Chapter 59: Statistic Variability (Spread)

Statistic Variability (also called spread, dispersion, or variation)

This is not a fancy or difficult concept — it’s actually very down-to-earth. It simply answers the question:

“How much do the values in my data differ from each other (or from the average)?”

If central tendency (mean/median) tells you “where the data is centered”, then variability tells you “how spread out or clustered the data is around that center”.

Let’s go through it slowly, with lots of everyday Hyderabad examples, pictures you can imagine, real numbers you can check yourself, and stories so it feels natural and useful — not like a textbook.

Step 1: Why Variability Matters More Than Most People Think

Imagine two tiffin centres near your home in Kukatpally:

• Tiffin Centre A Delivery time last 10 orders: 32, 33, 31, 34, 32, 33, 32, 31, 33, 32 minutes Average = 32.3 min
• Tiffin Centre B Delivery time last 10 orders: 18, 45, 22, 38, 29, 50, 15, 41, 27, 35 minutes Average = 32.0 min

Both have almost the same average delivery time (~32 minutes). But which one would you trust more?

Centre A: almost always 31–34 min → very predictable Centre B: anywhere from 15 to 50 min → very unpredictable

Variability is the difference between these two centres. It tells you consistency, risk, reliability — things the average alone can never show.

In real life we care about variability at least as much as we care about the average.

Step 2: The Most Common Ways to Measure Variability

Here are the four tools everyone uses — from simplest to most useful.

1. Range (easiest, but weakest)

   Range = maximum value – minimum value

   • Tiffin A: 34 – 31 = 3 minutes
   • Tiffin B: 50 – 15 = 35 minutes

   Range is simple, but one crazy late delivery (rain, traffic jam) can make it look terrible even if 99% of orders are fine.

2. Interquartile Range (IQR) (robust, ignores extremes)

   IQR = Q3 – Q1 (Q1 = 25th percentile, Q3 = 75th percentile)

   Middle 50% of the data — ignores the lowest 25% and highest 25%.

   • Tiffin A: very tight IQR (almost all values between 31–33)
   • Tiffin B: wide IQR (middle 50% spread over ~20–40 minutes)

   IQR is much better when there are outliers (sudden 120-minute delivery because rider got lost).

3. Variance (average squared distance from mean)

   Steps:

   • Find mean
   • Subtract mean from each value → get deviations
   • Square every deviation (so negative & positive don’t cancel)
   • Average those squared deviations → that’s variance

   Problem: variance is in squared units (minutes²) — hard to interpret.

4. Standard Deviation (√variance) — the most useful one

   • Same idea as variance, but square root → back to original units (minutes)
   • Most people use standard deviation when they talk about “typical variation”

   Rule of thumb (for roughly bell-shaped data):

   • ≈68% of values lie within ±1 standard deviation of mean
   • ≈95% within ±2 SD
   • ≈99.7% within ±3 SD

Step 3: Real Hyderabad Example – Delivery Times (Full Calculation)

Let’s take Tiffin Centre A’s 10 delivery times again:

28, 35, 42, 31, 27, 39, 45, 33, 29, 38

Step-by-step descriptive statistics

1. Sorted: 27, 28, 29, 31, 33, 35, 38, 39, 42, 45
2. Mean = sum ÷ 10 = 347 ÷ 10 = 34.7 minutes
3. Median = average of 5th & 6th = (33 + 35) ÷ 2 = 34 minutes
4. Range = 45 – 27 = 18 minutes
5. Quartiles Q1 (25th percentile) ≈ 29 Q3 (75th percentile) ≈ 39 IQR = 39 – 29 = 10 minutes
6. Variance (sample variance) Deviations from mean: 27–34.7 = –7.7 → (–7.7)² = 59.29 28–34.7 = –6.7 → 44.89 … (you square all 10, sum = 356.1, divide by n–1 = 9) Variance ≈ 39.57 minutes²
7. Standard Deviation = √39.57 ≈ 6.29 minutes

Interpretation: Most deliveries are within about 34.7 ± 6.3 minutes (28–41 min range for ~68% of orders).

That’s descriptive statistics telling the full story — not just the average.

Step 4: Quick Summary Table (Copy This!)

Final Teacher Words

Descriptive statistics of variability answers the question:

“How consistent or scattered is this data?”

It tells you whether you can trust the average:

• Low variability → average is very reliable (Tiffin A)
• High variability → average hides big differences (Tiffin B)

In Hyderabad 2026 you meet variability every day:

• “Delivery usually 30–40 min” (low spread → trust Swiggy time)
• “House rent in Kukatpally ₹15,000–₹50,000” (high spread → median more useful than mean)
• “My last 10 UPI spends ₹200–₹5,000” (huge spread → watch for fraud)

Variability is not noise — it’s information. It tells you risk, reliability, consistency, fairness, safety.

So next time someone only tells you the average (“average salary ₹80,000”), always ask the statistics question:

“What’s the spread? What’s the standard deviation? Are there big outliers?”

Because the average alone can lie — but variability tells the truth.

Understood the heart of variability now? 🌟

Want to go deeper?

• How to calculate standard deviation step-by-step (small numbers)?
• Real box-plot of Hyderabad flat rents (skewed + outliers)?
• Why standard deviation matters more than range?
• Difference between sample vs population standard deviation?

Just tell me — next class is ready! 🚀