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The probability distribution function of the normal distribution is the bell-curve formula that gives the density of a variable at each value, and its running total, the cumulative probability, is what you read to get an actual probability. The two are easy to confuse, and mixing them up is the most common error in worked examples. This guide separates them, computes both for the same number, and shows how a normal probability plot checks whether your data fit the curve at all.
Definition in one paragraph
A normal distribution is a continuous distribution described by two parameters: a location (the mean, μ) and a scale (the standard deviation, σ). Its probability density function (PDF) gives the height of the curve at each value x. Its cumulative distribution function (CDF) gives the probability that the variable is less than or equal to x, which is the area under the PDF from negative infinity up to x. Mean, median and mode all equal μ, skewness is 0 and kurtosis is 3 (NIST/SEMATECH e-Handbook). For the standardized version (μ = 0, σ = 1) and conversion by z-scores, see normal and standard normal distribution.
Normal distribution probability density function: the formula
The NIST/SEMATECH e-Handbook gives the density as:
f(x) = exp( -(x - μ)² / (2σ²) ) / ( σ × √(2π) )
- x: the value at which you evaluate the curve; it can be any real number, from negative to positive infinity.
- μ: the location parameter (the mean), which sets where the curve is centred.
- σ: the scale parameter (the standard deviation), which sets how wide the curve is. It must be positive.
- π: the constant 3.14159…, so √(2π) is about 2.50663.
- exp: the exponential function, e raised to the power in brackets.
When μ = 0 and σ = 1 the formula reduces to the standard normal density, exp(-x²/2)/√(2π). In practice μ and σ are estimated by the sample mean and sample standard deviation.
Normal distribution cumulative probability (the CDF)
The cumulative probability P(X ≤ x) is the integral of the density from negative infinity to x. According to NIST, this integral has no simple closed formula and is computed numerically. That is why you never evaluate it by hand: you use software or a standard normal table after converting to a z-score. The CDF always rises from 0 to 1, and the inverse of the CDF (the percent point function, or quantile function) turns a probability back into a value. For the general idea, see the guide to the cumulative distribution function.
Worked example: density versus probability
Suppose a test score is modelled as normal with μ = 100 and σ = 15. Compute the constant first: σ × √(2π) = 15 × 2.50663 = 37.599.
- Density at the mean. x = 100, so the exponent is 0 and exp(0) = 1. f(100) = 1 / 37.599 = 0.026596.
- Density at x = 115. z = (115 – 100)/15 = 1, so the exponent is -1²/2 = -0.5 and exp(-0.5) = 0.60653. f(115) = 0.60653 / 37.599 = 0.016131.
- Cumulative probability at x = 115. Using software for the integral: P(X ≤ 115) = 0.841345.
- Probability between two values. By symmetry P(X ≤ 85) = 0.158655, so P(85 < X ≤ 115) = 0.841345 – 0.158655 = 0.682689.
- Wider interval. P(70 < X ≤ 130) = 0.954500, and the 97.5th percentile is 129.40.
Now compare steps 2 and 3: the density at 115 is 0.0161, while the cumulative probability at 115 is 0.8413. They answer different questions. The density is a height, not a probability. A continuous variable has probability zero of equalling any single value; probabilities come from areas, that is, from differences of the CDF. Because the peak height is 1/(σ√(2π)), a density can even exceed 1: with σ = 0.1 the peak is 1/(0.1 × 2.50663) = 3.989, which is fine because the curve is narrow and the area still totals 1.
Interpretation: what each output means
| Function | What you give it | What it returns | Example (μ = 100, σ = 15) |
|---|---|---|---|
| a value x | curve height (a density, not a probability) | f(115) = 0.0161 | |
| CDF | a value x | P(X ≤ x) | P(X ≤ 115) = 0.8413 |
| Inverse CDF (percent point function) | a probability p | the value x with P(X ≤ x) = p | p = 0.975 gives x = 129.40 |
To answer “what is the chance a value falls between a and b”, subtract two CDF values. To answer “what cut-off leaves 2.5% above it”, use the inverse CDF.
Normal distribution probability plot: how to read it
Everything above assumes the data really are normal. A normal probability plot is the standard graphical check. NIST defines it as a graphical technique for assessing whether or not a data set is approximately normally distributed. The vertical axis holds the ordered response values; the horizontal axis holds the normal order statistic medians, which are computed from the normal percent point function (the inverse CDF). A reference line may be fitted through the points. NIST describes it as a special case of the probability plot: swap in another distribution’s percent point function to check a different distribution.
| What you see | What it indicates (NIST) |
|---|---|
| Points form an approximate straight line | The data are approximately normally distributed |
| Points depart from the straight line | Departures from normality |
| Points lie further from the line | A stronger indication of departure from normality |
For formal testing and related decisions, see normality of distribution. If a plot shows strong skew in positive-only data, the lognormal distribution may fit better.
Assumptions and common mistakes
- Reading a density as a probability. f(x) is not P(X = x). Use the CDF, or the difference of two CDF values.
- Forgetting the units of σ. The formula uses the standard deviation. Some software and some texts parameterize by variance, so check which one an argument expects.
- Applying the model without checking fit. The normal model is symmetric, with mean = median = mode and a range from negative to positive infinity. Strongly skewed or bounded data violate that; look at a probability plot first.
- Using sample estimates as exact. μ and σ are estimated by the sample mean and sample standard deviation, so probabilities computed from them inherit that uncertainty.
How to report it (APA example)
State the model, its parameters and the quantity you computed. In APA-style reporting, probabilities and proportions that cannot exceed 1 are written without a leading zero:
Scores were modelled as normally distributed (M = 100, SD = 15). The probability of a score at or below 115 was .84, and 68% of scores were expected to fall between 85 and 115.
Add that normality was assessed (for example, with a normal probability plot) when the analysis depends on it.
How to calculate it in R or Python
SciPy’s scipy.stats.norm takes loc as the mean and scale as the standard deviation, with pdf, cdf and ppf (the inverse of the CDF). R’s dnorm, pnorm and qnorm return the density, the cumulative distribution function and the quantile function, with mean and sd arguments.
Python
from scipy.stats import norm
norm.pdf(115, loc=100, scale=15) # 0.016131 (density)
norm.cdf(115, loc=100, scale=15) # 0.841345 (P(X <= 115))
norm.cdf(115, 100, 15) - norm.cdf(85, 100, 15) # 0.682689
norm.ppf(0.975, loc=100, scale=15) # 129.40R
dnorm(115, mean = 100, sd = 15) # 0.016131
pnorm(115, mean = 100, sd = 15) # 0.841345
pnorm(115, 100, 15) - pnorm(85, 100, 15) # 0.682689
qnorm(0.975, mean = 100, sd = 15) # 129.40
qqnorm(x); qqline(x) # normal probability (Q-Q) plot of a numeric vector xFAQ
What is the probability distribution function of the normal distribution?
For a continuous variable it is usually the probability density function, f(x) = exp(-(x – μ)²/(2σ²)) / (σ√(2π)). The cumulative form, the CDF, gives P(X ≤ x).
Is the PDF the same as the probability?
No. The PDF is a height; a probability is an area under it. At μ = 100, σ = 15 the density at 115 is 0.0161 while P(X ≤ 115) is 0.8413.
How do I find normal distribution cumulative probability?
Use the CDF in software (norm.cdf in SciPy, pnorm in R) or a standard normal table after converting to a z-score. The integral has no simple closed formula and is computed numerically.
How do I know if data fit a normal distribution?
Draw a normal probability plot. If the points form an approximate straight line, the data are approximately normal; departures from the line indicate departures from normality.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, “Normal Distribution” (checked 2026-10-29): itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
- NIST/SEMATECH e-Handbook of Statistical Methods, “Normal Probability Plot” (checked 2026-10-29): itl.nist.gov/div898/handbook/eda/section3/normprpl.htm
- SciPy documentation,
scipy.stats.norm(checked 2026-10-29): docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html - R documentation, The Normal Distribution and Quantile-Quantile Plots, stats package (checked 2026-10-29): stat.ethz.ch/R-manual/R-devel/library/stats/html/Normal.html
Worked-example values were computed with Python’s statistics.NormalDist; the arithmetic in steps 1 and 2 is shown so you can check it by hand.
