A normal distribution calculator turns a value into a probability. First convert the value to a z score, z = (x − μ) ÷ σ, then read the area under the bell curve to the left of z.

Example: leaf lengths have mean 52.44 mm and SD 4.19 mm. For x = 60, z = 1.80, so P(X ≤ 60) = 0.9643. About 96.4% of leaves are 60 mm or shorter.

Rule of thumb: 68% of values fall within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD. The total area under a normal curve is always exactly 1.

The normal distribution is a smooth, bell-shaped curve that describes how many natural measurements spread around their average. Most values sit close to the middle. Fewer values sit far away. The curve falls off the same way on both sides. It is also called the Gaussian distribution or the bell curve.

Normal distribution notation. We write X ~ N(μ, σ²). Read it as "X is normally distributed with mean μ and variance σ²". The mean sets the centre of the curve. The standard deviation σ sets its width. The special case N(0, 1) is the standard normal distribution, and its values are called z scores.

The equation of a normal distribution (its probability density function) is:

The height of this curve is not a probability. Probability is the area under the curve between two points. The total area under any normal curve is exactly 1, which is 100%.

Characteristics of a normal distribution:

• It is symmetric around the mean, so the mean, median and mode are all equal.
• It is unimodal: one single peak at μ.
• The tails never touch the x axis. They get closer forever but never reach zero.
• The inflection points, where the curve changes from bending down to bending up, are at μ − σ and μ + σ.
• Skewness is 0 and excess kurtosis is 0.
• The expected value of a normal distribution is μ, and its variance is σ².
• The sum of normal distributions is also normal. If X and Y are independent normals, X + Y ~ N(μ_X + μ_Y, σ²_X + σ²_Y).

A worked one-liner: leaf length has μ = 52.44 mm and σ = 4.19 mm. A 60 mm leaf has z = (60 − 52.44) ÷ 4.19 = 1.80, and the area to the left of z = 1.80 is 0.9643.

Who uses it: doctors use it to set reference ranges for blood tests. Teachers use it to turn raw marks into percentiles. Engineers use it for quality control and tolerance limits. Psychologists scale IQ tests to N(100, 15²). Ecologists use it for body size and measurement error. Finance uses it (with care) for returns. Nearly every t test, ANOVA and regression assumes normal errors.

Why it is everywhere: the central limit theorem says that the mean of many independent small effects tends toward a normal distribution, whatever the shape of each effect. That is why sample means are close to normal even when raw data are not.

Related ideas. A bivariate normal distribution describes two variables that are jointly normally distributed, such as height and weight. It adds a correlation ρ to the two means and two SDs, and its contour lines are ellipses. A non normal distribution is any shape that breaks the bell: skewed, heavy tailed, bounded or with two peaks.

Your probability question

4.1 Your probability, in plain words

A normal probability is a share of the whole population. P(X ≤ 60) = 0.9643 means that, if the variable really follows this normal curve, 96.43% of all values are 60 or less. It also means that one randomly chosen value has a 96.43% chance of being 60 or less. Both readings are correct. Use whichever is easier for your reader.

Always say which side of the value you mean. "The probability of 60" has no meaning for a continuous variable, because the chance of any single exact value is zero. You need "at most 60", "at least 60" or "between 45 and 60".

4.2 The z score: how unusual is this value?

The z score is the distance from the mean in standard deviation units. z = 0 is exactly average. z = 1 is one SD above the mean. z = −2 is two SDs below. A z score lets you compare values measured on different scales, such as a height in cm and a score out of 100.

4.3 Left tail, right tail, between and outside

These four areas are linked. Once you have one, the others follow:

• Right tail = 1 − left tail. P(X ≥ 60) = 1 − 0.9643 = 0.0357.
• Between = left tail at b minus left tail at a. P(45 ≤ X ≤ 60) = 0.9643 − 0.0381 = 0.9262.
• Outside = 1 − between = 0.0738.

For a continuous variable, ≤ and < give the same answer. P(X < 60) equals P(X ≤ 60). This is not true for counts, which is why the binomial approximation needs a continuity correction (Section 4.10).

4.4 Percentiles and the inverse normal

The inverse normal answers the reverse question: "Which value cuts off the lowest p%?" For leaf length, the 90th percentile is 57.81 mm. So 90% of leaves are shorter than 57.81 mm and 10% are longer. The formula is x = μ + z_p × σ, where z_p comes from the z table read backwards. Reference ranges in medicine are built this way: the central 95% runs from μ − 1.96σ to μ + 1.96σ.

4.5 The empirical rule, and what your data say about it

For a perfect normal distribution, 68.27% of values lie within 1 SD of the mean, 95.45% within 2 SD and 99.73% within 3 SD. Chart 4 compares your data with these targets. Small samples wobble. With 16 values, 62.5% in place of 68.3% is normal chance. But if far more than 68% sit within 1 SD, the curve is too peaked. If far fewer than 95% sit within 2 SD, the tails are too heavy. Both are signs of a non normal distribution.

4.6 How to tell if data is normally distributed: the Shapiro-Wilk test

The Shapiro-Wilk test asks: "Could these data come from a normal distribution?" W runs from 0 to 1. Values near 1 mean the sorted data match normal quantiles closely.

• p > α (usually 0.05): no clear evidence against normality. It does not prove the data are normal.
• p ≤ α: the data depart from normality. Look at the Q-Q plot to see how.

The test depends heavily on sample size. With 8 values it misses almost everything. With 2000 values it flags differences too small to matter. So always read the p value together with Charts 2 and 3.

4.7 Skewness and kurtosis

Skewness measures lopsidedness. 0 is symmetric. Positive means a long right tail (income, reaction time). Negative means a long left tail (easy exam scores). Excess kurtosis measures tail weight. 0 is normal. Positive means heavy tails and more outliers. Negative means light tails and a flat top.

A simple rule from Kim (2013): divide each by its standard error. For n under 50, |z| above 1.96 is a problem. For n from 50 to 300, use 3.29. For n above 300, look at the raw values: |skewness| above 2 or |excess kurtosis| above 7 is a real departure.

4.8 Reading the histogram and the Q-Q probability plot

Chart 2, the normal distribution histogram. A histogram of normal distribution data looks like a rough bell. The bars should follow the line. Count data with few values will look lumpy. That is fine. What matters is a clear lean to one side, two peaks, or a long tail.

Chart 3, the normal probability plot. Each point pairs a sorted value with the value a perfect normal would give. Points on the straight line mean the data are normally distributed. The shape of the bend tells you the problem:

4.9 Could the graph represent a variable with a normal distribution?

This is a common exam question. Answer "yes" only if all of these hold: the graph has one peak; it is roughly symmetric about that peak; it falls off smoothly on both sides; and it does not stop abruptly at a boundary such as zero. Answer "no" if the graph is skewed, has two humps, is flat, or is cut off on one side. The mean of the normal distribution shown is always the x value under the peak.

4.10 Normal approximation to the binomial distribution

A binomial count X (successes in n trials) has mean np and SD √(np(1 − p)). When np ≥ 5 and n(1 − p) ≥ 5, its shape is close to normal. Because counts are whole numbers, widen each bar by 0.5 on each side. That is the continuity correction: P(X ≤ 25) becomes P(Y ≤ 25.5). For n = 50 and p = 0.4, the approximation gives 0.9438 and the exact binomial is 0.9427. Without the correction you would get 0.9255, which is much worse.

4.11 Decision guide

4.12 Common mistakes this calculator prevents

• Reading the wrong side of the z table. The tool always states which area it found and shades it on Chart 1.
• Using the population SD by accident. The default is the sample SD, matching R and Excel STDEV.S.
• Trusting a normal probability for skewed data. Every run checks normality, skewness and kurtosis, and warns you.
• Forgetting the continuity correction for binomial counts. It is applied and shown.
• Treating "not significant" as proof of normality. The wording in Section 5 avoids that claim.
• Mixing up a negative z value with a negative probability. Probabilities are always between 0 and 1.

▶ Run the analysis above to auto-fill all five examples with your results.

A good write-up of a normal distribution result always has four parts: (1) the variable and its units, (2) the model, N(μ, σ²), with n if the parameters came from data, (3) the probability or percentile with the exact question, and (4) evidence that the normal model fits. Leave out part 4 and a reviewer will ask for it.

5.1 Phrases to use, and phrases to avoid

5.2 What to put in a table

For several groups, give one row per group with columns n, M, SD, W, p (Shapiro-Wilk) and the probability you computed. Add a note under the table: "Probabilities assume a normal distribution with the sample mean and SD; SD uses n − 1."

This flowchart shows how a normal distribution probability is calculated, step by step, from raw data to a checked answer. Every box uses the same worked example as the Excel, R and Python sections: 16 leaf lengths in mm, and the question "what share of leaves are 60 mm or shorter?"

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return a mean of 52.4375, an SD of 4.1947, z = 1.8029 and P(X ≤ 60) = 0.9643.

1. Enter your data. The default box takes comma-separated numbers, as the placeholder shows: 52, 48, 55, 61, 47, .... Spaces, tabs, semicolons and new lines also work, so a column pasted from Excel is fine.
2. Name each group. The name field above each box is editable. Type something like Site A and it appears in the table, the charts and the report.
3. Add groups to compare. Press Add group. Each group gets its own mean, SD, normal curve and normality test.
4. Or upload a file. On the Upload tab, choose a CSV or Excel file and click the column names you want. Each column you click becomes its own group.
5. Or skip the data. For a textbook question, set "Where do μ and σ come from?" to "I will type μ and σ myself". Use μ = 0 and σ = 1 for the standard normal distribution.
6. Pick your question. Left tail, right tail, between, outside, percentile, central interval or the binomial approximation.
7. Type the values. For example x = 60, or a = 45 and b = 60, or p = 90 for the 90th percentile.
8. Press Calculate Normal Distribution. Nothing is computed until you do. Changing any input clears old results so you never read stale numbers.
9. Read the answer and the checks. The top box gives the probability. The assumption table tells you whether the normal model fits your data.
10. Use the charts and export. Chart 1 shows the shaded area. Section 5 fills in five report styles. Download the text report or a print-ready PDF.

Excel has built-in functions for every normal distribution calculation: the left-tail probability, the density, the inverse and the z score. What trips people up is the TRUE or FALSE last argument, the right tail, and the fact that Excel has no Shapiro-Wilk test.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

• No Shapiro-Wilk in Excel. Neither the functions nor the Analysis ToolPak include it. Use this calculator, R or Python for the test, and use SKEW and KURT in Excel as a rough check (here 0.3853 and −0.4360).
• Normal probability plot in Excel. Sort the data in column A. In B2 type =NORM.S.INV((ROW()-1-0.375)/(COUNT($A$2:$A$17)+0.25)) and fill down. Insert a scatter chart of B (x axis) against A (y axis). Points on a straight line mean normal data.
• Histogram with a normal curve. The Analysis ToolPak Histogram tool (File → Options → Add-ins → Excel Add-ins → tick Analysis ToolPak) makes the bars. Add a second series of =NORM.DIST(bin_mid,m,s,FALSE)*n*bin_width and change it to a line.
• Normal approximation to the binomial. =NORM.DIST(25.5, 50*0.4, SQRT(50*0.4*0.6), TRUE) gives 0.9438. Compare with the exact =BINOM.DIST(25,50,0.4,TRUE), 0.9427.
• Old function names. NORMDIST, NORMSDIST, NORMINV and NORMSINV still work but are kept only for compatibility. Use the dotted versions.

9.4 Why your Excel answer might differ from R or Python

9.5 Drawing a normal distribution curve in Excel

1. In column F, make x values from μ − 4σ to μ + 4σ in small steps, for example =$D$2-4*$D$3+(ROW()-2)*$D$3/10, filled down 81 rows.
2. In G2 type =NORM.DIST(F2,$D$2,$D$3,FALSE) and fill down.
3. Select F and G, then Insert → Charts → Scatter with Smooth Lines.
4. To shade P(X ≤ 60), add column H: =IF(F2<=60,G2,NA()), add it as a series and change it to an Area chart.
5. Label the axes with the variable name and units. Chart 1 on this page draws the same figure automatically.

9.6 Excel errors you will probably hit

R has four normal functions built in: pnorm for probabilities, qnorm for percentiles, dnorm for the curve height and rnorm for random values. The script below estimates μ and σ, finds every probability, runs the Shapiro-Wilk test, prints a plain-English verdict and saves one figure at 300 dpi. It uses base R only.

10.1 The complete script

R, base only, saves normal_distribution_plot.png

# ============================================================
# Normal distribution calculator in R
# StatsUnlock - https://statsunlock.com/normal-distribution-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   x <- read.csv("data.csv")$column_name
x <- c(52, 48, 55, 61, 47, 50, 58, 53,
       49, 56, 51, 54, 46, 57, 50, 52)
x <- x[!is.na(x)]                 # drop missing values

label   <- "Leaf length"
units   <- "mm"
x_value <- 60                     # P(X <= x_value)
a <- 45; b <- 60                  # P(a <= X <= b)
pct <- 0.90                       # percentile to find

# ---- 2. Estimate mu and sigma ------------------------------
n     <- length(x)
mu    <- mean(x)
sigma <- sd(x)                    # n - 1, same as Excel STDEV.S

# ---- 3. Probabilities --------------------------------------
z         <- (x_value - mu) / sigma
p_left    <- pnorm(x_value, mean = mu, sd = sigma)
p_right   <- pnorm(x_value, mean = mu, sd = sigma, lower.tail = FALSE)
p_between <- pnorm(b, mu, sigma) - pnorm(a, mu, sigma)
x_pct     <- qnorm(pct, mean = mu, sd = sigma)

# ---- 4. Normality check ------------------------------------
sw <- shapiro.test(x)
d  <- x - mu
g1 <- mean(d^3) / mean(d^2)^1.5
g2 <- mean(d^4) / mean(d^2)^2 - 3
skew <- g1 * sqrt(n * (n - 1)) / (n - 2)                     # same as Excel SKEW
kurt <- ((n + 1) * g2 + 6) * (n - 1) / ((n - 2) * (n - 3))   # same as Excel KURT

# ---- 5. Print a plain-English report -----------------------
cat(sprintf("n = %d, mean = %.4f, SD = %.4f\n", n, mu, sigma))
cat(sprintf("z for x = %g: %.4f\n", x_value, z))
cat(sprintf("P(X <= %g) = %.4f\n", x_value, p_left))
cat(sprintf("P(X >= %g) = %.4f\n", x_value, p_right))
cat(sprintf("P(%g <= X <= %g) = %.4f\n", a, b, p_between))
cat(sprintf("%.0fth percentile = %.4f %s\n", pct * 100, x_pct, units))
cat(sprintf("Shapiro-Wilk W = %.4f, p = %.4f\n", sw$statistic, sw$p.value))
cat(sprintf("Skewness = %.4f, excess kurtosis = %.4f\n", skew, kurt))
if (sw$p.value > 0.05) {
  cat("Verdict: no evidence against normality, so the normal model is reasonable.\n")
  cat(sprintf("About %.1f%% of values are expected to be %g %s or less.\n",
              p_left * 100, x_value, units))
} else {
  cat("Verdict: the data depart from normality. Treat normal probabilities with caution.\n")
}

# ---- 6. One figure: histogram + normal curve + shaded area -
set.seed(1)                       # nothing random here, but seeded for habit
png("normal_distribution_plot.png", width = 7, height = 4.5, units = "in", res = 300)
xs <- seq(mu - 4 * sigma, mu + 4 * sigma, length.out = 400)
hist(x, freq = FALSE, col = "#bfdbfe", border = "#1e40af",
     xlim = range(xs), ylim = c(0, dnorm(mu, mu, sigma) * 1.15),
     main = sprintf("N(%.2f, %.2f^2), Shapiro-Wilk p = %.3f", mu, sigma, sw$p.value),
     xlab = sprintf("%s (%s)", label, units), ylab = "Density")
left <- xs[xs <= x_value]
polygon(c(min(left), left, max(left)), c(0, dnorm(left, mu, sigma), 0),
        col = adjustcolor("#16a34a", alpha.f = 0.3), border = NA)
lines(xs, dnorm(xs, mu, sigma), col = "#1d4ed8", lwd = 2)
abline(v = x_value, lty = 2, col = "#16a34a")
legend("topright", bty = "n",
       legend = c("Normal curve", sprintf("P(X <= %g) = %.4f", x_value, p_left)),
       col = c("#1d4ed8", "#16a34a"), lwd = c(2, 8))
dev.off()
cat("Saved normal_distribution_plot.png\n")

Printed output:

n = 16, mean = 52.4375, SD = 4.1947
z for x = 60: 1.8029
P(X <= 60) = 0.9643
P(X >= 60) = 0.0357
P(45 <= X <= 60) = 0.9262
90th percentile = 57.8133 mm
Shapiro-Wilk W = 0.9789, p = 0.9541
Skewness = 0.3853, excess kurtosis = -0.4360
Verdict: no evidence against normality, so the normal model is reasonable.
About 96.4% of values are expected to be 60 mm or less.

10.2 What each line does

10.3 What the figure shows

The saved figure has the histogram of your data (blue bars), the fitted normal curve (dark blue line) and the shaded area for P(X ≤ 60) (green). The title carries the model and the Shapiro-Wilk p value, so the figure stands on its own in a report. If the bars follow the curve, the normal model is a fair description.

10.4 Problems and fixes

10.5 One-liners

pnorm(1.96)                         # 0.9750, z table value
pnorm(-1.96)                        # 0.0250, negative z table
qnorm(0.95)                         # 1.6449, one-tailed 5% critical z
pnorm(130, 100, 15, lower.tail = FALSE)   # 0.0228, IQ above 130
pnorm(25.5, 20, sqrt(12))           # 0.9438, binomial approximation
qqnorm(x); qqline(x)                # normal probability plot

In Python, scipy.stats.norm does the work. cdf gives left-tail probabilities, sf gives right tails, ppf gives percentiles and pdf gives the curve height. The script below mirrors the R script line for line and gives the same numbers.

Install once with pip install numpy scipy matplotlib.

11.1 The complete script

Python 3, saves normal_distribution_plot.png

# ============================================================
# Normal distribution calculator in Python
# StatsUnlock - https://statsunlock.com/normal-distribution-calculator/
# Needs: numpy, scipy, matplotlib
# ============================================================
import numpy as np
from scipy import stats
import matplotlib
matplotlib.use("Agg")            # draw without a screen
import matplotlib.pyplot as plt

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   import pandas as pd
#   x = pd.read_csv("data.csv")["column_name"].dropna().to_numpy()
x = np.array([52, 48, 55, 61, 47, 50, 58, 53,
              49, 56, 51, 54, 46, 57, 50, 52], dtype=float)
x = x[~np.isnan(x)]              # drop missing values

label = "Leaf length"
units = "mm"
x_value = 60                     # P(X <= x_value)
a, b = 45, 60                    # P(a <= X <= b)
pct = 0.90                       # percentile to find

# ---- 2. Estimate mu and sigma ------------------------------
n = len(x)
mu = x.mean()
sigma = x.std(ddof=1)            # ddof=1 matches R sd() and Excel STDEV.S

# ---- 3. Probabilities --------------------------------------
z = (x_value - mu) / sigma
p_left = stats.norm.cdf(x_value, loc=mu, scale=sigma)
p_right = stats.norm.sf(x_value, loc=mu, scale=sigma)
p_between = stats.norm.cdf(b, mu, sigma) - stats.norm.cdf(a, mu, sigma)
x_pct = stats.norm.ppf(pct, mu, sigma)

# ---- 4. Normality check ------------------------------------
w, p_sw = stats.shapiro(x)
skew = stats.skew(x, bias=False)
kurt = stats.kurtosis(x, bias=False)     # excess kurtosis

# ---- 5. Print a plain-English report -----------------------
print(f"n = {n}, mean = {mu:.4f}, SD = {sigma:.4f}")
print(f"z for x = {x_value}: {z:.4f}")
print(f"P(X <= {x_value}) = {p_left:.4f}")
print(f"P(X >= {x_value}) = {p_right:.4f}")
print(f"P({a} <= X <= {b}) = {p_between:.4f}")
print(f"{pct*100:.0f}th percentile = {x_pct:.4f} {units}")
print(f"Shapiro-Wilk W = {w:.4f}, p = {p_sw:.4f}")
print(f"Skewness = {skew:.4f}, excess kurtosis = {kurt:.4f}")
if p_sw > 0.05:
    print(f"Verdict: no evidence against normality, so the normal model is reasonable.")
    print(f"About {p_left*100:.1f}% of values are expected to be {x_value} {units} or less.")
else:
    print("Verdict: the data depart from normality. Treat normal probabilities with caution.")

# ---- 6. One figure: histogram + normal curve + shaded area -
np.random.seed(1)                # nothing random here, but seeded for habit
fig, ax = plt.subplots(figsize=(7, 4.5))
ax.hist(x, bins="sturges", density=True, color="#bfdbfe", edgecolor="#1e40af", label="Data")
grid = np.linspace(mu - 4 * sigma, mu + 4 * sigma, 400)
ax.plot(grid, stats.norm.pdf(grid, mu, sigma), color="#1d4ed8", lw=2, label="Normal curve")
fill = grid[grid <= x_value]
ax.fill_between(fill, stats.norm.pdf(fill, mu, sigma), color="#16a34a", alpha=0.3,
                label=f"P(X <= {x_value}) = {p_left:.4f}")
ax.axvline(x_value, color="#16a34a", ls="--")
ax.set_xlabel(f"{label} ({units})")
ax.set_ylabel("Density")
ax.set_title(f"N({mu:.2f}, {sigma:.2f}^2), Shapiro-Wilk p = {p_sw:.3f}")
ax.legend(frameon=False)
fig.tight_layout()
fig.savefig("normal_distribution_plot.png", dpi=300)
print("Saved normal_distribution_plot.png")

Printed output: identical to the R output in Section 10.1, ending with Saved normal_distribution_plot.png.

11.2 What each line does

11.3 What the figure shows

The same three layers as the R figure: histogram, fitted normal curve and the shaded probability. matplotlib.use("Agg") lets the script run on a server or in a terminal with no screen.

11.4 Problems and fixes

11.5 One-liners

from scipy import stats
from scipy.stats import norm
norm.cdf(1.96)                  # 0.9750, z table value
norm.cdf(-1.96)                 # 0.0250, negative z table
norm.ppf(0.95)                  # 1.6449, one-tailed 5% critical z
norm.sf(130, 100, 15)           # 0.0228, IQ above 130
norm.cdf(25.5, 20, 12 ** 0.5)   # 0.9438, binomial approximation
stats.probplot(x, plot=plt)     # normal probability plot

12.1 Standard normal distribution table, negative z values (z table, left tail)

Each cell is Φ(z), the area to the left of z. Read the row for the first decimal and the column for the second. Example: z = −1.80 is row −1.8, column 0.00, giving 0.0359.

Conclusion: every value in the negative normal distribution table is below 0.5, because negative z values lie left of the mean.

12.2 Standard normal distribution table, positive z values

Example: z = 1.80 is row 1.8, column 0.00, giving 0.9641. For z = 1.96, row 1.9 and column 0.06 give 0.9750.

Conclusion: to get a right-tail area, subtract the table value from 1. P(Z ≥ 1.96) = 1 − 0.9750 = 0.0250.

12.3 Critical z values for common confidence levels

Conclusion: the famous 1.96 is the two-tailed 95% value. For a one-tailed 5% test use 1.645.

12.4 The empirical rule, exact percentages

Conclusion: "68-95-99.7" is rounded. Exact values are 68.27%, 95.45% and 99.73%.

12.5 Percentiles of the normal distribution

Conclusion: the 50th percentile is always the mean. Percentiles are symmetric: the 10th and 90th sit the same distance below and above it.

12.6 Skewness and kurtosis cut-offs by sample size

Conclusion: large samples make any z huge, so switch to raw values above n = 300 (Kim, 2013).

12.7 Normal approximation to the binomial: continuity correction

Conclusion: the rule is "widen to include the whole bar of every count you want". Valid when np ≥ 5 and n(1 − p) ≥ 5.

Eight worked examples, each with its own normal curve. They cover a textbook case, a tail probability, an interval, a percentile, a small sample, a large sample where the test is significant but the effect is small, a clear failure of normality, and the binomial approximation.

Design statement. To use a normal model you need one continuous measurement per independent unit, collected the same way for every unit, from a clearly defined population.

1. Define the population and the variable, with its units, before you collect anything.
2. Choose the measuring tool and its precision. Record to at least one more decimal than you plan to report.
3. Decide the sample size. Aim for 30 or more values per group for a stable μ and σ, and 50 or more if tail probabilities matter.
4. Sample at random, or as close to random as you can. Write down how units were chosen.
5. Measure each unit once. Repeated measures on one unit are not independent values.
6. Use the same person, instrument and conditions for every unit, or record who and what was used.
7. Record missing values as blank, never as 0.
8. Keep one row per unit and one column per group or variable (see the layout below).
9. Check for typing errors: sort each column and look at the smallest and largest values.
10. Run this calculator. Look at the Shapiro-Wilk p value, the Q-Q plot and the histogram together.
11. If normality fails, decide in advance whether you will transform, trim or use a non-normal method.

In the upload tab, click "Site A (mm)" and "Site B (mm)" and each becomes its own group. The blank cell for L003 is skipped.

Minimum sample. The Shapiro-Wilk test needs at least 3 values, but below about 20 it rarely detects anything. For reference ranges (the central 95%), guidelines often ask for 120 or more.

Pre-registration line. "We will estimate μ and σ from the sample (SD with n − 1), check normality with the Shapiro-Wilk test at α = 0.05 plus a Q-Q plot, and log-transform the data if the test fails."

Common mistakes: mixing two populations in one column; rounding so hard that values pile up on a few numbers; entering zeros for missing values; and collecting from a biased subgroup, such as only the largest plants.

Normal distribution real world examples:

• Medicine: the share of adults with systolic blood pressure above 140 mmHg.
• Education: the share of students scoring below a pass mark, or the mark needed for the top 10%.
• Manufacturing: the share of bolts inside a 9.9 to 10.1 mm tolerance band.
• Psychology: how rare an IQ of 130 is on a test scaled to N(100, 15²).

Design guidance. If each value is the sum or average of many small effects (height, measurement error, a test total), the normal model is usually a good fit. If values come from multiplying effects (growth, income), the log of the data is often normal instead.

Assumptions

1. The variable is continuous. Check: it can take any value in a range. If it fails: probabilities for exact values are wrong; for counts, use the continuity correction or the exact distribution.
2. The population is normal. Check: Shapiro-Wilk, Q-Q plot, histogram, skewness and kurtosis. If it fails with right skew: right-tail probabilities are underestimated and left-tail ones are overestimated. With heavy tails: probabilities beyond 2 SD are underestimated.
3. Values are independent. Check: each unit measured once, no clustering. If it fails: the SD is usually underestimated, so tail probabilities look smaller than they are.
4. μ and σ are well estimated. Check: n of at least 30. If it fails: with small n the estimated σ is noisy, and tail probabilities can be off in either direction.
5. One population per group. Check: one peak in the histogram. If it fails: the fitted curve is too wide and describes neither subgroup.

Limitations

• The tool treats estimated μ and σ as if they were exact. For a prediction interval for one new value, use the t distribution instead.
• The Shapiro-Wilk test is limited to 3 to 5000 values. For larger data, use the Q-Q plot or the Kolmogorov-Smirnov test.
• A normal model can never be proven, only shown to be a fair fit.
• It does not fit a bivariate normal distribution. For two correlated variables, use correlation or regression tools.
• Extreme tail probabilities (beyond 4 SD) are exact for the model but rarely true for real data. Use extreme value methods for rare events.

A normal distribution calculator answers one simple question: what share of values lies on one side of a cut-off, or between two cut-offs? The method is always the same. Convert each value to a z score, then find the area under the standard normal curve. That area is the probability.

The hard part is not the arithmetic. It is knowing whether the normal model fits. That is why this tool always shows a Shapiro-Wilk test, skewness, kurtosis, a histogram with the fitted curve, a Q-Q probability plot and an empirical rule check next to every probability. When these agree that the data are approximately normal, you can report the probability with confidence. When they disagree, the probability in the tails is the first thing to go wrong.

For counts, the normal curve is a good approximation to the binomial distribution once np and n(1 − p) both reach 5, as long as you apply the 0.5 continuity correction. For skewed data, a log transform or empirical percentiles usually serve better than forcing a bell curve onto the data.

Use the Excel, R and Python sections to repeat any result in your own software, and the reporting templates in Section 5 to write it up correctly. With the model checked and the question stated clearly, the normal distribution calculator gives an answer you can defend.

These peer-reviewed articles support the methods in this normal distribution calculator: the normal model itself, the Shapiro-Wilk and other normality tests, probability plots, the empirical rule, the binomial approximation and the numerical methods used for Φ and its inverse.

1. Altman, D. G., & Bland, J. M. (1995). Statistics notes: The normal distribution. BMJ, 310(6975), 298. https://doi.org/10.1136/bmj.310.6975.298
2. Anderson, T. W., & Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49(268), 765–769. https://doi.org/10.1080/01621459.1954.10501232
3. Cody, W. J. (1969). Rational Chebyshev approximations for the error function. Mathematics of Computation, 23(107), 631–637. https://doi.org/10.1090/S0025-5718-1969-0247736-4
4. D'Agostino, R. B., Belanger, A., & D'Agostino, R. B., Jr. (1990). A suggestion for using powerful and informative tests of normality. The American Statistician, 44(4), 316–321. https://doi.org/10.1080/00031305.1990.10475751
5. Das, K. R., & Imon, A. H. M. R. (2016). A brief review of tests for normality. American Journal of Theoretical and Applied Statistics, 5(1), 5–12. https://doi.org/10.11648/j.ajtas.20160501.12
6. Ghasemi, A., & Zahediasl, S. (2012). Normality tests for statistical analysis: A guide for non-statisticians. International Journal of Endocrinology and Metabolism, 10(2), 486–489. https://doi.org/10.5812/ijem.3505
7. Kim, H.-Y. (2013). Statistical notes for clinical researchers: Assessing normal distribution (2) using skewness and kurtosis. Restorative Dentistry & Endodontics, 38(1), 52–54. https://doi.org/10.5395/rde.2013.38.1.52
8. Kwak, S. G., & Kim, J. H. (2017). Central limit theorem: The cornerstone of modern statistics. Korean Journal of Anesthesiology, 70(2), 144–156. https://doi.org/10.4097/kjae.2017.70.2.144
9. Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association, 62(318), 399–402. https://doi.org/10.1080/01621459.1967.10482916
10. Lumley, T., Diehr, P., Emerson, S., & Chen, L. (2002). The importance of the normality assumption in large public health data sets. Annual Review of Public Health, 23, 151–169. https://doi.org/10.1146/annurev.publhealth.23.100901.140546
11. Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105(1), 156–166. https://doi.org/10.1037/0033-2909.105.1.156
12. Mishra, P., Pandey, C. M., Singh, U., Gupta, A., Sahu, C., & Keshri, A. (2019). Descriptive statistics and normality tests for statistical data. Annals of Cardiac Anaesthesia, 22(1), 67–72. https://doi.org/10.4103/aca.ACA_157_18
13. Pukelsheim, F. (1994). The three sigma rule. The American Statistician, 48(2), 88–91. https://doi.org/10.1080/00031305.1994.10476030
14. Royston, J. P. (1982). An extension of Shapiro and Wilk's W test for normality to large samples. Journal of the Royal Statistical Society: Series C (Applied Statistics), 31(2), 115–124. https://doi.org/10.2307/2347973
15. Royston, P. (1995). Remark AS R94: A remark on algorithm AS 181: The W-test for normality. Journal of the Royal Statistical Society: Series C (Applied Statistics), 44(4), 547–551. https://doi.org/10.2307/2986146
16. Schader, M., & Schmid, F. (1989). Two rules of thumb for the approximation of the binomial distribution by the normal distribution. The American Statistician, 43(1), 23–24. https://doi.org/10.1080/00031305.1989.10475601
17. Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3–4), 591–611. https://doi.org/10.1093/biomet/52.3-4.591
18. Stigler, S. M. (1982). A modest proposal: A new standard for the normal. The American Statistician, 36(2), 137–138. https://doi.org/10.1080/00031305.1982.10482818
19. Wichura, M. J. (1988). Algorithm AS 241: The percentage points of the normal distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 37(3), 477–484. https://doi.org/10.2307/2347330
20. Wilk, M. B., & Gnanadesikan, R. (1968). Probability plotting methods for the analysis of data. Biometrika, 55(1), 1–17. https://doi.org/10.1093/biomet/55.1.1