Data Spread: Center, Range, IQR, and MAD
Drag data points and watch how an outlier pulls the mean but barely nudges the median.
What makes a question "statistical"?
Not every question is a statistical question. 'How old is the school principal?' has one fixed answer — there's no variability, so it isn't statistical. But 'How many minutes did each 6th grader spend on homework last night?' anticipates variability: different students will report different numbers.
A statistical question is one where you expect the answers to vary — and that variability is exactly what you're trying to describe and measure.
Describing a distribution: center, spread, shape
Once you collect the answers to a statistical question, you have a distribution — the natural shape of a set of data. Every distribution can be described by three things:
- Center — a typical or representative value (mean or median).
- Spread — how much the values vary (range, IQR, or MAD).
- Shape — roughly symmetric, skewed to one side, or with outliers?
Reporting just one number ('the average is 12') is never the whole story. Spread tells you how much to trust that single number as 'typical.'
Mean — add every value and divide by the count. It uses every data point, but a single extreme value can pull it a long way.
Median — the middle value once the data is ordered (average the two middle values if the count is even). It only depends on position, so an outlier barely moves it.
Rule of thumb: in a roughly symmetric distribution with no big outliers, mean and median land close together. When the data is skewed or has an outlier, they pull apart — and the median is usually the more honest 'typical' value.
Three ways to measure spread
| Measure | How to compute it | Sensitivity to outliers |
|---|---|---|
| Range | max − min | Very high — uses only the 2 extreme points |
| IQR | Q3 − Q1 (spread of the middle 50%) | Low — ignores the extremes entirely |
| MAD | average of |value − mean| | Moderate — uses every point, but less swayed than the mean/range |
To find IQR: order the data, locate the median (Q2), then find the median of the lower half (Q1) and the median of the upper half (Q3). IQR = Q3 − Q1.
To find MAD: compute the mean, then find how far each value sits from that mean (dropping the sign), and average those distances.
Mean and MAD travel together — both use every value in their calculation, so they work best for roughly symmetric data. Median and IQR travel together — both are based on position in the ordered list, so they hold up when data is skewed or has outliers.
The display matters too (6.SP.4): a dot plot is great for small data sets, a histogram groups values into bars for larger sets, and a box plot shows the five-number summary (min, Q1, median, Q3, max) at a glance — perfect for comparing spread and skew quickly.
Range is easy to compute, but it depends entirely on the smallest and largest values — one unusual measurement can make a tight, consistent data set look wildly spread out. A class where everyone scored 80–90 except for one 20 has a huge range (70), even though nearly every student's score was close together.
Whenever an outlier is present, report IQR or MAD alongside (or instead of) range — they describe the spread of the typical data, not just its extremes.
- Mean: sum = 8+10+10+11+12+40 = 91; 91 ÷ 6 ≈ 15.2.
- Median: the data is already ordered. With 6 values, the median is the average of the 3rd and 4th: (10+11)/2 = 10.5.
- The mean (≈15.2) is pulled upward by the outlier of 40, while the median (10.5) stays close to where most of the data actually sits.
- Range = max − min = 20 − 3 = 17.
- The median (Q2) is the 5th value: 9.
- Lower half (3, 5, 6, 8) → Q1 = (5+6)/2 = 5.5.
- Upper half (11, 14, 16, 20) → Q3 = (14+16)/2 = 15.
- IQR = Q3 − Q1 = 15 − 5.5 = 9.5.
- Mean = (4+6+8+10+12) ÷ 5 = 40 ÷ 5 = 8.
- Absolute deviations from the mean: |4−8|=4, |6−8|=2, |8−8|=0, |10−8|=2, |12−8|=4.
- Average the deviations: (4+2+0+2+4) ÷ 5 = 12 ÷ 5 = 2.4.
Check your understanding
- A statistical question anticipates variability in the answers — 'how tall is Maria' isn't one, but 'how tall are the students in 6th grade' is.
- A distribution is described by its center, its spread, and its overall shape.
- Mean and median both measure center, but the median is far more resistant to outliers.
- Range (max − min), IQR (Q3 − Q1), and MAD (average distance from the mean) all measure spread — IQR and MAD are more informative than range alone.
- Pair mean with MAD for roughly symmetric data; pair median with IQR when the data is skewed or has outliers.