Common Descriptive Statistics

Introduction

Descriptive statistics summarize and describe the main features of a dataset. They provide a way to present large amounts of data in a meaningful and understandable way, forming the foundation for further statistical analysis.

The three main categories of descriptive statistics are measures of central tendency (where is the center?), measures of dispersion (how spread out is the data?), and measures of shape (what does the distribution look like?).

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Measures of Central Tendency

These describe the "typical" or "central" value in a dataset.

Mean (Average)

• Pros: Uses all data points; mathematically tractable
• Cons: Sensitive to outliers
• Use when: Data is symmetric without extreme outliers

Median

The middle value when data is sorted. For even number of observations, it's the average of the two middle values.

• Pros: Not affected by outliers
• Cons: Doesn't use all data points
• Use when: Data is skewed or has outliers

Mode

The most frequently occurring value.

• Pros: Works for categorical data; shows most common value
• Cons: May not exist or may have multiple modes
• Use when: Finding most common category or response

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Measures of Dispersion

These describe how spread out the data is.

Range

• Simple but affected by outliers
• Doesn't describe distribution within the range

Variance

• Uses all data points
• Units are squared (harder to interpret)

Standard Deviation

• Most commonly used measure of spread
• Same units as original data
• For normal distributions: ~68% within 1 SD, ~95% within 2 SD

Interquartile Range (IQR)

• Robust to outliers
• Used with median for skewed data

Measure	Formula	Sensitive to Outliers?
Range	Max - Min	Yes (very)
Variance	Σ(x-μ)²/n	Yes
Std Deviation	√Variance	Yes
IQR	Q3 - Q1	No

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Measures of Shape

Skewness

Measures asymmetry of the distribution.

• Positive skew: Tail extends to the right (Mean > Median)
• Negative skew: Tail extends to the left (Mean < Median)
• Zero skew: Symmetric distribution

Kurtosis

Measures "tailedness" of the distribution.

• High kurtosis: Heavy tails, more outliers
• Low kurtosis: Light tails, fewer outliers

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Choosing the Right Measure

Data Characteristic	Central Tendency	Dispersion
Symmetric, no outliers	Mean	Standard Deviation
Skewed or outliers	Median	IQR
Categorical data	Mode	N/A

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Conclusion