Inverse Normal Calculator

The inverse normal distribution finds the value x that has a given area p to its left under a normal curve: x = μ + σ·Φ⁻¹(p). It answers "which value cuts off the lowest 95%?" where the normal CDF answers "what share falls below this value?".

• Φ⁻¹(p) is the inverse of the standard normal CDF. Other names: the quantile function, probit and invNorm.
• For the standard normal (μ = 0, σ = 1), Φ⁻¹(0.95) = 1.6449 and Φ⁻¹(0.5) = 0.
• For IQ scores with μ = 100 and σ = 15, the value with 95% below it is 100 + 15 × 1.6449 = 124.67.

The area p must be strictly between 0 and 1. Φ⁻¹(0) = −∞ and Φ⁻¹(1) = +∞, so no finite value has all of the curve on one side.

To calculate the inverse normal, find z = Φ⁻¹(p) for the area to the left, then convert it with x = μ + zσ. For p = 0.95, μ = 100 and σ = 15:

1. Write the area as an area to the left. A left-tail area of 0.95 stays 0.95.
2. Find z: Φ⁻¹(0.95) = 1.6449 (a z-table gives 1.64 or 1.65).
3. Scale by σ: 1.6449 × 15 = 24.67.
4. Add the mean: x = 100 + 24.67 = 124.67.

Check: z = (124.67 − 100) / 15 = 1.6449, and Φ(1.6449) = 0.95.

For a right-tail area, use 1 − p first: the top 5% starts at Φ⁻¹(0.95), the same 124.67.

Pick the area type, type the area p, the mean μ and the standard deviation σ, and the calculator returns x as you type. The steps:

1. Choose Normal (μ, σ) for real data, or Standard normal (z) for a z-score.
2. Pick the area type: left tail, right tail, between (a central area) or outside (2 tails).
3. Type p as a decimal (0.95), a percent (95%) or a fraction (19/20).
4. Read x (or x₁ and x₂), the z-score, the areas on each side, the percentile and the density.
5. Open the bell curve or the CDF graph, and the worked steps.

The result can be copied, shared as a link, printed or kept in the history list.

The inverse normal formula is x = μ + σ·Φ⁻¹(p), where Φ⁻¹ is the inverse of the standard normal CDF.

• Left tail P(X ≤ x) = p: z = Φ⁻¹(p).
• Right tail P(X ≥ x) = p: z = Φ⁻¹(1 − p) = −Φ⁻¹(p).
• Between P(x₁ ≤ X ≤ x₂) = p: z = Φ⁻¹((1 + p)/2), x = μ ± zσ.
• Outside, p split over 2 tails: z = Φ⁻¹(1 − p/2), x = μ ± zσ.
• Symmetry: Φ⁻¹(1 − p) = −Φ⁻¹(p), so Φ⁻¹(0.05) = −1.6449.

Φ⁻¹ has no closed form. Software uses rational approximations such as Acklam's algorithm, refined to about 15 significant digits.

To find z from a z-table, search the body of the table for the area closest to p, then read z from the row (first decimal) and the column (second decimal). For p = 0.95:

1. Look for 0.9500 in the body. The 2 closest entries are 0.9495 and 0.9505.
2. 0.9495 sits in row 1.6, column .04, so z = 1.64.
3. 0.9505 sits in row 1.6, column .05, so z = 1.65.
4. 0.95 lies halfway, so z ≈ 1.645. The exact value is 1.6449.

A standard z-table lists areas to the left. For a right-tail area, look up 1 − p. For a negative z, use the negative half of the table or the rule Φ⁻¹(p) = −Φ⁻¹(1 − p).

The area type decides which area is converted to z: left tail uses p, right tail uses 1 − p, between uses (1 + p)/2 and outside uses 1 − p/2. With p = 0.9 on the standard normal:

• Left tail: 90% below x gives z = 1.2816.
• Right tail: 90% above x gives z = −1.2816.
• Between: the middle 90% runs from −1.6449 to 1.6449.
• Outside: 90% in the 2 tails leaves the middle 10%, from −0.1257 to 0.1257.

On a TI-84 Plus CE, invNorm takes a 4th input for the tail: LEFT, CENTER or RIGHT. invNorm(0.9, 0, 1, CENTER) returns −1.6449 and 1.6449.

The critical value z* for a confidence level C is Φ⁻¹((1 + C)/2): 1.645 for 90%, 1.960 for 95% and 2.576 for 99%.

A 95% confidence interval for a mean is x̄ ± 1.96 × σ/√n. For n = 25 and σ = 10, the margin of error is 1.96 × 2 = 3.92.

About 68% of a normal distribution lies within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. The exact areas are 68.27%, 95.45% and 99.73%.

• μ ± 1σ: 68.27%, so 31.73% lies outside.
• μ ± 2σ: 95.45%. The exact 95% band is μ ± 1.96σ.
• μ ± 3σ: 99.73%, so about 1 value in 370 lies outside.

The inverse normal runs the rule backwards: for a central area of 0.6827 it returns z = ±1, and for 0.95 it returns z = ±1.96.

The graph of z = Φ⁻¹(p) is an S-shaped curve that rises from −∞ at p = 0 to +∞ at p = 1 and passes through (0.5, 0).

• Odd about the centre: Φ⁻¹(1 − p) = −Φ⁻¹(p), so the curve is symmetric about the point (0.5, 0).
• Vertical asymptotes at p = 0 and p = 1.
• The slope is 1/φ(z), the reciprocal of the bell-curve height: √(2π) ≈ 2.5066 at p = 0.5, rising fast near the edges.
• Mirror image of the normal CDF Φ(z) across the line y = x.

Every statistics tool has an inverse normal function: invNorm on a TI-84, NORM.INV in Excel, norm.ppf in Python and qnorm in R.

• TI-84: press 2nd, VARS (DISTR), choose 3: invNorm(, type area, μ, σ and press ENTER. invNorm(0.95, 100, 15) returns 124.6728.
• Casio fx-991: MENU, Distribution, Inverse Normal, then enter the area, σ and μ.
• Excel and Google Sheets: =NORM.INV(0.95, 100, 15). For the standard normal, =NORM.S.INV(0.95) returns 1.644853627.
• Python: scipy.stats.norm.ppf(0.95, loc=100, scale=15), or statistics.NormalDist(100, 15).inv_cdf(0.95) with no extra package.
• R: qnorm(0.95, mean = 100, sd = 15); qnorm(0.05, lower.tail = FALSE) gives a right-tail z.

Excel, TI and Python all expect the area to the left. Convert a right-tail area with 1 − p first.

The inverse normal table lists z = Φ⁻¹(p) for common areas to the left, from the 0.1st to the 99.9th percentile. Click a row to load it into the calculator.

Values to know: Φ⁻¹(0.9) = 1.2816, Φ⁻¹(0.95) = 1.6449, Φ⁻¹(0.975) = 1.9600 and Φ⁻¹(0.99) = 2.3263.

The normal CDF turns a value into an area; the inverse normal turns an area back into a value. Φ(1.6449) = 0.95 and Φ⁻¹(0.95) = 1.6449.

The 2 functions cancel: Φ(Φ⁻¹(p)) = p for every area p, and Φ⁻¹(Φ(z)) = z for every z.

The inverse normal turns a percentile or a target share into a cut-off value, for test scores, quality limits, sizes and risk.

• Test scores: with μ = 100 and σ = 15, the 90th percentile is 100 + 15 × 1.2816 = 119.2.
• Grading: if marks follow N(70, 10), the top 10% starts at 70 + 10 × 1.2816 = 82.8.
• Quality control: bolts of N(50, 0.2) mm keep 98% between 49.53 mm and 50.47 mm (μ ± 2.3263σ).
• Sizing: a door height that clears 99.9% of people sits 3.09 standard deviations above the mean height.
• Finance: a 95% value at risk uses z = 1.6449 on the return distribution.