Definition 1: Let -∞ ≤ a < b ≤ ∞. Then the pdf of the truncated normal distribution with mean μ and variance σ² constrained by  is

where φ is the pdf of the normal distribution and Φ is the cdf of the normal distribution.

We assume that if x < a or x = -∞ then φ(x, µ, σ) = 0 and Φ(x, µ, σ) = 0. If x > b or x = ∞ then φ(x, µ, σ) = 0 and Φ(x, µ, σ) = 1.

Thus, in Excel if a and b are finite then

f(x) = NORM.DIST(x, µ, σ, FALSE)/(NORM.DIST(b, µ, σ)– NORM.DIST( a, µ, σ))

The cdf of this distribution is

The inverse distribution function is

Observation: We now present some key statistical properties, but first we define

Here, we assume that if b = ∞ then Φ(b, µ, σ) = 1 and (b–µ)^kφ(b, µ, σ) = 0. Similarly, if a = -∞ then Φ(a, µ, σ) = 0 and (a–µ)^kφ(a, µ, σ) = 0.

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions.

TNORM_DIST(x, μ, σ, cum, a, b) = the probability density function value f(x) for the truncated normal distribution N(μ, σ², a, b) when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

TNORM_INV(p, μ, σ, a, b) = the value x such that TNORM_DIST(x, μ, σ, TRUE, a, b) = p, i.e. inverse of TNORM_DIST(x, μ, σ, TRUE, a, b).

TNORM_PARAM(μ, σ, a, b, lab): array function that returns a column array with the following parameters for the truncated normal distribution N(μ, σ², a, b): mean, median, mode, variance, skewness, kurtosis.

If a is omitted then it defaults to -∞; if b is omitted then it defaults to ∞; if lab = TRUE (default FALSE), then an extra column of labels is appended to the output.