A box-and-whisker plot floating above a number line, with measuring rulers showing the spread of a data set.
Statistics · 6.SP

Lesson 6.3 — Range, IQR & MAD (Spread)

An average tells you the "middle" — but spread tells you how stretched out or bunched up the data really is.

🎙️ Narration script

Hi again! Last time we found the center of our data. Today we're looking at something just as important: the spread. Spread tells you how stretched out or bunched up your data really is.

Here's why it matters. Two data sets can have the exact same average and still look totally different. One might be tightly clustered, and the other all over the place. Think of two cities that both average seventy degrees. One ranges from sixty-five to seventy-five, nice and mild. The other swings from forty to a hundred. Same average, wildly different spread!

We've got three ways to measure spread. The first is the range. That's just the biggest value minus the smallest value, max minus min. It tells you the full width of your data.

The second is the interquartile range, or IQR. To find it, you sort the data, find the median, then find Q1, the median of the lower half, and Q3, the median of the upper half. The IQR is Q3 minus Q1. It measures the width of the middle fifty percent of your data. The nice thing about the IQR is that it ignores the extremes, so one weird outlier won't blow it up.

The third is the mean absolute deviation, or MAD. That's the average distance each value sits from the mean.

Let's try the MAD with two, four, six, eight, ten. The mean is six. Now find each value's distance from six: four, two, zero, two, four, always positive. Add those up to get twelve. Divide by five values, and the MAD is two point four. So on average, each value sits two point four away from the mean.

Quick recap. Range is max minus min. IQR is the middle-half width, steadier against outliers. MAD is the average distance from the mean. And always sort first!

1 Core idea

Two data sets can have the same average and still look totally different — one tightly clustered, the other all over the place. Spread (also called variability) measures how far apart the values are. We have three ways to measure it: the range (full width), the IQR (width of the middle half), and the MAD (the average distance from the mean). Bigger spread = data is more scattered.

🧩 Think of it like… spread is the spacing of runners in a race. The center (average) tells you roughly where the pack is; spread tells you whether they're bunched in a tight clump or strung out across the whole track — two races can finish at the same average time yet look completely different.
Where it breaks: the range is a gap you can literally see (last runner minus first), but the MAD isn't the distance between any two specific runners — it's the average gap from the pack's balance point, so you have to compute it rather than just eyeball it.

2 Key terms

Range
Biggest value minus smallest value: max − min.
Median
The middle value when the data is sorted in order.
Quartiles (Q1, Q3)
Q1 is the median of the lower half; Q3 is the median of the upper half. They split the data into four parts.
IQR (Interquartile Range)
The spread of the middle 50% of the data: Q3 − Q1.
MAD (Mean Absolute Deviation)
The average distance each value sits from the mean.

3 Real-life examples

  • Weather: Two cities both average 70°F, but one ranges 65–75 (small spread) and one ranges 40–100 (huge spread).
  • Test scores: A class with scores all near 85 has a small MAD; a class with scores from 50–100 has a large MAD.
  • Heights: The IQR of the soccer team's heights tells you how spread out the "typical middle" players are.
  • Allowance: If most kids get $10 but one gets $200, the range jumps a lot — IQR barely moves.
🤔 Pause & think: In the allowance example, most kids get $10 but one gets $200. Why does the range explode while the IQR barely moves?
Reveal the thinking
Range = max − min uses only the two most extreme values, so that lone $200 instantly becomes the max and blows the range wide open. IQR = Q3 − Q1 looks only at the middle half and throws away the outer quarters — the $200 sits out in the discarded top quarter and never enters the calculation, so it stays steady.

4 Common doubts

What's the difference between range and IQR?

Range uses the two most extreme values, so one weird outlier can blow it up. IQR ignores the outer quarters and measures just the middle half, so it's steadier.

Do I sort the data first?

Yes — always put values in order before finding the median, quartiles, or range.

Is the MAD ever negative?

No. We use absolute distances (always positive), so the smallest MAD possible is 0 — which means every value equals the mean.

Why divide by the number of values in MAD?

Because MAD is an average distance — you add up all the distances, then split evenly among all the data points.

5 Step-by-step

  1. Sort the data from least to greatest.
  2. Range = max − min.
  3. Find the median (middle value), then Q1 (median of lower half) and Q3 (median of upper half).
  4. IQR = Q3 − Q1.
  5. MAD: find the mean, take each value's distance from the mean (always positive), add them, then divide by the count.

📊 See it · box-and-whisker plot

Minutes of exercise over 11 days, sorted: 3, 5, 7, 8, 10, 12, 14, 15, 18, 20, 22

0 4 8 12 16 20 24 min 3 Q1 7 median 12 Q3 18 max 22 range = 22 − 3 = 19

Now read the spread straight off the plot:

Range = max − min = 22 − 3 = 19 → full width of the data
IQR = Q3 − Q1 = 18 − 7 = 11 → width of the middle 50% (the box)

A quick MAD example with a smaller set 2, 4, 6, 8, 10 (mean = 30 ÷ 5 = 6):

distances from 6: 4, 2, 0, 2, 4 → |2−6|, |4−6|, … always positive
sum = 4 + 2 + 0 + 2 + 4 = 12 → add the distances
MAD = 12 ÷ 5 = 2.4 → on average, each value is 2.4 from the mean
✅ Check yourself
  1. For 4, 6, 8, 10, 12, find the range and the MAD.
    answer Range = 12 − 4 = 8. Mean = 40 ÷ 5 = 8; distances are 4, 2, 0, 2, 4 (sum 12), so MAD = 12 ÷ 5 = 2.4.
  2. A data set has IQR = 0 but range = 20. What does that tell you about the data?
    answer The middle 50% are all identical (Q1 = Q3), so the bulk of the data is bunched at one value — yet the min and max still sit 20 apart, meaning a few extreme values stretch the edges.
⚡ Quick recap. Spread shows how scattered data is. Range = max − min (sensitive to outliers). IQR = Q3 − Q1, the middle-50% width (steadier). MAD = the average distance from the mean. Always sort first, and remember distances in MAD are always positive.

Grounded in CA CCSS-M, Grade 6 · 6.SP (summarizing & describing distributions), California Department of Education. Hero image generated with Gemini Nano Banana Pro.