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Descriptive Statistics & Outlier Detector

Calculate variance, standard deviation, quartiles, IQR, and detect statistical outliers with box plots.

Formula Verified by Dr. Elena Rostova, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Inputs

Dataset Values (Comma-separated)

Results

Sample Mean

19.3000

Median (Q2)

16.50

Sample Standard Dev (s)9.592242

Sample Variance (s²)92.011111

Population Std Dev (σ)9.100000

Population Variance (σ²)82.810000

Five-Number Summary[Min: 12, Q1: 14.00, Med: 16.50, Q3: 21.00, Max: 45]

Interquartile Range (IQR)7.0000

Outliers Detected

Five-Number Summary Box Plot

Box plot: Blue boxrepresents IQR (Q1 to Q3) | Orange lineis Median | Red dots are Outliers

Step-by-Step Descriptive Solution

Sort the dataset of size $N = 10$ in ascending order:

\[\text{Sorted Data} = \{ 12,\; 13,\; 14,\; 15,\; 16,\; 17,\; 18,\; 21,\; 22,\; 45 \}\]

Calculate arithmetic Mean (average):

\[\bar{x} = \frac{\sum x_i}{N} = \frac{193}{10} = 19.3000\]

Calculate Five-Number Summary:

1. Minimum: $12$

2. First Quartile (Q1): $14.00$

3. Median (Q2): $16.50$

4. Third Quartile (Q3): $21.00$

5. Maximum: $45$

Calculate Interquartile Range (IQR):

\[\text{IQR} = Q_3 - Q_1 = 21.00 - 14.00 = 7.00\]

Detect outliers using the Tukey outlier rule (boundaries $[Q1 - 1.5\text{IQR}, \; Q3 + 1.5\text{IQR}]$):

\[\text{Lower Bound} = 14.00 - 1.5(7.00) = 3.50\]

\[\text{Upper Bound} = 21.00 + 1.5(7.00) = 31.50\]

Detected Outliers: $ \{ 45 \} $.

Bookmark this Statistics Calculator

Save this tool to analyze dataset summaries, check variance and standard deviation, and detect outliers.

Formula

Descriptive moments formulas

s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N - 1}}, \quad \text{IQR} = Q_3 - Q_1

Sample variance uses Bessel's correction dividing by N-1. Standard deviations measure average offset magnitudes.

Step by step

How the calculation works

1Input the dataset coordinates separated by commas.
2Review the standard central tendencies: mean, median, modes.
3Study box plot quartiles (Q1, Q3), variance, and Tukey outlier detections.

Related guides

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Guide

Standard Deviation Guide: Measure Data Spread Clearly

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Std Dev

Standard Deviation

Evaluate standard deviations only.

Average Solver

Average Calculator

Evaluate general dataset averages.

Results

Sample Mean

19.3000

Median (Q2)

16.50

Sample Standard Dev (s)9.592242

Sample Variance (s²)92.011111

Population Std Dev (σ)9.100000

Population Variance (σ²)82.810000

Five-Number Summary[Min: 12, Q1: 14.00, Med: 16.50, Q3: 21.00, Max: 45]

Interquartile Range (IQR)7.0000

Outliers Detected

Five-Number Summary Box Plot

Box plot: Blue boxrepresents IQR (Q1 to Q3) | Orange lineis Median | Red dots are Outliers

Step-by-Step Descriptive Solution

Sort the dataset of size $N = 10$ in ascending order:

\[\text{Sorted Data} = \{ 12,\; 13,\; 14,\; 15,\; 16,\; 17,\; 18,\; 21,\; 22,\; 45 \}\]

Calculate arithmetic Mean (average):

\[\bar{x} = \frac{\sum x_i}{N} = \frac{193}{10} = 19.3000\]

Calculate Five-Number Summary:

1. Minimum: $12$

2. First Quartile (Q1): $14.00$

3. Median (Q2): $16.50$

4. Third Quartile (Q3): $21.00$

5. Maximum: $45$

Calculate Interquartile Range (IQR):

\[\text{IQR} = Q_3 - Q_1 = 21.00 - 14.00 = 7.00\]

Detect outliers using the Tukey outlier rule (boundaries $[Q1 - 1.5\text{IQR}, \; Q3 + 1.5\text{IQR}]$):

\[\text{Lower Bound} = 14.00 - 1.5(7.00) = 3.50\]

\[\text{Upper Bound} = 21.00 + 1.5(7.00) = 31.50\]

Detected Outliers: $ \{ 45 \} $.

How to understand and use this calculator

Introduction to Descriptive Statistics and Summaries

Descriptive statistics consist of coefficients that summarize a given dataset, which can be either a representation of the entire population or a sample of a population. They are broken down into measures of central tendency (which describe the center of the data) and measures of variability (which describe the spread of the data). Descriptive statistics are the first step in data analysis, providing an overview of data before performing complex inferential hypothesis tests.

Central tendency metrics include the mean (the arithmetic average), the median (the middle value when data is sorted), and the mode (the most frequent value). Measures of variability include the range (difference between maximum and minimum), variance, and standard deviation (which measure how far numbers are spread out from the mean).

This calculator evaluates descriptive statistics for any comma-separated list of numbers. It computes all measures of central tendency, sample and population variance, the five-number summary, the Interquartile Range (IQR), and automatically identifies any statistical outliers using the standard IQR rule.

Measures of Variability: Variance and Standard Deviation

Standard deviation measures the average distance of data points from the mean. We distinguish between sample standard deviation $s$ (used when data represents a subset of a population) and population standard deviation $\sigma$ (used when data represents the entire group):

1. Sample Variance ($s^2$): Uses Bessel's correction (dividing by $N-1$): \[s^2 = \frac{\sum (x_i - \bar{x})^2}{N - 1}\] Where $\bar{x}$ is the sample mean and $N$ is the sample size.

2. Population Variance ($\sigma^2$): Divides by the total count $N$: \[\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}\]

Standard deviation is the square root of the variance ($s = \sqrt{s^2}$ or $\sigma = \sqrt{\sigma^2}$).

Five-Number Summary, IQR, and Outlier Detection

The five-number summary provides a descriptive overview of sorted data:

1. Minimum: The smallest value. 2. First Quartile (Q1): The 25th percentile (median of lower half). 3. Median (Q2): The 50th percentile. 4. Third Quartile (Q3): The 75th percentile (median of upper half). 5. Maximum: The largest value.

The Interquartile Range is calculated as: $\text{IQR} = Q3 - Q1$.

Under the Tukey outlier fence method, outliers are defined as any data points that fall outside the boundaries:

\[\text{Lower Bound} = Q1 - 1.5 \times \text{IQR}\]

\[\text{Upper Bound} = Q3 + 1.5 \times \text{IQR}\]

This calculator identifies outliers, lists them separately, and explains the boundary calculations.

Related concepts

Bessel's Correction

The math adjustment dividing by N-1 for sample variances.

Tukey Fences

The 1.5*IQR bounds used to identify outlier values.

Five-Number Summary

The box plot values representing range and quartiles.

Frequently Asked Questions

What are descriptive statistics?+

Coefficients that summarize and describe the main characteristics of a dataset.

What is the difference between mean and median?+

The mean is the arithmetic average of all numbers. The median is the middle value when data is sorted in order.

When is the median preferred over the mean?+

The median is preferred when the dataset contains extreme outliers or skewness, as the mean is highly sensitive to outliers.

What is the difference between sample and population standard deviation?+

Sample standard deviation divides by $N-1$ (Bessel's correction) to correct for bias. Population standard deviation divides by $N$.

What is the Interquartile Range (IQR)?+

The difference between the third quartile ($Q3$) and the first quartile ($Q1$), representing the spread of the middle 50% of the data.

How is Q1 and Q3 calculated?+

Q1 is the median of the lower half of the sorted dataset. Q3 is the median of the upper half of the sorted dataset.

How do you detect outliers using the IQR rule?+

Any value below $Q1 - 1.5 \times IQR$ or above $Q3 + 1.5 \times IQR$ is classified as an outlier.

What is standard deviation?+

Standard deviation is a measure of the amount of variation or dispersion of values in a dataset.

What is the range?+

The range is the difference between the largest (maximum) and smallest (minimum) values in the dataset.

What is Bessel's correction?+

The use of $N-1$ instead of $N$ in the denominator when calculating sample variance, which corrects the bias in estimating population variance.

Can standard deviation be negative?+

No, standard deviation is always a non-negative number because it is the square root of variance, which sums squared distances.

What does a standard deviation of 0 mean?+

A standard deviation of 0 means that all values in the dataset are identical to the mean (no variation).

How to understand and use this calculator

Introduction to Descriptive Statistics and Summaries

Measures of Variability: Variance and Standard Deviation

1. Sample Variance ($s^2$): Uses Bessel's correction (dividing by $N-1$): \[s^2 = \frac{\sum (x_i - \bar{x})^2}{N - 1}\] Where $\bar{x}$ is the sample mean and $N$ is the sample size.

2. Population Variance ($\sigma^2$): Divides by the total count $N$: \[\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}\]

Standard deviation is the square root of the variance ($s = \sqrt{s^2}$ or $\sigma = \sqrt{\sigma^2}$).

Five-Number Summary, IQR, and Outlier Detection

The five-number summary provides a descriptive overview of sorted data:

The Interquartile Range is calculated as: $\text{IQR} = Q3 - Q1$.

Under the Tukey outlier fence method, outliers are defined as any data points that fall outside the boundaries:

\[\text{Lower Bound} = Q1 - 1.5 \times \text{IQR}\]

\[\text{Upper Bound} = Q3 + 1.5 \times \text{IQR}\]

This calculator identifies outliers, lists them separately, and explains the boundary calculations.

Related concepts

Bessel's Correction

The math adjustment dividing by N-1 for sample variances.

Tukey Fences

The 1.5*IQR bounds used to identify outlier values.

Five-Number Summary

The box plot values representing range and quartiles.

Frequently Asked Questions

What are descriptive statistics?+

Coefficients that summarize and describe the main characteristics of a dataset.

What is the difference between mean and median?+

The mean is the arithmetic average of all numbers. The median is the middle value when data is sorted in order.

When is the median preferred over the mean?+

The median is preferred when the dataset contains extreme outliers or skewness, as the mean is highly sensitive to outliers.

What is the difference between sample and population standard deviation?+

Sample standard deviation divides by $N-1$ (Bessel's correction) to correct for bias. Population standard deviation divides by $N$.

What is the Interquartile Range (IQR)?+

The difference between the third quartile ($Q3$) and the first quartile ($Q1$), representing the spread of the middle 50% of the data.

How is Q1 and Q3 calculated?+

Q1 is the median of the lower half of the sorted dataset. Q3 is the median of the upper half of the sorted dataset.

How do you detect outliers using the IQR rule?+

Any value below $Q1 - 1.5 \times IQR$ or above $Q3 + 1.5 \times IQR$ is classified as an outlier.

What is standard deviation?+

Standard deviation is a measure of the amount of variation or dispersion of values in a dataset.

What is the range?+

The range is the difference between the largest (maximum) and smallest (minimum) values in the dataset.

What is Bessel's correction?+

The use of $N-1$ instead of $N$ in the denominator when calculating sample variance, which corrects the bias in estimating population variance.

Can standard deviation be negative?+

No, standard deviation is always a non-negative number because it is the square root of variance, which sums squared distances.

What does a standard deviation of 0 mean?+

A standard deviation of 0 means that all values in the dataset are identical to the mean (no variation).

Descriptive Statistics & Outlier Detector

Inputs

Results

Five-Number Summary Box Plot

Step-by-Step Descriptive Solution

Bookmark this Statistics Calculator

Descriptive moments formulas

How the calculation works

Learn more about this calculation

Standard Deviation Guide: Measure Data Spread Clearly

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Standard Deviation

Average Calculator

Descriptive Statistics & Outlier Detector

Inputs

Results

Five-Number Summary Box Plot

Step-by-Step Descriptive Solution

Bookmark this Statistics Calculator

Key output metrics

Descriptive Central metrics

How to understand and use this calculator

Introduction to Descriptive Statistics and Summaries

Measures of Variability: Variance and Standard Deviation

Five-Number Summary, IQR, and Outlier Detection

Related concepts

Frequently Asked Questions

Descriptive moments formulas

How the calculation works

Learn more about this calculation

Standard Deviation Guide: Measure Data Spread Clearly

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Standard Deviation

Average Calculator

Key output metrics

Descriptive Central metrics

How to understand and use this calculator

Introduction to Descriptive Statistics and Summaries

Measures of Variability: Variance and Standard Deviation

Five-Number Summary, IQR, and Outlier Detection

Related concepts

Frequently Asked Questions