> to convert scores to percentiles instantly.

Troubleshooting & tips

σ must be positive. If SD is 0, all values equal the mean and percentiles aren’t defined.
Units don’t matter for z. Whether points or dollars, z-scores are unitless.
Rounding: Report percentiles to 1–2 decimals for clarity.
Quality check: Plot a histogram or QQ-plot to check normality before reporting “precise” percentiles.

Final thoughts

Knowing how to calculate percentile from mean and standard deviation lets you quickly turn raw scores into meaningful, comparable ranks. When your data are roughly normal, the z-score method is fast, transparent, and easy to implement in spreadsheets. When in doubt—or if you just want a quick answer—use the embedded infographic above or the dedicated

percentile from mean and standard deviation calculator.percentile from mean and standard deviation calculator.

FAQ (quick answers)

Q1. How do you calculate percentile from mean and standard deviation?

Compute z=(x−μ)/σz=(x-\mu)/\sigmaz=(x−μ)/σ then percentile =Φ(z)×100%=\Phi(z)\times 100\%=Φ(z)×100% (normal model). This is the standard method people mean when they ask how to calculate percentile from mean and standard deviation.

Q2. Can I do this in Excel or Google Sheets?

Yes. =NORM.DIST(x, mean, sd, TRUE)*100 returns the percentile. For a target percentile, use =NORM.INV(p, mean, sd).

Q3. What if my data aren’t normal?

Use empirical percentiles from the actual dataset (rank-based), or transform the data first. Otherwise, the “percentile from mean and standard deviation” will be an approximation.

Q4. What’s the difference between percentile and percentile rank?

They’re often used interchangeably. Strictly, percentile rank is a percentage, while “the 80th percentile” refers to the value below which 80% of observations fall.

Q5. How accurate is the z-table approach?

For near-normal distributions, very accurate. For skewed or heavy-tailed data, accuracy declines—state that you used the normal approximation.