Descriptive Statistics

Paste a column of data and get the mean, median, standard deviation, quartiles, skewness, kurtosis, box-plot and histogram.

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Tip: paste a column from Excel or drag a .csv

Values

Central tendency

Mean (x̄): 5
Median: 4.5
Mode: 4
Sum (Σx): 40

Spread

Sample std dev (s): 2.138
Population std dev (σ): 2
Sample variance (s²): 4.571
Population variance (σ²): 4
Coefficient of variation (CV): 42.76 %
Standard error (SEM): 0.7559
MAD: 0.5
Range: 7
Interquartile range (IQR): 1.5

Position & quartiles

Minimum: 2
Q1 (25%): 4
Q3 (75%): 5.5
Maximum: 9

Shape

Skewness (g₁): 0.8185
Excess kurtosis (g₂): 0.9406
Outliers (1.5·IQR): 1

Minimum: 2Q1: 4Md: 4.5Q3: 5.5Maximum: 9

MeanOutliers (1)

Summary

Mean

Median

4.5

Std dev (s)

2.138

Positive skew (g₁>0): right tail; typically mean > median.

Leptokurtic (g₂>0): heavier tails than the normal.

1 value(s) beyond the 1.5·IQR range (flagged in the box-plot).

To model this data as Normal, use μ̂ = 5 and σ̂ = 2.138 in Probability Distributions.

Fundamentals & Explanation

Fundamentals of Descriptive Statistics

Descriptive statistics provides the mathematical methods to organize, summarize, and visualize a dataset. Its purpose is to extract key information regarding the central tendency, dispersion, and shape of the distribution, facilitating a rigorous understanding of the sample prior to applying inferential probability techniques.

1. Measures of Central Tendency

These define the value around which data points cluster. The selection among different metrics depends heavily on the nature of the distribution and the presence of outliers.

Arithmetic Mean

\bar{x} = \frac{\sum x_i}{n}

It represents the center of gravity of the data. It is mathematically comprehensive but exhibits high sensitivity to extreme values.

Median

It is the 50th percentile of the distribution. It constitutes a robust estimator, ideal for asymmetric distributions or those with strong outliers.

2. Measures of Dispersion

These quantify the variability of the data relative to its center. Low dispersion indicates homogeneous data, whereas high dispersion signals sample heterogeneity.

s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}

The Sample Standard Deviation (s) measures the average deviation from the mean. The denominator $n-1$ (Bessel's correction) is used to obtain an unbiased estimator of the population variance.

As a robust alternative, the Interquartile Range (IQR) evaluates the spread of the central 50% of the data ( $Q_3 - Q_1$ ), mitigating the statistical effect of the tails.

3. Analysis of Shape (Skewness and Kurtosis)

Skewness (g₁) evaluates the lack of symmetry; a positive value indicates a prolonged tail to the right. Excess Kurtosis (g₂) quantifies the weight of the tails relative to a perfect Normal distribution (g₂ = 0); positive values suggest a higher frequency of extreme events (leptokurtic distribution).

4. Density Analysis (Histograms)

A histogram groups data into intervals (bins) to empirically visualize the probability density function. The choice of binning algorithm drastically alters visual perception:

Sturges' Rule: Ideal for normal distributions and continuous samples. It depends on the sample size ( $n$ ).
Freedman-Diaconis: Highly robust against outliers, as it uses the Interquartile Range (IQR) to define the bin width.
Scott's Rule: Minimizes the integrated standard error, but is sensitive to extreme values since it relies on the standard deviation ( $s$ ).

The Power of the Box-Plot

The Box-Plot is the ultimate tool for spotting outliers at a glance:

The central box wraps the middle 50% of your data (from Q1 to Q3).
The solid line cutting through it is the Median.
The whiskers show the normal reach of the data.
Any loose point outside is a statistical alarm (outlier).

Tip: If your mean (◆) falls very far from your median, you have heavily skewed data.

Descriptive Statistics

Central tendency

Spread

Position & quartiles

Shape

Summary

Fundamentals & Explanation

Fundamentals of Descriptive Statistics

1. Measures of Central Tendency

2. Measures of Dispersion

3. Analysis of Shape (Skewness and Kurtosis)

4. Density Analysis (Histograms)

The Power of the Box-Plot

Suggested Resources

You may also like:

Fundamentals of Descriptive Statistics

1. Measures of Central Tendency

2. Measures of Dispersion

3. Analysis of Shape (Skewness and Kurtosis)

4. Density Analysis (Histograms)

The Power of the Box-Plot

Suggested Resources