Descriptive Statistics

Descriptive statistics condense large datasets into simple summaries. They help you understand the shape, centre, and spread of your data before moving on to more complex analyses.

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Measures of Central Tendency

Central tendency describes where the "middle" of a dataset lies.

Mean (Arithmetic Average)

The mean is the sum of all values divided by the number of values:

x̄ = (x₁ + x₂ + ... + xₙ) / n

Example: For the data set {4, 8, 6, 5, 3, 7, 9}

x̄ = (4 + 8 + 6 + 5 + 3 + 7 + 9) / 7 = 42 / 7 = 6

Note: The mean is sensitive to outliers. A single extreme value can pull the mean far from the typical value.

Median

The median is the middle value when data is sorted in ascending order.

• If n is odd, the median is the centre value.
• If n is even, the median is the average of the two centre values.

Example: Sorted data {3, 4, 5, 6, 7, 8, 9} → Median = 6

Mode

The mode is the value that appears most frequently. A dataset can be:

• Unimodal — one mode
• Bimodal — two modes
• Multimodal — more than two modes
• No mode — all values appear equally often

When to Use Which?

Measure	Best For	Limitation
Mean	Symmetric data without outliers	Sensitive to extreme values
Median	Skewed data or data with outliers	Ignores the magnitude of values
Mode	Categorical data or identifying peaks	May not be unique or may not exist

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Measures of Spread (Dispersion)

Knowing the centre alone is not enough. You also need to know how spread out the data is.

Range

Range = Maximum − Minimum

Simple but highly affected by outliers.

Interquartile Range (IQR)

The IQR measures the spread of the middle 50% of the data:

IQR = Q₃ − Q₁

Where:

• Q₁ (25th percentile) — the median of the lower half
• Q₃ (75th percentile) — the median of the upper half

The IQR is robust against outliers.

Variance

Variance measures the average squared deviation from the mean:

Population variance:  σ² = Σ(xᵢ − μ)² / N
Sample variance:      s² = Σ(xᵢ − x̄)² / (n − 1)

Why n − 1 for samples? Dividing by (n − 1) instead of n corrects for bias and provides an unbiased estimate of the population variance. This is known as Bessel's correction.

Standard Deviation

Standard deviation is the square root of the variance. It is in the same units as the original data, making it easier to interpret: