Module 03: Spread - Statistics Visualizer

Range

Range: Max - Min

Min: 12

Max: 89

Range = 89 - 12 = 77

Range is the simplest measure of spread: maximum value minus minimum value. It tells you the total span of your data—how far apart the extremes are.

Formula: Range = Max - Min

Sensitivity to Outliers

Data: 10, 12, 14, 15, 18, 20

Range = 20 - 10 = 10

Data: 10, 12, 14, 15, 18, 20, 150

Range = 150 - 10 = 140

Range is extremely sensitive to outliers. One extreme value completely changes the range, making it unreliable for data with outliers. It uses only 2 of n data points.

One outlier made range 14× larger!

Appropriate Uses

✓ Good For

• Quick assessment
• Quality control limits
• Small, clean datasets
• When extremes matter

✗ Not Good For

• Data with outliers
• Comparing groups
• Inferential statistics
• Large datasets

Range is useful for quick checks and when you specifically care about extremes (temperature ranges, stock price swings). But for serious analysis, use IQR or standard deviation.

Range: simple but limited

Interquartile Range (IQR)

Understanding Quartiles

25%

MinQ1Q2 (Median)Q3Max

Quartiles divide sorted data into four equal parts. Q1 is the 25th percentile, Q2 (median) is the 50th, and Q3 is the 75th. Each quartile contains 25% of the data.

Q1, Q2, Q3 split data into fourths

The Interquartile Range

Lower 25%

IQR (Middle 50%)

Upper 25%

IQR = Q3 - Q1

IQR measures the spread of the middle 50% of data. By ignoring the bottom and top 25%, it's immune to outliers. It's like range, but focused on the 'core' of your data.

IQR = Q3 - Q1

Step-by-Step Example

Data (sorted): 2, 4, 5, 7, 8, 10, 12, 15, 18

245 ←Q178 ←Q21012 ←Q31518

IQR = 12 - 5 = 7

1. Sort the data. 2. Find Q1 (median of lower half): 5. 3. Find Q3 (median of upper half): 12. 4. IQR = Q3 - Q1 = 12 - 5 = 7. The middle 50% spans 7 units.

IQR represents spread of middle 50%

Detecting Outliers with IQR

Outlier Rule

Value is outlier if:

Below

Q1 - 1.5 × IQR

Above

Q3 + 1.5 × IQR

Example: Q1=5, Q3=12, IQR=7
Lower fence: 5 - 1.5(7) = -5.5
Upper fence: 12 + 1.5(7) = 22.5
Values below -5.5 or above 22.5 are outliers

The 1.5×IQR rule identifies outliers. Calculate fences at Q1 - 1.5×IQR and Q3 + 1.5×IQR. Any value beyond these fences is considered an outlier. This is what box plots use.

1.5×IQR rule is standard for outlier detection

Variance

Average Squared Deviation

Data: 2, 4, 4, 4, 5, 5, 7, 9 Mean = 5

(2-5)²=9

(4-5)²=1

(5-5)²=0

(7-5)²=4

(9-5)²=16

Variance measures how far values spread from the mean. For each value, calculate the squared difference from the mean. Then average these squared differences. Squaring ensures all deviations are positive.

Why square? Makes all deviations positive

Population vs Sample Variance

Population (σ²)

σ² = Σ(x-μ)² / N

Divide by N (total count)

Sample (s²)

s² = Σ(x-x̄)² / (n-1)

Divide by n-1 (Bessel's correction)

Population variance divides by N. Sample variance divides by n-1 (Bessel's correction) to give an unbiased estimate of population variance. In practice, almost always use n-1.

n-1 corrects for sample bias

Step-by-Step Example

Data: 2, 4, 4, 4, 5, 5, 7, 9 (n=8)

Step 1: Mean = 40/8 = 5

Step 2: Squared deviations:
(2-5)²=9, (4-5)²=1, (4-5)²=1, (4-5)²=1,
(5-5)²=0, (5-5)²=0, (7-5)²=4, (9-5)²=16

Step 3: Sum = 9+1+1+1+0+0+4+16 = 32

Sample Variance: s² = 32/(8-1) = 32/7 = 4.57

1. Calculate the mean. 2. Subtract mean from each value and square. 3. Sum all squared deviations. 4. Divide by n-1 for sample variance. The result (4.57) is in squared units.

Variance units are squared (e.g., cm²)

Problems with Variance

Units Problem

If data is in cm, variance is in cm².

Hard to interpret: 'average 4.57 cm² from mean' means what?

Scale Problem

Variance can be huge.

Heights in mm: variance might be 10,000+, making comparison difficult.

Solution: Take the square root → Standard Deviation

Variance is hard to interpret because it's in squared units. If heights are in centimeters, variance is in square centimeters. To return to original units, we take the square root.

This is why we use Standard Deviation

Standard Deviation

Square Root of Variance

Standard Deviation = √Variance

Variance

4.57

cm²

→

Std Dev

2.14

Standard deviation (SD or σ) is the square root of variance. It returns the spread measure to original units, making it directly interpretable: 'typical deviation is about 2.14 cm from the mean.'

SD = √Variance

What SD Tells You

-2σ-1σMean+1σ+2σ

Low SD = tightly clustered | High SD = widely spread

SD describes the typical distance from the mean. Small SD means data clusters tightly around the mean. Large SD means data is spread widely. It's the most commonly used spread measure.

SD is in the same units as your data

The Empirical Rule

■ 68% (±1σ)■ 95% (±2σ)■ 99.7% (±3σ)

For normal distributions: ~68% of data falls within 1 SD of the mean, ~95% within 2 SDs, and ~99.7% within 3 SDs. This 68-95-99.7 rule helps interpret SD in context.

Only applies to normal distributions

Comparing Distributions

Group A: Mean = 100, SD = 5

Group B: Mean = 100, SD = 15

Same mean (100), different SDs. Group A (SD=5) is highly consistent—most values near 100. Group B (SD=15) is more variable—values spread further from 100. SD quantifies this difference.

SD lets you compare variability across groups

Choosing a Spread Measure

All Measures at a Glance

Measure	Formula	Robust?	Best For
Range	Max - Min	No	Quick check
IQR	Q3 - Q1	Yes	Skewed data
Variance	Σ(x-x̄)²/(n-1)	No	Calculations
Std Dev	√Variance	No	General use

Range is simplest but least reliable. IQR is robust for skewed data. Variance is used in formulas. Standard deviation is the go-to measure for most purposes.

SD is most commonly used; IQR for outliers

Match Center with Spread

Symmetric Data

Center: Mean

Spread: Standard Deviation

Skewed / Outliers

Center: Median

Spread: IQR

Always pair center and spread measures: Mean goes with SD (both affected by outliers). Median goes with IQR (both robust). Mixing them (mean with IQR) makes less sense.

Mean + SD or Median + IQR

Which Measure to Choose?

Is data skewed or has outliers?

No → Use SD

Report: Mean ± SD

Yes → Use IQR

Report: Median (IQR)

Simple decision: Check for skewness/outliers. If data is symmetric and well-behaved, use SD. If skewed or outliers exist, use IQR. When in doubt, report both.

Report 'Mean ± SD' or 'Median (IQR)'

Interactive Spread Calculator

101214151820

Click a number to remove it

Mean14.83

Median14.50

Variance11.47

Std Dev3.39

Range10

IQR6.00

Count6

Add your own data to see all spread measures calculated live! Try adding an outlier (like 100) to see how Range and SD change dramatically while IQR stays relatively stable. This demonstrates why IQR is 'robust'.

Try adding an extreme value to see robustness in action

★

Real-World Examples

Apply your knowledge to practical scenarios

Manufacturing Quality Control

A factory produces steel rods with a target length of 100mm. Two machines produce rods with the same mean length (100mm), but Machine A has SD = 0.5mm and Machine B has SD = 2.0mm. Quality specifications require rods between 98mm and 102mm.

Sample Data

{
  "machineA": {
    "mean": 100,
    "sd": 0.5,
    "range": "98.5-101.5mm typical"
  },
  "machineB": {
    "mean": 100,
    "sd": 2,
    "range": "96-104mm typical"
  }
}

Q Using the 68-95-99.7 rule, what percentage of Machine A's rods fall within specification? +

A Approximately 99.99%. The specs (98-102mm) are ±4 SDs from the mean for Machine A (4/0.5 = 8 SDs), which includes virtually all production.

Q What percentage of Machine B's rods are likely out of specification? +

A About 32%. The specs are ±1 SD from the mean for Machine B (2/2 = 1 SD). Only 68% fall within ±1 SD, so ~32% are out of spec.

Q If defective rods cost $10 each to scrap and you produce 10,000 rods/day, what's the daily cost difference? +

A Machine A: ~0 defects = ~$0. Machine B: ~3,200 defects = ~$32,000/day. The SD difference creates massive cost implications!

Key Insights

Same mean, vastly different outcomes—spread matters enormously in quality control
SD directly translates to defect rates and costs
Six Sigma methodology focuses on reducing SD to 1/6 of tolerance

Salary Negotiation Intelligence

You're negotiating a salary for a data analyst position. You've gathered salary data: Mean = $75,000, Median = $68,000, SD = $15,000, IQR = $18,000. The range is $45,000 to $150,000.

Q What does the difference between mean ($75K) and median ($68K) tell you? +

A The distribution is right-skewed (mean > median). A few high earners pull up the average. Most analysts earn less than the 'average' salary.

Q Which measure should you use as your target: mean or median? +

A Median ($68K) better represents 'typical'—you have a 50% chance of getting at least this. Mean ($75K) is inflated by outliers.

Q The range is huge ($45K-$150K). Why is IQR ($18K) more useful? +

A IQR shows the middle 50% spans just $18K, indicating most salaries cluster in a narrow band. The extreme range is driven by outliers (interns vs. senior leads).

Q If offered $60K, how does this compare statistically? +

A Q1 would be around $68K - $9K = $59K. So $60K is roughly at the 25th percentile—below average. You have leverage to negotiate up.

Key Insights

Median + IQR are better for skewed salary data than mean + SD
Understanding spread helps you gauge where an offer falls
Large range with small IQR indicates outlier-driven extremes

The Suspicious Test Scores

A teacher analyzes test scores: 72, 75, 78, 80, 82, 85, 88, 90, 92, 98. One student scored 45 on the same test. Should this be flagged as an outlier?

Sample Data

{
  "originalScores": [
    72,
    75,
    78,
    80,
    82,
    85,
    88,
    90,
    92,
    98
  ],
  "suspiciousScore": 45
}

Q Calculate Q1, Q3, and IQR for the original 10 scores. +

A Q1 = 77 (between 75 and 78), Q3 = 91 (between 90 and 92), IQR = 91 - 77 = 14

Q What are the outlier fences using the 1.5×IQR rule? +

A Lower fence: 77 - 1.5(14) = 77 - 21 = 56. Upper fence: 91 + 1.5(14) = 91 + 21 = 112.

Q Is the score of 45 a statistical outlier? +

A Yes! 45 < 56 (lower fence), so it's officially an outlier by the 1.5×IQR rule.

Q What should the teacher do with this information? +

A Investigate—don't automatically remove! Was the student sick? Did they miss material? Is there a data entry error? Outliers require context.

Key Insights

IQR method provides objective outlier detection
Statistical outliers need human judgment—detection ≠ deletion
Outliers may be the most important data points (fraud, errors, special cases)

← Central Tendency Next: Distributions →

Range

Range: Max - Min

Sensitivity to Outliers

Appropriate Uses

Interquartile Range (IQR)

Understanding Quartiles

The Interquartile Range

Step-by-Step Example

Detecting Outliers with IQR

Variance

Average Squared Deviation

Population vs Sample Variance

Step-by-Step Example

Problems with Variance

Standard Deviation

Square Root of Variance

What SD Tells You

The Empirical Rule

Comparing Distributions

Choosing a Spread Measure

All Measures at a Glance

Match Center with Spread

Which Measure to Choose?

Interactive Spread Calculator

Real-World Examples

Manufacturing Quality Control

Sample Data

Key Insights

Salary Negotiation Intelligence

Key Insights

The Suspicious Test Scores

Sample Data

Key Insights

Spread Measures Quiz

Quiz Results