What is the difference between sample and population standard deviation?

Sample standard deviation applies Bessel's correction by dividing the sum of squares by (n-1) instead of n. This corrects bias in estimating population parameters from limited samples. Population standard deviation divides by the full count n because the entire data universe is known.

How does the Tukey method identify outliers?

The Tukey method calculates the Interquartile Range (IQR = Q3 - Q1). It then establishes two fences: a lower fence at Q1 - 1.5*IQR and an upper fence at Q3 + 1.5*IQR. Any data point that falls outside these fences is classified as an outlier.

How is the median calculated if the dataset size is even?

If the number of values (n) is even, there is no single middle number. Instead, the median is calculated by taking the two middle values (at positions n/2 and n/2 + 1 after sorting) and averaging them.

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Descriptive Statistics Calculator

Analyze datasets instantly. Get Mean, Median, Mode, Variance, Standard Deviation, and Quartiles. Detect statistical outliers and view a dynamic SVG Box Plot.

Input Dataset values

Separate your numbers using commas, spaces, or line breaks.

Data Scope level

Custom Percentile to Solve

Pre-loaded Datasets

Mean (Average)

Median

4.5

Count (n)

Range

SVG Box-and-Whisker Plot

Central Tendency metrics

Mean (Average)5

Median (Middle)4.5

Mode4

Mid-Range5.5

Dispersion & Variability

Standard Deviation (Sample)2.13809

Variance (Sample)4.571429

Mean Absolute Dev (MAD)1.5

Sum of Squares (SS)32

Quartiles & Outlier Fences

Quartile 1 (Q₁)4

Quartile 3 (Q₃)5.5

Interquartile Range (IQR)1.5

Lower Fence (Q₁ - 1.5×IQR)1.75

Upper Fence (Q₃ + 1.5×IQR)7.75

Outliers Detected9

90th Percentile Value7.6

Descriptive Workings Logs

Parsed dataset containing 8 numeric items.

Sorted dataset: [2, 4, 4, 4, 5, 5, 7, 9]

Calculated Sum = 40, Mean (μ) = 40 / 8 = 5.000000

Solved Median = 4.500000

Found Mode(s): [4] with frequency of occurrence = 3

Sum of Squares (SS) = Σ(x_i - Mean)² = 32.000000

Variance calculations:

- Sample Variance (s²) = SS / (n - 1) = 4.571429

- Sample Std Dev (s) = √s² = 2.138090

- Population Variance (σ²) = SS / n = 4.000000

- Population Std Dev (σ) = √σ² = 2.000000

Quartiles & Outlier Fences:

- Q₁ (25th percentile) = 4.0000

- Q₃ (75th percentile) = 5.5000

- IQR = Q₃ - Q₁ = 1.5000

- Lower Tukey Fence = Q₁ - 1.5 * IQR = 1.7500

- Upper Tukey Fence = Q₃ + 1.5 * IQR = 7.7500

- Detected Outliers outside fences: [9]

Solved custom 90th percentile = 7.600000

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descriptive statistics in data analysis

Descriptive statistics summarize and organize the features of a dataset. They provide basic information about the variables in a study, highlighting central trends and spread characteristics.

Central Tendency Metrics

These metrics describe the center of a distribution:

Mean (Arithmetic Average): The sum of all values divided by the total number of items ($n$).
Median: The middle value when the data points are ordered from smallest to largest. Unlike the mean, the median is robust against extreme outliers.
Mode: The most frequently occurring value in the dataset. A distribution can be unimodal, multimodal (multiple modes), or have no mode at all.

Dispersion and Spread Metrics

These numbers define how spread out or clustered the data points are:

Variance ($s^2$ or $\sigma^2$): The average of the squared deviations from the mean.
Standard Deviation ($s$ or $\sigma$): The square root of the variance. It measures spread in the same unit as the original data.
Interquartile Range (IQR): The range covered by the middle $50\%$ of the data ($Q_3 - Q_1$).