Statistics Calculator – Mean, Median, Mode, Variance, Standard Deviation & More

Descriptive Statistics

Data (comma separated)

Numbers separated by commas. Spaces optional.

Variance / Standard Deviation

Sample (n‑1)Population (n)

Enter numbers, choose variance type, then click "Calculate".

Example: 78, 85, 92, 88, 76

The Statistics Calculator computes the most common descriptive statistics: mean, median, mode, range, variance, standard deviation, quartiles, and IQR. It supports both sample and population variance. This tool is perfect for students, data analysts, and researchers who need quick, accurate summaries of their data. The step‑by‑step output explains each calculation, making it an excellent learning aid.

Formulas Used

Mean = Σx / n

Median = middle value (or average of two middle values when n is even)

Mode = value(s) with highest frequency

Variance (sample) = Σ(x – x̄)² / (n – 1)

Variance (population) = Σ(x – μ)² / n

Standard deviation = √variance

Quartiles = median of lower/upper halves (Tukey method)

Applications

Education: Analysing test scores and class performance.
Business: Understanding sales variation, customer metrics.
Research: Summarising experimental data.
Quality control: Monitoring process stability.

Choosing Between Sample and Population

If your data is the entire group (e.g., all students in a class), use population formulas. If it's a subset of a larger group (e.g., a random sample of voters), use sample formulas to get an unbiased estimate of the population parameters.

Descriptive Statistics Summary & Formula Comparison Reference Table

Dataset Scenario	Raw Data Sample	Mean (X̄)	Median	Mode(s)	Sample Variance (s²)	Sample Std Dev (s)	IQR (Q3 - Q1)
Test Scores (n=5)	78, 85, 92, 88, 76	83.8000	85.0000	None	45.2000	6.7231	10.0000
Even Set (n=6)	10, 20, 30, 40, 50, 60	35.0000	35.0000	None	350.0000	18.7083	30.0000
Multimodal Set (n=10)	3, 7, 5, 13, 20, 23, 39, 23, 40, 23	19.6000	21.5000	23	174.4889	13.2094	18.0000
Small Sample (n=4)	10, 20, 30, 40	25.0000	25.0000	None	166.6667	12.9099	20.0000

Understanding Descriptive Statistics in Depth

Descriptive statistics summarise and organise data so it becomes easier to understand. The most fundamental measures are central tendency (mean, median, mode) and dispersion (range, variance, standard deviation, IQR). The mean provides the arithmetic average, but it is sensitive to outliers. The median, being the middle value, is robust against extreme values. The mode identifies the most frequent value(s), which is especially useful for categorical data or when multiple peaks exist.

Variance and standard deviation quantify how spread out the data points are around the mean. A small variance indicates that most numbers are close to the mean; a large variance indicates wide dispersion. The standard deviation is in the same units as the original data, making it more interpretable. The interquartile range (IQR) measures the spread of the middle 50% and is often used to identify outliers: any point below Q1 – 1.5×IQR or above Q3 + 1.5×IQR is considered a mild outlier.

Quartiles split the data into four equal parts. Q1 (first quartile) is the 25th percentile, Q2 is the median (50th percentile), and Q3 is the 75th percentile. Box plots (box‑and‑whisker plots) are built from these five numbers: min, Q1, median, Q3, max. Understanding these statistics helps in comparing datasets, detecting skewness, and making data‑driven decisions.

Our calculator also distinguishes between sample and population variance. The population variance formula divides by n, but this underestimates the true variance when you only have a sample. The sample variance divides by n‑1 (Bessel's correction) to provide an unbiased estimator of the population variance. Always choose the correct version based on whether your data represents the whole population or just a sample.

Real‑world applications of descriptive statistics are everywhere. In business, you might analyse daily sales: the mean tells you average revenue, the standard deviation tells you how much sales fluctuate, and quartiles help you understand seasonal patterns. In healthcare, patient vitals are summarised using mean and standard deviation to set normal ranges. In sports, batting averages (means) and consistency (standard deviation) are regularly reported. Our calculator gives you all these metrics in one click, with full transparency of the calculations.

Step‑by‑Step Manual Example

Data: 78, 85, 92, 88, 76

Sorted: 76, 78, 85, 88, 92

Mean = (76+78+85+88+92)/5 = 419/5 = 83.8

Median = 85 (middle value)

Mode = none (all values appear once)

Range = 92‑76 = 16

Variance (sample) = Σ(x‑83.8)² / 4 = (60.84+33.64+1.44+17.64+67.24)/4 = 180.8/4 = 45.2

Std Dev = √45.2 ≈ 6.72

Frequently Asked Questions about Statistics

What is the difference between sample and population variance?

Population variance divides by n (total data points in complete population); sample variance divides by n - 1 (Bessel's correction) to eliminate estimation bias. Use sample variance when your dataset is a sample drawn from a larger population.

How are quartiles calculated in this statistics calculator?

Q1 is the median of the lower half of sorted data (excluding the overall median if count n is odd). Q3 is the median of the upper half. This method is the standard Tukey method.

What does the Interquartile Range (IQR) tell you?

Interquartile range (IQR = Q3 - Q1) measures the spread of the middle 50% of the dataset. Unlike range or standard deviation, IQR is resistant to extreme outliers.

Can I enter decimal numbers and negative numbers?

Yes. The calculator accepts all real numerical inputs (decimals, integers, and negative numbers) separated by commas, spaces, or line breaks.

What is the difference between Mean, Median, and Mode?

The Mean is the arithmetic average (Sum/n). The Median is the physical middle value in a sorted list. The Mode is the most frequently occurring value(s) in the dataset.

What is standard deviation and why is it preferred over variance?

Standard deviation is the square root of variance ($s = \sqrt{s^2}$). It is preferred in practical reports because it is measured in the exact same units as your original data points.

How does the calculator handle multimodal datasets?

If a dataset has multiple numbers tied for highest frequency (e.g. 23 appears three times and 40 appears three times), the calculator reports all multimodal values.

What is Bessel's Correction in sample statistics?

Bessel's correction replaces $n$ with $n - 1$ in the sample variance formula. It accounts for the loss of one degree of freedom when estimating the true population mean from sample data.