Statistics Calculator – Mean, Median, Mode, Standard Deviation, Variance, Skewness & Kurtosis

  • Adam
  • November 4, 2025

Free online statistics calculator for descriptive analysis. Calculate mean, median, mode, variance, standard deviation, skewness, kurtosis, quartiles, and frequency distributions with complete step-by-step explanations.

Statistics Calculator

Advanced Tool for Computing Descriptive Statistics and Data Analysis

Quick Navigation

Descriptive Statistics

Enter data values separated by commas

Distribution Analysis

Analyze data distribution shape and outliers

Grouped Data Statistics

Analyze frequency-grouped data

Frequency Distribution

Build frequency distribution and cumulative frequency

What is Descriptive Statistics?

Descriptive statistics summarizes and describes important features of data. Unlike inferential statistics (which makes predictions about populations), descriptive statistics organize and present data in meaningful ways. It answers "What does this data look like?" rather than "What does this tell us about the population?"

Key measures include measures of central tendency (mean, median, mode showing typical values), measures of spread (range, variance, standard deviation showing data variability), and measures of shape (skewness, kurtosis showing distribution form). Together these statistics provide complete picture of dataset characteristics, helping identify patterns, outliers, and data distribution.

Descriptive statistics are fundamental for data exploration, quality control, reporting, and decision-making. Before advanced statistical analysis, always start with descriptive statistics to understand your data thoroughly. This calculator provides comprehensive analysis of any numeric dataset.

Key Concept: Descriptive statistics transform raw data into meaningful summaries. Five numbers (minimum, Q1, median, Q3, maximum) often describe entire dataset effectively, revealing patterns invisible in raw data.

Key Features & Capabilities

This comprehensive statistics calculator provides complete data analysis:

📊 Descriptive Stats Mean, median, mode, range
📈 Distribution Analysis Skewness, kurtosis, outliers
🔢 Grouped Data Frequency-grouped calculations
📋 Frequency Table Distribution with percentages
📊 Variance & SD Population and sample measures
📈 Quartiles Q1, Q2, Q3, IQR calculation
🎯 Outlier Detection Identify unusual values
📊 Summary Stats All important measures
📋 Detailed Breakdown Step-by-step explanations
📋 Copy Results Export data easily
🎓 Educational Learn statistics concepts
📱 Responsive Works on all devices

How to Use This Calculator

Step-by-Step Guide

  1. Choose Analysis Type: Select Descriptive, Distribution, Grouped Data, or Frequency.
  2. Format Data: Enter values separated by commas or each value on new line.
  3. Optional Parameters: For frequency analysis, specify number of bins/classes.
  4. Click Calculate: Press Calculate to compute all statistics.
  5. Review Results: See comprehensive statistical analysis in table format.
  6. Understand Measures: Read explanations to interpret each statistic.
  7. Copy or Export: Copy results to use in reports or presentations.

Tips for Accurate Analysis

  • Data Quality: Ensure no typos or non-numeric values. Remove headers and labels.
  • Decimal Values: Use decimals (10.5) not commas (10,500). Ensure consistent formatting.
  • Sample vs Population: Calculator provides both. Use sample SD for data subsets.
  • Outlier Investigation: Unusual values may indicate data entry errors or important findings.
  • Distribution Shape: Positive skew = tail right. Negative skew = tail left. Symmetric = skew near 0.

Complete Formulas Guide

Mean (Average)

Mean Formula
Mean = Σx / n

Sum all values and divide by count
Example: {10, 20, 30}
Mean = (10+20+30)/3 = 60/3 = 20

Median

Median Formula
Median = middle value (sorted data)

Sort data, find middle value
Odd count: exact middle value
Even count: average of two middle values

Variance

Variance Formulas
Population Variance (σ²) = Σ(x - μ)² / N
Sample Variance (s²) = Σ(x - x̄)² / (n - 1)

Average squared deviation from mean
Sample uses n-1 (Bessel's correction)

Standard Deviation

Standard Deviation Formula
SD = √Variance

Square root of variance
Returns to original data units
More interpretable than variance

Skewness

Skewness Formula
Skewness = Σ(x - x̄)³ / (n × s³)

Measures distribution asymmetry
Negative = left tail
Positive = right tail
Zero = symmetric

Kurtosis

Kurtosis Formula
Kurtosis = Σ(x - x̄)⁴ / (n × s⁴) - 3

Measures tail heaviness/peakedness
Negative = flat distribution
Positive = peaked distribution
Zero = normal distribution

Understanding Statistical Concepts

Central Tendency

Measures of central tendency describe typical or average value. Mean (average) works well for symmetric data. Median (middle) resists outlier effects. Mode (most frequent) useful for categorical data. Choose based on data type and distribution.

Variability and Spread

Measures of spread show data consistency. Range is simple but affected by outliers. Interquartile range (IQR) is robust. Standard deviation is most important—shows typical distance from mean. Higher SD means more variation.

Skewness

Skewness measures distribution asymmetry. Right-skewed (positive): tail extends right, mean > median. Left-skewed (negative): tail extends left, mean < median. Symmetric: skewness near zero, mean ≈ median. Affects which measures to report.

Kurtosis

Kurtosis measures how peaked or flat distribution is. Leptokurtic (positive): peaked with heavy tails—extreme values more likely. Platykurtic (negative): flat with light tails. Mesokurtic (zero): normal distribution. Indicates tail behavior.

Outliers and Robustness

Outliers are unusually extreme values. Mean and SD are sensitive to outliers. Median and IQR are robust—less affected by outliers. Always investigate outliers: legitimate variation or data errors? Boxplots effectively identify outliers visually.

Worked Examples

Example 1: Simple Descriptive Statistics

Problem: Find statistics for test scores: 75, 85, 90, 78, 92

Solution:
Sorted: 75, 78, 85, 90, 92

Mean = (75+78+85+90+92)/5 = 420/5 = 84
Median = 85 (middle value)
Mode = none (all unique)
Range = 92-75 = 17
Min = 75, Max = 92

Example 2: Standard Deviation

Problem: Calculate SD for: 2, 4, 6, 8, 10

Solution:
Mean = 6

Deviations: -4, -2, 0, 2, 4
Squared: 16, 4, 0, 4, 16
Sum = 40

Sample Variance = 40/(5-1) = 10
Sample SD = √10 ≈ 3.16

Example 3: Identifying Outliers

Problem: Detect outliers in: 10, 12, 14, 15, 16, 18, 20, 100

Solution:
Sorted: 10, 12, 14, 15, 16, 18, 20, 100
Median = (15+16)/2 = 15.5
Q1 = 12.5, Q3 = 19
IQR = 19 - 12.5 = 6.5

Outlier bounds: [Q1-1.5×IQR, Q3+1.5×IQR]
= [2.25, 29.25]

100 is outside bounds → OUTLIER

Example 4: Grouped Data

Problem: Calculate mean for grouped data

Solution:
Class | Midpoint | Frequency
0-10 | 5 | 3
10-20 | 15 | 7
20-30 | 25 | 5

Mean = Σ(midpoint × frequency) / Σfrequency
= (5×3 + 15×7 + 25×5) / 15
= (15 + 105 + 125) / 15
= 245 / 15 ≈ 16.33

Example 5: Distribution Analysis

Problem: Analyze shape of data: 1, 2, 3, 4, 5, 6, 7, 8, 100

Solution:
Data is right-skewed (tail extends right)
Mean ≈ 15 > Median ≈ 5
Large positive skewness ≈ 2.5
High kurtosis due to outlier

Recommendation: Use median, not mean
Investigate outlier value 100

Frequently Asked Questions

❓ When should I use mean vs median?
Use mean for symmetric data without outliers. Use median for skewed data or when outliers present. Median is more resistant to extreme values. Always report both for complete picture.
❓ What's the difference between population and sample SD?
Population SD (σ) describes complete population. Sample SD (s) estimates from sample subset using n-1 divisor (Bessel's correction). Sample SD slightly larger, providing unbiased estimate.
❓ How do I identify outliers?
Use IQR method: values outside [Q1-1.5×IQR, Q3+1.5×IQR] are outliers. Alternatively, values beyond ±3 standard deviations are extreme outliers. Always investigate causes.
❓ Why is standard deviation more useful than variance?
Standard deviation is in original data units, making it interpretable. Variance is squared units—harder to understand. SD directly shows typical distance from mean. Both convey same information differently.
❓ What does positive skewness mean?
Positive skewness = right-skewed distribution. Tail extends to the right. Mean > median. More values concentrated on left. Example: income distribution (most earn less, few earn much).
❓ Can mode be missing?
Yes. Mode is absent when all values appear equally (no value repeats). Calculator will note "no mode" in such cases. Multimodal when multiple values tied for highest frequency.
❓ What is IQR used for?
IQR (Interquartile Range = Q3 - Q1) shows middle 50% spread. Robust measure resisting outlier effects. Used in boxplots and outlier detection. More stable than full range.
❓ How are percentiles different from quartiles?
Quartiles divide into 4 equal parts (Q1=25th, Q2=50th, Q3=75th percentile). Percentiles divide into 100 parts (1st to 99th). Quartiles are special case of percentiles.
❓ Why use grouped data statistics?
Grouped data appears in frequency tables where individual values unknown. Common in historical data or large datasets. Estimates statistics from grouped information using midpoints.
❓ What is the five-number summary?
Five-number summary: minimum, Q1, median, Q3, maximum. Describes data concisely. Forms basis of boxplot visualization. Provides robust data description resistant to outliers.

Analyze Your Data

Whether you're analyzing test scores, business metrics, scientific measurements, or any dataset, this comprehensive statistics calculator provides instant complete analysis. Understand data distribution, identify outliers, and extract meaningful insights. Fast, accurate, and completely free.

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