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To identify the distribution of your data, work in a fixed order: decide whether the variable is discrete or continuous, draw a histogram to read its shape, compare it with candidate distributions on a probability plot, and use a goodness-of-fit test only as a final check. This guide follows the order the NIST/SEMATECH e-Handbook of Statistical Methods uses for exploratory data analysis, with a worked example and short code for Excel, R and Python.
What does it mean to identify the distribution of your data?
A distribution describes which values a variable takes and how often. Identifying it means finding a named model (normal, lognormal, exponential, Poisson and so on) whose shape matches the data well enough to support the analysis you plan. NIST frames the question as “What kind of population distribution do the data come from?” and “Which distribution best fits my data?” The answer decides which summaries, tests and intervals are appropriate. For the broader vocabulary see descriptive versus inferential statistics.
Step 1: Decide what type of variable you have
Start with how the variable is measured, before any plot. A variable that counts events (0, 1, 2, …) is discrete and points toward count families such as the Poisson or binomial; see binomial distribution parameters. A variable measured on a continuous scale (time, mass, concentration) points toward families such as the normal, lognormal, exponential or Weibull. This step is an editorial rule of thumb: it narrows the candidate list, and the plots below then test the shortlist.
Step 2: How do you describe the distribution of the data?
Describe four features: centre, spread, skewness and the number of peaks (modes), plus any outliers. The NIST handbook lists exactly these as what a histogram shows, and says these features “provide strong indications of the proper distributional model for the data.”
Skewness: mean versus median
OpenStax Introductory Statistics 2e (section 2.6) gives the reading guide below, and adds that it is “not true for every data set.”
| Shape | Mean and median | Mode |
|---|---|---|
| Symmetric | Approximately equal | Near the centre |
| Right-skewed (long right tail) | Mean often greater than median | Smallest of the three |
| Left-skewed (long left tail) | Mean less than median | Largest of the three |
The mean is pulled by outliers, which is why it moves toward the tail. See right-skewed histograms and left-skewed histograms.
Bimodal and multimodal data
More than one peak in the histogram (a bimodal distribution) often signals two mixed groups, so check whether the data should be split before fitting a single family. See multimodal distributions.
Step 3: What is the best chart to show distribution?
The histogram is the standard first chart: NIST describes it as showing centre, spread, skewness, outliers and multiple modes. A box plot condenses the same information into quartiles and flags outliers, and a probability plot (Q-Q plot) tests a specific candidate. For the probability plot, NIST says that if the data follow the distribution, “the points should form approximately a straight line.” To compare several candidates, NIST suggests plotting each and choosing the one with the highest correlation coefficient (the straightest plot). Guides: histograms in Matplotlib and Q-Q plots.
Worked example: ten values, right skew, lognormal check
The data below are a constructed illustration, not field data: 2, 3, 3, 4, 4, 5, 6, 8, 12, 25 (n = 10), such as ten response times in minutes.
- Mean = (2+3+3+4+4+5+6+8+12+25) / 10 = 72 / 10 = 7.2.
- Median = average of the 5th and 6th ordered values = (4+5) / 2 = 4.5.
- Squared deviations from 7.2 sum to 429.6, so the sample variance is 429.6 / 9 = 47.73 and the SD is 6.91.
- Third central moment m3 = 5384.16 / 10 = 538.42; second central moment m2 = 429.6 / 10 = 42.96; skewness g1 = m3 / m21.5 = 538.42 / 281.58 = 1.91.
The mean (7.2) is well above the median (4.5), and g1 is clearly positive, so the data are right-skewed, consistent with the OpenStax guide. Taking natural logs gives a roughly symmetric set (skewness 0.78, Shapiro-Wilk W = 0.942, p = 0.58), whereas the raw values are not normal (W = 0.714, p = 0.0013). That pattern makes the lognormal a sensible candidate to check on a probability plot. With only ten values, a test has little power, so the plot carries most of the weight.
Interpretation: using goodness-of-fit tests
NIST describes the Anderson-Darling test as a modification of the Kolmogorov-Smirnov test that gives more weight to the tails. The null hypothesis is that the data follow the specified distribution; the alternative is that they do not. Critical values depend on the distribution tested (NIST gives tables for normal, uniform, lognormal, exponential, Weibull, extreme value type I, generalized Pareto and logistic), and the test is usually run in software. You reject the null when the A2 statistic exceeds the critical value.
Note the reversed logic: here you hope for a large p-value, because a small one says the model fits poorly. A large p-value does not prove the distribution; it only fails to reject it, and with small samples that is easy to achieve. For normality-specific procedures see normality of distribution.
Assumptions and common mistakes
- Choosing a family from the histogram alone; confirm with a probability plot for each candidate.
- Trusting a test with very small or very large samples: small samples rarely reject, large samples reject trivial departures.
- Fitting one distribution to mixed groups (look for two modes first).
- Ignoring outliers, which move the mean and inflate the standard deviation.
- Reading a non-significant test as proof of normality.
How do you use the 68-95-99.7 rule?
If the data are approximately normal, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. It applies only after you have established that the shape is close to normal. Full explanation and examples: the empirical rule for the normal distribution.
What is the 10% rule for sampling distributions?
The 10% rule is a check on independence, not on shape. OpenIntro Statistics (Diez, Barr and Çetinkaya-Rundel, section 2.4) states that when the sample is only a small fraction of the population (under 10%), observations are nearly independent even when sampling without replacement. Check it when you sample from a finite population before using formulas for a sampling distribution that assume independent draws.
How to report it (APA style example)
State the descriptive statistics, the shape and any transformation or test, with italic M, SD, W and p. Example: “Response times were positively skewed (skewness = 1.91; M = 7.20, Mdn = 4.50, SD = 6.91), and a Shapiro-Wilk test rejected normality, W = 0.71, p = .001; log-transformed values did not depart from normality, W = 0.94, p = .58.”
How to calculate it in Excel, R or Python
Excel
=AVERAGE(A1:A10) → 7.2
=MEDIAN(A1:A10) → 4.5
=SKEW(A1:A10) → 2.27 (sample-adjusted skewness)Excel’s SKEW applies a small-sample adjustment, so it returns 2.27 rather than the 1.91 computed above; both indicate positive skew.
R
x <- c(2,3,3,4,4,5,6,8,12,25)
hist(x)
qqnorm(log(x)); qqline(log(x))
shapiro.test(x)Python
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
x = np.array([2,3,3,4,4,5,6,8,12,25])
print(x.mean(), np.median(x), stats.skew(x)) # 7.2 4.5 1.91
print(stats.shapiro(np.log(x))) # W≈0.942, p≈0.58
print(stats.anderson(np.log(x), dist='norm'))
stats.probplot(np.log(x), dist='norm', plot=plt)FAQ
What are the main types of distribution in statistics?
Discrete distributions describe counts (binomial, Poisson); continuous distributions describe measured values (normal, lognormal, exponential, Weibull). The NIST handbook catalogues these with equations and parameter estimation methods.
Which test identifies a distribution?
A goodness-of-fit test such as Anderson-Darling or Shapiro-Wilk (normality only). Use it after the histogram and probability plot, not instead of them.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Histogram, Probability Plot, Anderson-Darling Test (read 2026-11-07).
- OpenStax, Introductory Statistics 2e, 2.6 Skewness and the Mean, Median, and Mode (read 2026-11-07).
- Diez, Barr and Çetinkaya-Rundel, OpenIntro Statistics, 2.4 Sampling from a Small Population (read 2026-11-07).
