📊 Pre-Lab: Descriptive Statistics

Master the fundamentals before your lab session

MSTA112 Medical Biostatistics — Week 4

🎯 Why This Matters

Imagine a clinic where the mean HbA1c is 7.5% — looks controlled. But half the patients are at 5.5% (great) and half at 9.5% (danger). The average hid the problem.

📌 The center is only half the story. You must also measure the spread.

📈 Central Tendency — Finding the Center

Mean (Average)

Mean = Σx / n

Sum all values, divide by count. Uses every data point — powerful but easily pulled by outliers.

✅ Use when: data is Normal
⚠️ Sensitive to outliers

Median (Middle Value)

Middle value when ordered

The 50th percentile. Not affected by extreme values. Represents the "typical" patient.

✅ Use when: data is Skewed

Mode (Most Frequent)

Value with highest frequency

The only measure that works for categorical data (blood type, gender).

✅ Use when: data is Categorical

🔀 Distribution Shapes

The relationship between mean and median reveals the shape:

Normal (Symmetric)

Mean = Median = Mode

Bell-shaped. All measures align at center. Example: height in a population.

Positive Skew (Right Tail)

Mode < Median < Mean

Mean pulled right by high values. Example: income, BMI, length of stay.

Negative Skew (Left Tail)

Mean < Median < Mode

Mean pulled left by low values. Example: easy exam scores.

💡 Shortcut: Skew is named for the TAIL direction, not the peak. If mean ≠ median → data is skewed.

📏 Dispersion — Measuring the Spread

Range

Range = Max − Min

Quick snapshot but uses only 2 values. One outlier ruins it.

Use for: quick glance only

IQR (Interquartile Range)

IQR = Q3 − Q1

Spread of the middle 50%. Robust to outliers — like the median.

✅ Use when: data is Skewed

Standard Deviation (SD)

SD = √[Σ(x−x̄)² / (n−1)]

Average distance from the mean. Foundation of the 68-95-99.7 rule.

✅ Use when: data is Normal

Coefficient of Variation (CV)

CV = (SD / Mean) × 100%

Compares variability across different units. Essential for lab quality control.

Use for: comparing different measurements

🔗 The Golden Pairing Rules

These measures are always reported together. Mixing them is a mistake.

Mean + SD

✅ For Normal Data

Median + IQR

✅ For Skewed Data

Mean + IQR

❌ Wrong combination

Median + SD

❌ Wrong combination

📊 The 68-95-99.7 Rule

For normally distributed data:

68%

of data falls within Mean ± 1 SD — the typical range

95%

of data falls within Mean ± 2 SD — the "normal" reference range

99.7%

of data falls within Mean ± 3 SD — beyond this → investigate

Example: Mean BP = 130 mmHg, SD = 15. Normal range (±2 SD) = 100–160 mmHg.

🔍 Detecting Outliers

IQR Method (Box Plot)

Below Q1 − 1.5×IQR
Above Q3 + 1.5×IQR

Values outside the fences are flagged as outliers on box plots.

SD Method

Beyond Mean ± 3 SD

Only 0.3% of normal data falls here. Very rare = likely unusual.

⚠️ Never delete outliers without investigation!

A cholesterol of 15 mmol/L could be a lab error OR familial hypercholesterolemia — a real disease. Investigate first. Always.

🌳 Decision Tree: Which Measure Should I Use?

❓ Is your data categorical (blood type, gender, diagnosis)?

→ Use Mode. It's the only measure that works for categories.

❓ Is your data numeric but skewed or has outliers?

→ Use Median + IQR. They resist extreme values.

❓ Is your data numeric and normally distributed?

→ Use Mean + SD. Maximum information, mathematically powerful.

🧮 Step-by-Step Practice

Work through these problems before checking answers. Grab a pen!

Problem 1: Saudi BMI Data

BMI values (kg/m²) for 5 patients: 26, 28, 29, 31, 33

Order the data
Already ordered: 26, 28, 29, 31, 33

Mean = (26+28+29+31+33)/5 = 147/5 = 29.4 kg/m²
Falls in "Overweight" (WHO: 25–29.9)

Median = 3rd value = 29
Mean (29.4) ≈ Median (29) → approximately symmetric ✓

Mode = No mode (all values appear once)

Range = 33 − 26 = 7

Problem 2: Hospital Length of Stay

Days in hospital: 2, 3, 3, 4, 150

⚠️ Notice the outlier: 150 days

Mean = (2+3+3+4+150)/5 = 32.4 days ← misleading!

Median = 3rd value = 3 days ← represents the typical patient

Mode = 3 (appears twice)

💡

Conclusion: Mean (32.4) >> Median (3) → Highly right-skewed
→ Report as: Median = 3 days (IQR), NOT Mean ± SD

🧩 Problem 3: Your Turn!

Patient group: 4, 4, 7, 2, 8

Try calculating mean, median, and mode, then check your answer:

📊 Outlier Impact Simulator

Drag the slider to add an outlier to this BMI dataset. Watch how Mean shifts dramatically while Median barely moves.

Mean

Median

Mean

Median

Mode

Range

IQR

Outlier BMI: 30 kg/m²

20Normal range200

🎯 Drag & Match

Drag concepts on the left to their correct match on the right.

Match each concept → correct statistical measure or description. Green = correct!

Score: 0/5

📈 Distribution Shape Explorer

See how Mean, Median, and Mode shift across different distribution shapes.

Mean

Median

Mode

Normal distribution: Mean = Median = Mode at the center. Common for height, BP, and HbA1c in healthy populations.

🎮 Simulate tab features contributed by Mistral Large — integrated by Endoglin 🧬

📊 Descriptive Statistics Calculator

Enter your data and get instant analysis with visualizations and interpretation.

💡 Try example datasets:

📈 Central Tendency

n (Sample Size)-

Mean-

Median-

Mode-

📏 Dispersion

Range-

IQR-

Std Deviation (SD)-

CV-

🎯 Five-Number Summary

Minimum-

Q1 (25th)-

Q3 (75th)-

Maximum-

📊 Pre-Lab: Descriptive Statistics

🎯 Why This Matters

📈 Central Tendency — Finding the Center

Mean (Average)

Median (Middle Value)

Mode (Most Frequent)

🔀 Distribution Shapes

Normal (Symmetric)

Positive Skew (Right Tail)

Negative Skew (Left Tail)

📏 Dispersion — Measuring the Spread

Range

IQR (Interquartile Range)

Standard Deviation (SD)

Coefficient of Variation (CV)

🔗 The Golden Pairing Rules

📊 The 68-95-99.7 Rule

🔍 Detecting Outliers

IQR Method (Box Plot)

SD Method

🌳 Decision Tree: Which Measure Should I Use?

🧮 Step-by-Step Practice

Problem 1: Saudi BMI Data

Problem 2: Hospital Length of Stay

🧩 Problem 3: Your Turn!

📊 Outlier Impact Simulator

🎯 Drag & Match

📈 Distribution Shape Explorer

📊 Descriptive Statistics Calculator

📈 Central Tendency

📏 Dispersion

🎯 Five-Number Summary

📊 Distribution

🔍 Outlier Detection

✅ Pre-Lab Quiz