Describing Distributions

Introduction

Looking at a graph is just the first step. To fully understand data, we must describe what we see using precise statistical language. The framework SOCS (Shape, Outliers, Center, Spread) provides a systematic approach to describing any distribution.

Shape

Shape describes the overall pattern of the distribution.

Symmetry

Symmetric Distribution:

• Left side mirrors right side
• Mean ≈ Median
• Balanced around center

Examples:

• Normal (bell-shaped) distributions
• Uniform distributions
• Heights of adult males

How to identify: If you fold the distribution at the center, both sides match

Skewness

Right-Skewed (Positively Skewed):

• Tail extends to the right
• Mean > Median
• Most data on left, few high values pull mean right

Examples:

• Income (most people earn moderate amounts, few earn very high)
• Home prices
• Test scores when test is easy (most score high, few score low)

Visual: Peak on left, tail stretches right

Left-Skewed (Negatively Skewed):

• Tail extends to the left
• Mean < Median
• Most data on right, few low values pull mean left

Examples:

• Age at death (most live to old age, few die young)
• Test scores when test is hard (most score low, few score high)

Visual: Peak on right, tail stretches left

Memory aid: Skewness direction = direction of the tail (not the peak!)

Modality

Number of peaks (modes) in distribution:

Unimodal: One clear peak

• Most common pattern
• Examples: heights, standardized test scores

Bimodal: Two distinct peaks

• Suggests two different groups
• Examples: Heights of adults (male peak and female peak)

Multimodal: More than two peaks

• Multiple distinct groups
• Less common

Uniform: No peaks, all values equally likely

• Flat distribution
• Example: Rolling a fair die

How to determine: Count prominent "humps" in the distribution

Special Shapes

Normal (Bell-Shaped):

• Symmetric
• Unimodal
• Mean = Median = Mode
• Most data near center, decreasing towards extremes
• Follows empirical rule (68-95-99.7)

Uniform:

• All values equally likely
• Rectangular shape
• No mode

Exponential:

• Decreasing pattern
• Extremely right-skewed
• Many small values, few large values

Outliers

Outliers are observations that fall notably far from the overall pattern.

Identifying Outliers

Visual method:

• Look for isolated points
• Values separated from main cluster

1.5 × IQR Rule (for boxplots):

• Calculate $IQR = Q3 - Q1$
• Lower fence: $Q1 - 1.5 \times IQR$
• Upper fence: $Q3 + 1.5 \times IQR$
• Outliers fall beyond fences

Example:

• Q1 = 65, Q3 = 85
• IQR = 85 - 65 = 20
• Lower fence: 65 - 1.5(20) = 65 - 30 = 35
• Upper fence: 85 + 1.5(20) = 85 + 30 = 115
• Values below 35 or above 115 are outliers

Standard deviation method:

• Outliers > 2 or 3 standard deviations from mean
• Less commonly used
• Appropriate for symmetric distributions

Reporting Outliers

Always:

• Note their presence: "There is one outlier at 150"
• Give actual values if possible
• Consider potential causes

Potential causes:

• Measurement error: Mistake in recording
• Data entry error: Typo when entering data
• Legitimate extreme value: Unusual but real observation
• Different population: Doesn't belong in this group

What to do:

• Investigate cause if possible
• Report with and without outliers (if they affect conclusions)
• Don't automatically delete (unless proven error)

Center

Center describes the "typical" or "middle" value.

Mean vs. Median

When to use each:

Mean ($\bar{x}$):

• Symmetric distributions
• No outliers
• Want to use all data values
• Mathematical properties needed

Median:

• Skewed distributions
• Presence of outliers
• Want resistant measure
• Ordinal data

Relationship to shape:

• Symmetric: Mean ≈ Median
• Right-skewed: Mean > Median (mean pulled right by tail)
• Left-skewed: Mean < Median (mean pulled left by tail)

Mode

Definition: Most frequently occurring value

When reported:

• Categorical data
• Describing bimodal distributions
• Identifying popular values

Limitations:

• May not exist (all values occur once)
• May not be unique (multiple modes)
• Not useful for continuous data with no repeated values

Spread

Spread describes the variability or dispersion of data.

Range

Definition: Maximum - Minimum

Formula: $Range = Max - Min$

Advantages:

• Easy to calculate
• Easy to understand
• Gives sense of total spread

Disadvantages:

• Affected by outliers
• Ignores distribution between extremes
• Only uses two values

Example:

• Data: 12, 15, 18, 20, 22, 25, 100
• Range = 100 - 12 = 88
• Dominated by outlier (100)

Interquartile Range (IQR)

Definition: Range of middle 50% of data

Formula: $IQR = Q3 - Q1$

Advantages:

• Resistant to outliers
• Focuses on middle of distribution
• Useful for boxplots

Disadvantages:

• Ignores lowest 25% and highest 25%
• Less intuitive than range

Example:

• Q1 = 65, Q3 = 85
• IQR = 85 - 65 = 20
• Middle 50% of data spans 20 points

Interpretation: "Half the data falls within [IQR] points"

Standard Deviation

Definition: Average distance from the mean

Interpretation: Typical deviation from mean

Advantages:

• Uses all data values
• Has important mathematical properties
• Basis for many statistical methods

Disadvantages:

• Affected by outliers
• Less intuitive than range
• Only meaningful for roughly symmetric distributions

When to report:

• Symmetric distributions
• No extreme outliers
• Want to use standard statistical methods

Context Matters!

Units

Always include units in descriptions:

❌ "The mean is 68"
✓ "The mean height is 68 inches"

❌ "The standard deviation is 3.5"
✓ "The standard deviation of test scores is 3.5 points"

Comparison

Describe in context of:

• What you'd expect
• Other groups
• Previous studies

Examples:

• "Students averaged 85%, which is higher than last year's 78%"
• "The standard deviation of 15 points shows high variability"

Complete Description Template

A complete distribution description includes:

Shape: "The distribution of [variable] is [symmetric/right-skewed/left-skewed] and [unimodal/bimodal/etc.]"

Outliers: "There is/are [number] outlier(s) at [value(s)]" or "There are no apparent outliers"

Center: "The [mean/median] [variable] is [value with units]"

Spread: "The [variable] ranges from [min] to [max] [units]" or "The standard deviation is [value] [units]"

Example:

Data: Test scores in AP Statistics class

"The distribution of test scores is slightly right-skewed and unimodal with one outlier at 45%. The median score is 82%, indicating that half the students scored below 82%. Scores range from 45% to 98%, with an IQR of 12 percentage points, meaning the middle 50% of students scored within a 12-point range. The outlier at 45% is notably below the main cluster of scores between 70% and 98%."

Common Patterns and Interpretations

What Shape Tells Us

Symmetric:

• Process or measurement is balanced
• Natural variation around center
• Use mean and standard deviation

Right-skewed:

• Floor effect (minimum limit)
• Most values small, few very large
• Use median and IQR

Left-skewed:

• Ceiling effect (maximum limit)
• Most values large, few very small
• Use median and IQR

Bimodal:

• Two distinct groups mixed together
• Consider separating and analyzing separately

What Outliers Tell Us

Potential meanings:

• Errors (investigate and possibly correct)
• Unusual but legitimate cases
• Different population mixed in
• Rare but important events

Impact:

• Affect mean more than median
• Affect standard deviation more than IQR
• Can change conclusions if not addressed

What Spread Tells Us

Large spread:

• High variability
• Data quite different from typical value
• Less predictability

Small spread:

• Low variability
• Data close to typical value
• More consistency, predictability

Comparing Distributions

When comparing two or more distributions:

Address each of SOCS:

Shape:

• "Group A is symmetric while Group B is right-skewed"

Outliers:

• "Both groups have outliers, but Group A's are more extreme"

Center:

• "Group A has a higher median (75) than Group B (68)"

Spread:

• "Group A shows more variability (SD = 12) than Group B (SD = 8)"

Example comparison:

"Both male and female height distributions are roughly symmetric and unimodal. Males have a higher mean height (70 inches) compared to females (64 inches), a difference of 6 inches. Both distributions have similar spreads, with standard deviations of approximately 3 inches. Neither distribution shows outliers."

Common Mistakes

❌ Confusing skewness direction (it's the tail, not the peak!)
❌ Using mean with skewed data (median is more appropriate)
❌ Reporting center without spread (both are needed!)
❌ Ignoring units (always include them)
❌ Incomplete descriptions (use full SOCS framework)
❌ Not describing in context (relate to actual situation)
❌ Confusing SD and IQR (they measure spread differently)

Quick Reference

SOCS Framework:

• Shape: Symmetric? Skewed (which direction)? Unimodal/bimodal?
• Outliers: Present? Where? How many?
• Center: Mean or median (with units!)
• Spread: Range, IQR, or SD (with units!)

Mean vs. Median:

• Symmetric, no outliers → Use mean
• Skewed or outliers → Use median

SD vs. IQR:

• Symmetric, no outliers → Use SD
• Skewed or outliers → Use IQR

Skewness:

• Right-skewed: Mean > Median, tail to right
• Left-skewed: Mean < Median, tail to left
• Symmetric: Mean ≈ Median

Remember: A complete description tells the story of the data. Don't just report numbers — interpret them in context and explain what they mean!