What is the difference between descriptive and inferential statistics?

Descriptive statistics summarise and describe a dataset you already have — using measures like mean, median, standard deviation, and range. Inferential statistics use a sample to draw probability-based conclusions about a larger population — using tools like confidence intervals, hypothesis tests, and p-values. If you measured the height of every student in one class, you would use descriptive statistics. If you want to infer the average height of all students in a country from that class, you would use inferential statistics.

When should I use mean vs median vs mode?

Use mean when data is symmetric and has no extreme outliers — for example, test scores in a normal class. Use median when data is skewed or contains outliers — for example, household incomes in a region with a few very high earners. Use mode when you need the most common value, especially for categorical data — for example, the most popular product size or the most frequent response in a survey.

What is a z-score and why does it matter?

A z-score (also called a standard score) measures how many standard deviations a value is from the mean of its distribution. Formula: z = (x − μ) / σ. A z-score of 0 means the value equals the mean; +1 means one standard deviation above the mean; −2 means two standard deviations below. Z-scores let you compare values from different datasets on the same scale — for example, comparing a student's performance across subjects with different score ranges.

What is the difference between variance and standard deviation?

Variance is the average squared deviation from the mean. Standard deviation is the square root of the variance. Both measure the spread of data, but standard deviation is expressed in the same units as the original data, making it easier to interpret. For example, if data is in kilograms, the standard deviation is in kilograms, but the variance is in kilograms². Most practical reporting uses standard deviation; variance is useful for mathematical derivations and analysis of variance (ANOVA).

What does a p-value tell you?

A p-value is the probability of observing a result as extreme as your data — or more extreme — assuming the null hypothesis is true. A p-value of 0.03 means there is a 3% chance the observed result occurred by random chance if the null hypothesis were true. By convention, a p-value below 0.05 is considered "statistically significant." However, p-value alone does not measure the size or practical importance of an effect — always report effect size alongside p-values.

How do I calculate the mean of grouped data?

For grouped data (frequency tables), use the midpoint of each class interval as the representative value. Multiply each midpoint by its frequency, sum the products, then divide by the total frequency. Formula: Mean = Σ(midpoint × frequency) / Σfrequency. For example, if 10 students scored in the range 60–70 (midpoint 65), the contribution is 65 × 10 = 650. Sum all contributions and divide by total students.

What is the difference between descriptive and inferential statistics?

Descriptive statistics summarise and describe a dataset you already have — using measures like mean, median, standard deviation, and range. Inferential statistics use a sample to draw probability-based conclusions about a larger population — using tools like confidence intervals, hypothesis tests, and p-values. If you measured the height of every student in one class, you would use descriptive statistics. If you want to infer the average height of all students in a country from that class, you would use inferential statistics.

When should I use mean vs median vs mode?

Use mean when data is symmetric and has no extreme outliers — for example, test scores in a normal class. Use median when data is skewed or contains outliers — for example, household incomes in a region with a few very high earners. Use mode when you need the most common value, especially for categorical data — for example, the most popular product size or the most frequent response in a survey.

What is a z-score and why does it matter?

A z-score (also called a standard score) measures how many standard deviations a value is from the mean of its distribution. Formula: z = (x − μ) / σ. A z-score of 0 means the value equals the mean; +1 means one standard deviation above the mean; −2 means two standard deviations below. Z-scores let you compare values from different datasets on the same scale — for example, comparing a student's performance across subjects with different score ranges.

What is the difference between variance and standard deviation?

Variance is the average squared deviation from the mean. Standard deviation is the square root of the variance. Both measure the spread of data, but standard deviation is expressed in the same units as the original data, making it easier to interpret. For example, if data is in kilograms, the standard deviation is in kilograms, but the variance is in kilograms². Most practical reporting uses standard deviation; variance is useful for mathematical derivations and analysis of variance (ANOVA).

What does a p-value tell you?

A p-value is the probability of observing a result as extreme as your data — or more extreme — assuming the null hypothesis is true. A p-value of 0.03 means there is a 3% chance the observed result occurred by random chance if the null hypothesis were true. By convention, a p-value below 0.05 is considered "statistically significant." However, p-value alone does not measure the size or practical importance of an effect — always report effect size alongside p-values.

How do I calculate the mean of grouped data?

For grouped data (frequency tables), use the midpoint of each class interval as the representative value. Multiply each midpoint by its frequency, sum the products, then divide by the total frequency. Formula: Mean = Σ(midpoint × frequency) / Σfrequency. For example, if 10 students scored in the range 60–70 (midpoint 65), the contribution is 65 × 10 = 650. Sum all contributions and divide by total students.

Statistics Calculators

Free statistics tools with formula explanations, step-by-step examples, and real-world context. From mean and median to confidence intervals and p-values — every calculator includes the math behind the result.

Maintained by CalcMulti Editorial Team · Last updated: February 2026

What Is Statistics?

Statistics is the branch of mathematics that deals with collecting, organising, analysing, interpreting, and presenting data. It underpins virtually every scientific field — from clinical trials and economic forecasting to machine learning and quality control. Statistics gives us the tools to extract meaningful conclusions from data that would otherwise be noise.

Descriptive Statistics

Summarises and describes the data you already have. It does not generalise beyond the sample.

• Mean, median, mode — central tendency
• Range, variance, standard deviation — spread
• Percentiles, quartiles — position
• Histograms, box plots — distribution shape

Inferential Statistics

Uses a sample to draw probability-based conclusions about a larger population.

• Confidence intervals — range for a parameter
• Hypothesis testing — p-values, t-tests
• Regression — relationships between variables
• ANOVA — comparing multiple group means

A critical distinction: when you have data for an entire population, you use population formulas (divide by n). When you have a sample, you use sample formulas (divide by n − 1). This correction — known as Bessel's correction — ensures sample estimates are unbiased estimators of population parameters.

Measures of Central Tendency

Mean · Median · Mode

Central tendency describes the centre of a dataset — a single representative value that summarises where most values cluster. There are three main measures, each suited to different data types and distributions.

Mean

x̄ or μ

Σx / n

Best for: Uses all data points; best for symmetric distributions
Limitation: Sensitive to outliers
Example use: Average exam score in a class

Open Mean Calculator →

Median

Middle value when sorted

Best for: Robust to outliers; best for skewed data
Limitation: Ignores magnitude of extreme values
Example use: Median household income

Open Median Calculator →

Mode

Most frequent value

Best for: Works for categorical data
Limitation: May not be unique; not useful for continuous data
Example use: Most popular shoe size

Open Mode Calculator →

When to Use Mean vs Median vs Mode

Decision guide — pick the right measure for your data

Choosing the wrong measure of central tendency produces misleading summaries. The decision depends on three factors: the level of measurement, the shape of the distribution, and whether outliers are present.

Situation	Mean	Median	Mode
Symmetric distribution, no outliers	✅ Best	✅ OK	—
Skewed distribution (e.g. incomes)	⚠️ Misleading	✅ Best	—
Data has extreme outliers	⚠️ Pulled by outliers	✅ Best	—
Categorical data (colours, sizes)	❌ Not valid	❌ Not valid	✅ Best
Finding most popular item	—	—	✅ Best
Normal (bell curve) distribution	✅ Best	✅ Same	✅ Same
Bimodal distribution (two peaks)	⚠️ Misleading	⚠️ Misleading	✅ Both modes
Small dataset (< 10 values)	✅ OK	✅ OK	⚠️ Unstable

Quick Decision Rule

→

Is data categorical? Use mode.

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Is data numerical with outliers or a skewed distribution? Use median. Report mean as supplemental information.

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Is data numerical, roughly symmetric, no extreme outliers? Use mean. It is the most mathematically useful measure.

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Not sure? Report both mean and median. If they differ significantly, the distribution is likely skewed.

Measures of Spread

Central tendency tells you where the data is centred. Spread (also called variability or dispersion) tells you how far values typically deviate from that centre. Two datasets can have identical means but completely different spreads — and that difference matters enormously in practice.

Measure	Formula	Units	Use when	Calculator
Range	Max − Min	Same as data	Quick, rough estimate of spread	Calculate →
Variance (σ²)	Σ(x − μ)² / n	Squared units	Mathematical derivations, ANOVA	Calculate →
Standard Deviation (σ)	√Variance	Same as data	Most practical reporting	Calculate →
Coefficient of Variation	(σ / μ) × 100	%	Comparing spread across different scales	Calculate →
Interquartile Range (IQR)	Q3 − Q1	Same as data	Robust to outliers; used in box plots	Calculate →

Standard deviation is by far the most commonly reported measure of spread because it is in the same units as the data. Variance is useful internally but rarely reported to non-technical audiences. When comparing datasets measured in different units (e.g., height in cm vs weight in kg), use the coefficient of variation — it expresses spread as a percentage of the mean, making comparison valid.

Position Measures — Where Does a Value Rank?

Position measures describe where a specific value sits within a distribution — relative to all other values. Unlike central tendency (where the data clusters) or spread (how wide the data is), position answers: how does this particular value compare to the rest?

Z-Score

z = (x − μ) / σ

Expresses how many standard deviations a value is from the mean. A z-score of +1.5 means the value is 1.5 standard deviations above average. Negative z-scores fall below the mean.

Best for:: Comparing values across different datasets (different units, different scales)
Example:: Comparing a student's performance in Math vs English on different scoring systems

Open Z-Score Calculator →

Percentile Rank

P = B / n × 100

Tells you what percentage of the dataset falls at or below a given value. The 75th percentile means 75% of values are at or below that point. Percentiles are used in standardised tests, growth charts, and salary benchmarks.

Best for:: Ranking a value within a real dataset (no assumption of normal distribution required)
Example:: Determining which percentile of test-takers a score falls in

Open Percentile Calculator →

Z-Score vs Percentile — Which to Use?

Condition	Z-Score	Percentile Rank
Data is approximately normally distributed	✅ Preferred	✅ OK
Data is skewed or non-normal	⚠️ Use with caution	✅ Preferred
Comparing across two different datasets	✅ Best (unit-free)	⚠️ Only if same reference group
Communicating to a non-technical audience	⚠️ Less intuitive	✅ "You scored higher than X%"
Population σ is known	✅ Use z = (x−μ)/σ	—
Working from raw data only	⚠️ Need mean + σ first	✅ Calculate directly from data

Probability & Inferential Statistics

Inferential statistics bridges the gap between a sample and the larger population it represents. The foundation is probability theory — which quantifies uncertainty mathematically.

Z-Score & Normal Distribution

The z-score converts any value to the number of standard deviations from its distribution's mean. This allows comparison across different datasets. Under a normal distribution, approximately 68% of values fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ (the empirical rule).

Z-Score Calculator →

Confidence Intervals

A 95% confidence interval means: if you repeated the sampling process 100 times, approximately 95 of the resulting intervals would contain the true population parameter. It quantifies the precision of an estimate — a wide interval means high uncertainty; a narrow interval means the sample provides strong evidence.

Confidence Interval Calculator →

P-Values & Hypothesis Testing

A p-value is the probability of observing your result (or more extreme) if the null hypothesis were true. A small p-value (< 0.05 by convention) is evidence against the null hypothesis. Important: statistical significance ≠ practical importance. Always pair p-values with effect sizes.

P-Value Calculator →

Conditional Probability & Bayes

Conditional probability asks: given that event A occurred, what is the probability of B? P(B|A) = P(A ∩ B) / P(A). Bayes' theorem reverses this: it lets you update a prior belief with new evidence. It is the foundation of Bayesian statistics and is used in spam filters, medical diagnosis, and machine learning.

Probability Calculator →

All Statistics Calculators

Mean CalculatorNew

Arithmetic, weighted, geometric, and harmonic mean.

Central Tendency

Median CalculatorNew

Middle value of any dataset — robust to outliers.

Central Tendency

Mode CalculatorNew

Most frequent value; bimodal and multimodal support.

Central Tendency

Standard Deviation Calculator

Population and sample standard deviation with variance.

Spread

Variance CalculatorNew

Population and sample variance from raw data.

Spread

Range CalculatorNew

Max − min — the simplest measure of spread.

Spread

Coefficient of VariationNew

Relative variability as a % of the mean.

Spread

Z-Score CalculatorNew

Standardise any value in standard deviation units.

Position

Percentile CalculatorNew

Percentile rank of a value within a dataset.

Position

Probability Calculator

Basic, conditional, and complement probability.

Probability

Normal Distribution CalculatorNew

P(X < x), P(X > x), P(a < X < b) for any mean and σ.

Inference

Sample Size CalculatorNew

Minimum sample size for surveys and experiments.

Inference

T-Test CalculatorNew

One-sample and two-sample t-test with p-value.

Inference

Correlation CalculatorNew

Pearson r and R² for two paired variables.

Relationships

Linear Regression CalculatorNew

Slope, intercept, R² and predictions for y = mx + b.

Relationships

Chi-Square CalculatorNew

Goodness-of-fit χ² statistic with p-value.

Inference

Standard Error CalculatorNew

SE of the mean, 95% and 99% confidence intervals.

Inference

Confidence Interval Calculator

95% and 99% CIs for means and proportions.

Inference

P-Value Calculator

One-tail and two-tail p-values from z and t scores.

Inference

Formula Guides

Deep-dive explanations — where each formula comes from, how to apply it, and common mistakes to avoid.

Mean Formula Explained

How arithmetic, weighted, geometric, and harmonic mean are derived.

Population vs Sample Variance

Why we divide by n for population but n−1 for a sample (Bessel's correction).

Z-Score Formula Guide

Standardisation, normal distribution, and when to use z vs t.

Probability Formula Explained

Probability rules, conditional probability, independence, and Bayes theorem.

Comparisons

Mean vs Median

When each measure is the right choice — with skewed data examples.

Variance vs Standard Deviation

Same information, different units — which to report and why.

Common Statistical Errors

Confusing correlation with causation

Two variables moving together (correlation) does not mean one causes the other. Ice cream sales and drowning rates both rise in summer — but ice cream does not cause drowning; both are driven by hot weather. Always look for confounding variables before inferring causation.

Using mean for skewed data

The arithmetic mean is pulled toward outliers. When a dataset is right-skewed — such as income distributions, housing prices, or response times — report the median. A small number of extremely high values inflate the mean, making it unrepresentative of the typical case.

Misinterpreting p < 0.05 as proof

Statistical significance means the result is unlikely under the null hypothesis — it does not confirm the alternative hypothesis is true. A p-value of 0.04 means there is a 4% chance of this result if the null hypothesis were true. Multiple comparisons compound this problem: run 20 tests and expect one to be "significant" by chance at p < 0.05.

Dividing by n instead of n−1 for sample variance

When computing variance from a sample (not the full population), the denominator must be n−1, not n. This is Bessel's correction — it produces an unbiased estimate of the population variance. Most calculators and software default to the correct formula, but be aware of which convention a tool uses.

Averaging percentages directly

You cannot take the arithmetic mean of percentages unless the sample sizes are equal. If store A has 20% return rate on 1,000 sales and store B has 80% return rate on 10 sales, the combined rate is not 50% — it is (200 + 8) / 1,010 ≈ 20.6%. Use weighted mean with sample sizes as weights.

Frequently Asked Questions

Educational use only. All calculators on this page use standard mathematical formulas from academic and public domain sources. Content is reviewed for accuracy by the CalcMulti Editorial Team. For research, clinical, or professional decisions, verify results with qualified software and subject-matter expertise. Last updated: February 2026.