SamCalculator

Statistics Summary

Quick mean, median, SD, range, and min/max snapshot of any dataset.

Your Dataset

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What Is Descriptive Statistics?

Descriptive statistics reduce a dataset to a handful of numbers that describe its centre, spread, shape, and tail behaviour. Centre is captured by the mean, median, and mode; spread by the variance, standard deviation, and range; shape by skewness and kurtosis; and tail behaviour by quartiles, percentiles, and outlier fences. A complete descriptive summary tells you what a dataset typically looks like in a single screen, and it's the entry point to any inferential test — confidence intervals, hypothesis tests, regression, machine-learning baselines — all start by summarising what you have.

This Statistics Calculator bundles six analysis modes onto a single dataset: a Quick Summary that returns every classic statistic in one click; a Descriptive Statistics table with formulas and confidence intervals; a Frequency Distribution with histogram and cumulative-frequency chart; a Percentile & Outlier view with quartiles, percentiles, and an SVG box plot; a Distribution Shape mode with normal-curve overlay and 68-95-99.7 rule; and a Probability mode with per-value Z-scores and interval probabilities. It pairs naturally with the standard deviation calculator, the sample size calculator, the z-score calculator, and the probability calculator.

How the Calculator Works

Paste, type, or upload your dataset

Drop a list of numbers separated by commas, spaces, tabs, semicolons, or newlines. Scientific notation, negatives, and decimals are all parsed. You can also upload a CSV, TSV, or TXT file straight from your machine — the calculator extracts every numeric token and flags any that fail to parse.

Pick population or sample

Statistics that involve variance change formula based on whether you have the entire population (divide by n, σ²) or a sample drawn from one (divide by n − 1, s²). The toggle at the top of the input card switches both the variance and standard-deviation outputs site-wide.

One click computes 20+ statistics

Press Calculate Statistics and the calculator returns mean, median, mode, range, midrange, variance, standard deviation, standard error, coefficient of variation, geometric mean, harmonic mean, RMS, mean absolute deviation, skewness, excess kurtosis, quartiles, percentiles, IQR, outliers, and 90/95/99/99.9% confidence intervals.

Six views of the same dataset

Switch tabs to drill into different perspectives — frequency tables, histogram, cumulative CDF, box plot, normal-curve overlay, 68-95-99.7 rule, Z-scores, and probability lookups. Every view stays in sync with the dataset you entered.

6 Ways to Use the Statistics Calculator

1

Get a one-screen overview of any dataset

The Summary tab returns 16 statistics in a single grid — perfect for a quick read on a homework dataset, A/B test sample, or sensor batch. No scrolling, no exporting to a spreadsheet.

2

Write a methods section with confidence intervals

The Descriptive tab includes a four-row CI table at 90%, 95%, 99%, and 99.9%. Copy the row that matches your paper's reporting standard straight into the manuscript.

3

Build a frequency table for a presentation

The Frequency tab returns value, count, relative frequency, cumulative count, percentage, and cumulative percentage — ready to drop into a deck or report.

4

Hunt for outliers before modelling

The Percentile tab applies both the 1.5·IQR and 3·IQR fences automatically. Mild outliers appear amber, extreme outliers rose — see exactly which observations might bias a regression.

5

Sanity-check normality assumptions

The Distribution tab overlays a fitted normal curve on the histogram and reports skewness and excess kurtosis with plain-English interpretation — leptokurtic, mesokurtic, or platykurtic.

6

Look up a probability for a value

The Probability tab converts any value into a Z-score and reports tail probabilities P(X < x), P(X > x), and the interval P(a < X < b) — no separate Z-table needed.

Best Practices for Reporting Descriptive Statistics

Always report the centre and the spread together. A mean without a standard deviation is uninterpretable — you don't know whether the value is typical or coincidental. The conventional report is x̄ ± s or x̄ (s) for sample data; the calculator displays both side by side so you can copy them directly.

Pick the right measure of centre for the shape of your data. The mean is the natural choice for symmetric distributions; the median is far more robust for skewed data such as incomes, response times, and home prices, where a few extreme values pull the mean away from where most observations actually sit. The calculator prints both, and the Distribution tab tells you which is more representative.

Decide whether your data is a sample or the entire population before you read the spread. The sample standard deviation s uses Bessel's correction (divides by n − 1) and is the right choice for almost every real research dataset. The population formula σ (divides by n) is only correct when you genuinely have every member of the population — a rare situation outside textbook exercises.

Flag and report outliers — don't silently delete them. Use the IQR fence as a first pass, look at each flagged value, and decide whether it's a true reading, a data-entry error, or a meaningful tail event. Drop only the values you can defend dropping, and document the decision in your methods section.

Why Descriptive Statistics Matter

Foundation for inference

Every hypothesis test, confidence interval, and machine-learning baseline reads in the same descriptive summary as a first step. Getting the mean, SD, and outliers right protects every downstream conclusion you draw.

Communication of uncertainty

Reporting only a mean implies there is no spread — a deceptive simplification. A median, range, IQR, and confidence interval together communicate how much the underlying value could realistically be, given the data you collected.

Quality control and safety

Manufacturing, healthcare, and clinical settings rely on out-of-control signals defined in standard deviations. Calculating SD and Z-scores correctly is what separates a controlled process from a recall.

Honest data storytelling

Skewness, kurtosis, and outliers are exactly the features that visual summaries hide. Quoting them alongside the mean and median makes a chart honest rather than misleading.

Where Descriptive Statistics Get Tricky

Heavily skewed data

For income, runtime, and reaction-time data the mean lies far above the median because the right tail is long. Report both — and consider a log transform or the geometric mean for multiplicative effects. The calculator returns the geometric mean automatically when all values are positive.

Small samples (n < 10)

Skewness, kurtosis, and outlier fences become unstable for tiny samples — a single extra observation can flip the verdict. Treat the descriptive summary as preliminary and lean on the median, range, and confidence interval rather than the moment-based statistics.

Heaping and rounding

Datasets where every value is rounded to the nearest 5 or 10 will look bimodal or multimodal even when the underlying process is unimodal. Check the frequency table for suspicious clumping before reading into the modes.

Mixed populations

Two genuinely different groups (e.g. weekend and weekday traffic) mashed into one dataset will produce a bimodal distribution and a misleading single mean. Use the Frequency and Histogram tabs to spot multimodality, then split the data before summarising.

The Core Statistics Formulas

Mean (arithmetic average)

x̄ = Σx ÷ n

Sum of every value divided by the number of values.

Median

P50 = middle value of sorted x

Average of the two middle values when n is even.

Mode

argmax frequency(x)

Value (or values) that occurs most often.

Range

R = max(x) − min(x)

Crudest measure of spread; affected only by extremes.

Population variance

σ² = Σ(x − x̄)² ÷ n

Use only when you have every member of the population.

Sample variance

s² = Σ(x − x̄)² ÷ (n − 1)

Bessel correction — the unbiased estimator for σ².

Standard deviation

σ = √σ² s = √s²

Reported in the same units as the underlying data.

Standard error of mean

SEM = s ÷ √n

Standard deviation of the sampling distribution of x̄.

Coefficient of variation

CV = s ÷ |x̄| × 100

Spread expressed as a percentage of the mean.

Quartiles (linear interpolation)

Qp = sorted[(p/100)·(n − 1)]

Standard Tukey method — used here for all percentiles.

Interquartile range

IQR = Q3 − Q1

Robust measure of spread that ignores outliers.

IQR outlier fence

Q1 − 1.5·IQR … Q3 + 1.5·IQR

Tukey fence — values outside are flagged as mild outliers.

Skewness (moment)

g₁ = m₃ ÷ σ³

Positive = right-tailed; negative = left-tailed.

Excess kurtosis

g₂ = m₄ ÷ σ⁴ − 3

Positive = heavier tails than normal; negative = lighter.

Z-score

z = (x − x̄) ÷ s

Standard deviations away from the mean for any value.

Confidence interval (mean)

x̄ ± Z* · s ÷ √n

Standard normal critical value Z*; for 95% Z* ≈ 1.960.

Geometric mean

GM = (∏ xᵢ)^(1/n)

Right metric for ratios and rates of return; positive values only.

Harmonic mean

HM = n ÷ Σ(1 ÷ xᵢ)

Right metric for averaging rates per unit time.

Root mean square

RMS = √(Σx² ÷ n)

Used in physics and signal processing for magnitudes.

Mean absolute deviation

MAD = Σ|x − x̄| ÷ n

Robust alternative to standard deviation.

Standard normal CDF

Φ(z) = ½ [1 + erf(z/√2)]

Maps any Z-score to a cumulative probability.

Common Mistakes to Avoid

  1. 1

    Confusing mean and median for skewed data

    For salaries, response times, or home prices the mean sits above the median because of a long right tail. Quoting the mean alone for skewed data overstates a typical observation. Report the median (and the IQR) for skewed distributions.

  2. 2

    Using the population SD on sample data

    If you only collected a slice of the underlying population, divide by n − 1 (sample SD), not n (population SD). Using the population formula understates the true variability and produces confidence intervals that are too narrow.

  3. 3

    Ignoring outliers because they&apos;re inconvenient

    Outliers can be data-entry errors, sensor faults, or legitimate tail events. Never delete them silently — the calculator's IQR fence shows them so you can investigate. Document any deletions in writing.

  4. 4

    Reading variance in the wrong units

    Variance is reported in squared units of the underlying data — $² for dollars, ms² for milliseconds. Take the square root to get the standard deviation, which is reported in the original units and is comparable to the mean.

  5. 5

    Treating a 95% CI as a probability statement about the parameter

    A 95% confidence interval is a procedure that captures the true mean 95 times out of 100 across repeated samples — it's not the probability that this particular interval contains the true mean. The distinction matters for Bayesian vs frequentist reporting.

  6. 6

    Misinterpreting percentile rank

    Being at the 90th percentile means 90% of values are ≤ you, not that you scored 90%. The calculator's percentile table makes this distinction explicit.

  7. 7

    Assuming normality without checking

    A lot of intro-stats results — z-tests, naive confidence intervals, parametric ANOVAs — require approximate normality. Use the Distribution tab's histogram + normal overlay and the skewness/kurtosis numbers to check the assumption before relying on it.

Worked Examples

Example 1 — full descriptive summary for a tiny dataset

Dataset: {15, 22, 18, 31, 27, 19, 24, 29, 35, 20}

  1. 1Sort: {15, 18, 19, 20, 22, 24, 27, 29, 31, 35}.
  2. 2Sum = 240, n = 10, so x̄ = 240 / 10 = 24.
  3. 3Median = (22 + 24) / 2 = 23 (average of the two middle values for even n).
  4. 4Mode: every value appears once — no mode.
  5. 5Range = 35 − 15 = 20; midrange = (35 + 15) / 2 = 25.
  6. 6Σ(x − x̄)² = 9² + 6² + 5² + 4² + 2² + 0² + 3² + 5² + 7² + 11² = 81 + 36 + 25 + 16 + 4 + 0 + 9 + 25 + 49 + 121 = 366.
  7. 7Sample variance s² = 366 / 9 ≈ 40.667 — so s ≈ 6.377.
  8. 8SEM = s / √n = 6.377 / √10 ≈ 2.017; 95% CI = 24 ± 1.96·2.017 ≈ [20.05, 27.95].

Example 2 — IQR outlier detection on a wage dataset

Dataset: {55, 62, 70, 58, 95, 64, 72, 68, 75, 60, 88, 66, 73, 71, 67, 245} ($k salaries)

  1. 1Sort and find quartiles: Q1 ≈ 63, Q3 ≈ 74.5, IQR ≈ 11.5.
  2. 2Lower fence = 63 − 1.5·11.5 = 45.75; upper fence = 74.5 + 1.5·11.5 = 91.75.
  3. 3The value 245 lies well above the upper fence — flagged as a mild outlier.
  4. 4Apply the 3·IQR test: extreme upper = 74.5 + 3·11.5 = 109 — 245 is also an extreme outlier.
  5. 5Decision: investigate the 245k entry before computing the mean and SD; consider reporting the median ($69) alongside or instead of the mean.

Example 3 — 95% confidence interval for a measurement series

Dataset: {12.4, 12.6, 12.3, 12.5, 12.7, 12.4, 12.5, 12.6, 12.4, 12.5, 12.6, 12.3, 12.5, 12.4, 12.5} mm

  1. 1Sum = 187.2; n = 15; x̄ = 12.48 mm.
  2. 2Sample standard deviation s ≈ 0.117 mm.
  3. 3SEM = 0.117 / √15 ≈ 0.0302 mm.
  4. 495% CI = 12.48 ± 1.96·0.0302 ≈ [12.421, 12.539] mm.
  5. 5Report: the true mean diameter is in [12.42, 12.54] mm with 95% confidence — comfortably inside any 12.5 ± 0.1 mm tolerance band.

Example 4 — Z-score probability for a single observation

Dataset: x̄ = 75, s = 8, query value x = 90

  1. 1z = (90 − 75) / 8 = 1.875.
  2. 2P(X < 90) = Φ(1.875) ≈ 0.9696 — about 96.96% of values lie at or below 90.
  3. 3P(X > 90) = 1 − Φ(1.875) ≈ 0.0304 — about 3.04% of values lie above 90.
  4. 4Interpretation: a score of 90 is roughly the top 3% of the distribution.

Real-World Applications

Business analytics

Daily-revenue dashboards almost always quote a mean and a standard deviation. Add the median and IQR to spot whether a recent revenue dip is a single bad day or a sustained pattern.

Market research

Survey data lives or dies by descriptive statistics — mean satisfaction scores, percentage in each segment, modal preference. Use the frequency table for categorical-style rating scales.

Academic research

Methods sections begin with descriptive statistics for every key variable: mean, SD, range, and n at minimum. Confidence intervals are increasingly expected for any primary outcome.

Machine learning

Feature engineering starts with descriptive statistics — outliers, skewness, and scale all change how a model behaves. Run every numeric column through this calculator before you scale, transform, or fit.

Healthcare

Vitals are reported as means with confidence intervals; outliers are clinically actionable. The IQR fence is the standard rule of thumb for flagging measurements worth a second look.

Finance

Return distributions are notoriously fat-tailed — descriptive stats with skewness and kurtosis are the entry point to risk modelling. Coefficient of variation is the standard way to compare volatility across instruments.

Manufacturing

Statistical process control sets specification limits at x̄ ± 3·σ. Daily descriptive summaries are how engineers spot a drifting process before it produces scrap.

Sports analytics

Per-game stats, season averages, percentile rankings, and outlier games all rely on the same descriptive toolkit. The frequency table is perfect for shot-distance distributions.

Quality control

Acceptance sampling and Six Sigma compute defect rates, process capability indices, and out-of-control rules in terms of mean and SD — the calculator returns every input those formulas need.

Government statistics

Census data, economic indicators, and public-health releases publish full descriptive distributions — quartiles, percentiles, IQR — because the median and IQR are robust to the extreme tails of real-world data.

Statistics Reference Guide

Mean

The arithmetic average, x̄ = Σx ÷ n. Uses every observation, so a single extreme value pulls it noticeably. The natural measure of centre for roughly symmetric data.

Median

The middle value of the sorted dataset (or the average of the two middle values for even n). Unaffected by extreme values, so the right choice for skewed distributions like incomes or response times.

Mode

The value (or values) that appears most often. A dataset can be unimodal, bimodal, multimodal, or have no mode at all when every observation is unique.

Variance

Mean of the squared deviations from the mean. Reported in squared units of the underlying data — variance of dollars is in dollars². Take the square root to get the standard deviation.

Standard deviation

Square root of the variance. Reported in the same units as the data, so it's directly comparable to the mean. The most widely-used measure of spread.

Sample standard deviation

Uses Bessel's correction — divides Σ(x − x̄)² by n − 1 instead of n. The unbiased estimator of the population SD when you only have a sample.

Range

Maximum minus minimum. Trivial to compute but driven entirely by the two most extreme values — use it as a quick sanity check, not as a serious measure of spread.

Quartiles

Q1 (25th percentile), Q2 (median, 50th percentile), and Q3 (75th percentile) split the sorted data into four equal pieces.

Percentiles

The value below which a given percentage of observations falls — the 90th percentile means 90% of values are ≤ that point. Linear interpolation between order statistics is the standard method.

Outliers

Values flagged by the IQR fence (Q1 − 1.5·IQR, Q3 + 1.5·IQR). Extreme outliers use the 3·IQR fence. Always investigate before deleting.

Skewness

Third standardised moment. Zero means symmetric; positive means a long right tail; negative means a long left tail. Values above |1| indicate strong asymmetry.

Kurtosis

Fourth standardised moment minus 3 (excess kurtosis). Zero matches the normal distribution; positive indicates fatter tails; negative indicates lighter tails.

Confidence interval

A range computed from the data that captures the true population parameter a known fraction of the time across repeated samples. 95% is the conventional reporting standard.

Normal distribution

The bell-shaped continuous distribution determined entirely by its mean and standard deviation. Many natural processes are approximately normal — and many statistical tests assume it.

Z-score

The number of standard deviations a value lies away from the mean. Standardises across datasets — a Z of 2 means "two SD above the mean" regardless of units.

Probability

Long-run proportion of observations expected at or below a value (left tail), above (right tail), or between two values, under the assumed distribution.

Methodology you can verify

Quartile and percentile values use linear interpolation between order statistics (Tukey method). Variance and standard deviation report both the population (divide by n) and sample (divide by n − 1, Bessel correction) formulas. Skewness and excess kurtosis are computed from the third and fourth central moments. Normal-distribution probabilities use the Abramowitz & Stegun erf approximation, accurate to ≈ 1.5 × 10⁻⁷. Read more on the methodology and editorial policy pages.

Frequently Asked Questions

Descriptive statistics are the small set of numbers that summarise a dataset — centre, spread, shape, and tail behaviour. Centre is captured by the mean, median, and mode; spread by the range, variance, and standard deviation; shape by skewness and kurtosis; and tail behaviour by quartiles, percentiles, and outlier fences. A complete descriptive summary fits on one screen and tells you what a dataset typically looks like before you run any test or model. Every inferential procedure — confidence intervals, hypothesis tests, regression, machine-learning baselines — begins by reading the same descriptive summary, so getting it right protects every downstream conclusion.

Standard deviation measures how far the values in a dataset typically sit from the mean. Compute the deviation of each value from the mean, square it, average the squared deviations (population uses n; sample uses n − 1), then take the square root. The result is in the same units as the original data, so it is directly comparable to the mean. A small standard deviation says the dataset is tightly clustered; a large one says values are spread widely. For roughly normal data, about 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD — the rule of thumb behind statistical process control limits.

Population standard deviation σ assumes you have every member of the population and divides by n. Sample standard deviation s assumes your data is a sample drawn from a larger population and divides by n − 1 (Bessel's correction). The correction makes s an unbiased estimator of σ — without it, the formula systematically underestimates the true population spread. In practice almost every real research dataset is a sample, so the sample formula is the right default. The toggle at the top of the Statistics Calculator switches between the two.

Variance is the mean of the squared deviations from the mean — the standard deviation squared. Population variance σ² uses Σ(x − x̄)² ÷ n; sample variance s² uses Σ(x − x̄)² ÷ (n − 1). Variance is mathematically tidy because it preserves additivity for independent random variables, which is why it appears throughout probability theory and inferential statistics. The downside is that it is reported in squared units of the underlying data — dollars² for income, ms² for response times — which makes the numbers hard to interpret directly. For human-readable spread, take the square root and quote the standard deviation instead.

Sort the dataset from smallest to largest. If n is odd, the median is the single middle value at position (n + 1) ÷ 2. If n is even, the median is the average of the two middle values at positions n ÷ 2 and n ÷ 2 + 1. For {12, 16, 21, 18, 30, 25, 17, 28} sorted to {12, 16, 17, 18, 21, 25, 28, 30}, the median is the average of the 4th and 5th values: (18 + 21) ÷ 2 = 19.5. The calculator handles odd and even cases automatically and shows the sorted dataset alongside the median for verification.

Quartiles split a sorted dataset into four equal pieces. Q1 (the first quartile) is the value below which 25% of the data falls; Q2 is the median (50%); Q3 (the third quartile) is the value below which 75% falls. The interquartile range — IQR = Q3 − Q1 — captures the middle 50% of the data and is the basis of the box plot and the standard outlier-fence rule. The calculator uses linear interpolation between order statistics (the Tukey method) to handle datasets where the quartile position doesn't land exactly on an integer index.

A percentile is the value below which a given percentage of observations falls. The 90th percentile is the value such that 90% of the dataset is at or below it. Quartiles are just special percentiles — Q1 = P25, Q2 = P50, Q3 = P75. The calculator returns P10, P25, P50, P75, P90, P95, and P99 in a single table, which is exactly the panel reporters and methods sections most often need. The numerical recipe matches the same linear-interpolation method used for the quartiles.

Outliers are values that lie far from the rest of the dataset. The most common detection rule is Tukey's IQR fence: any value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR is flagged as a mild outlier; the 3·IQR fence flags extreme outliers. The calculator applies both fences automatically and lists the flagged values alongside the lower and upper bounds. Always investigate before deleting — outliers can be data-entry errors, sensor faults, or legitimate tail events. Document any decisions you make about them so your analysis stays reproducible.

Skewness measures the asymmetry of the distribution. Zero means symmetric; positive means a long right tail (a few high values pull the mean above the median); negative means a long left tail. Values between roughly −0.5 and +0.5 are considered approximately symmetric; values beyond ±1 indicate strong asymmetry. The calculator returns the moment-based skewness g₁ = m₃ ÷ σ³ and classifies the distribution as left-skewed, right-skewed, or symmetric — useful when deciding whether to report the mean or the median as the typical value.

Kurtosis measures the heaviness of the tails relative to a normal distribution. The calculator reports excess kurtosis (g₂ = m₄ ÷ σ⁴ − 3), so 0 corresponds to a normal distribution, positive values are leptokurtic (heavier tails, sharper peak — outliers are more likely), and negative values are platykurtic (lighter tails, flatter peak — values cluster more uniformly). For financial returns and reaction-time data, expect strongly positive excess kurtosis; for uniform or bounded data, expect negative.

Use sample standard deviation s (divides by n − 1) whenever your data is a subset drawn from a larger population — which is essentially every real research dataset. Use population standard deviation σ (divides by n) only when you genuinely have every member of the population, like a complete census or the entire output of a fixed production batch. If in doubt, choose Sample — the formula is unbiased and slightly larger, which makes any subsequent confidence interval correctly wider, not falsely narrow.

Population statistics describe the complete group; sample statistics describe a subset drawn from it. Mean and median are computed the same way in both cases, but variance and standard deviation use different divisors (n vs n − 1) to keep the sample estimate unbiased. Confidence intervals and hypothesis tests exist precisely to bridge the gap — they let you state, with explicit uncertainty, what the population value probably is given the sample you collected. The Statistics Calculator returns both sample (s, s²) and population (σ, σ²) versions, plus 90%, 95%, 99%, and 99.9% confidence intervals for the mean.

A confidence interval is a procedure for constructing a range from sample data that captures the true population parameter a known fraction of the time across repeated samples. A 95% confidence interval means that if you repeated the experiment 100 times and built the interval each time, about 95 of those intervals would contain the true value. The interval is centred on the sample mean and extends by a margin of error of Z* · SEM, where Z* depends on the confidence level (1.96 for 95%, 2.576 for 99%) and SEM = s ÷ √n. Larger samples produce tighter intervals because SEM shrinks with √n.

The coefficient of variation (CV) is the sample standard deviation divided by the absolute value of the mean, expressed as a percentage. It standardises spread, letting you compare variability across datasets with different units or different scales — a CV of 5% is small for any dataset; a CV of 80% is high regardless of units. CV is only meaningful when the mean is well away from zero; for data centred on zero, the metric explodes and should not be quoted. The calculator returns CV automatically and warns when the mean is too small to use it.

Geometric mean is the n-th root of the product of n positive values. It is the right measure of centre for ratios, rates of return, and multiplicative processes — for example, average annual investment growth or average price-ratio across periods. Computing it requires every value to be positive (logs of zero or negative values are undefined), and it always lies between the harmonic mean and the arithmetic mean. The calculator returns it automatically when all values are positive and shows a dash when they are not.

The interquartile range — IQR = Q3 − Q1 — captures the middle 50% of the dataset. It is a robust measure of spread because, unlike the standard deviation, it is not affected by extreme outliers. The IQR is also the basis of the box plot and the outlier-fence rule. A small IQR relative to the range says the bulk of the data is tightly clustered with a few far-flung values; a large IQR says the data is spread broadly even in its core. Report the IQR alongside the median for skewed distributions where the mean and SD would be misleading.

A Z-score expresses how many standard deviations a value lies away from the mean. A Z of +1 means one SD above the mean; a Z of −2 means two SD below. Under a normal distribution, |Z| > 2 corresponds to roughly the outer 5% of values (P(|Z| > 2) ≈ 4.55%), and |Z| > 3 to the outer 0.27%. The Probability tab of the calculator standardises every value into a Z, returns left- and right-tail probabilities, and lets you compute interval probabilities P(a < X < b) without a separate Z-table.

Related Statistics Calculators

Pair the statistics calculator with related statistics and probability tools.