10.4 Statistical Questions, Variability, & Describing Distributions
Key Takeaways
- A statistical question is one that anticipates variability in the answers; "How tall am I?" is not statistical, but "How tall are the students in this class?" is.
- Every distribution is described by three things together: center, spread, and shape — reporting a mean without a spread is an incomplete answer.
- Mean absolute deviation measures typical distance from the mean, and a larger MAD means a more spread-out distribution.
- Shape is described as symmetric, skewed right, or skewed left, and the direction of skew is named for the long tail, not the bulk of the data.
Statistical Questions, Variability, & Describing Distributions
Sections 9.3 and 9.4 computed mean, median, mode, range, quartiles, and IQR. This section covers the framing the DRC specification wraps around them: recognizing a statistical question, understanding a data set as a distribution, and summarizing that distribution "in relation to its context."
What Makes a Question Statistical
A statistical question is one that anticipates variability in the answers.
That single criterion decides every item of this type.
| Question | Statistical? | Why |
|---|---|---|
| How old am I? | no | one answer, no variability |
| How old are the students in this program? | yes | answers vary from person to person |
| What did the thermometer read at 8 a.m. today? | no | a single reading |
| What are the daily high temperatures in July? | yes | 31 values that vary |
| How much does this box weigh? | no | one object, one weight |
| How much do the boxes on this pallet weigh? | yes | a distribution of weights |
The tell is usually plural: a group of people, a set of days, a batch of parts. A question about one thing at one time is not statistical, and it does not need a mean.
A Data Set Has a Distribution
A distribution describes how the values are spread across their possible range. Three features describe it, and TABE expects all three:
| Feature | Question it answers | Statistics used |
|---|---|---|
| Center | What is a typical value? | mean, median, mode |
| Spread | How much do values differ? | range, interquartile range, mean absolute deviation |
| Shape | How are values arranged? | symmetric, skewed right, skewed left, gaps, clusters, outliers |
The reporting rule. A center without a spread is misleading. Two crews averaging 40 hours per week are very different if one crew ranges 38–42 and the other ranges 12–68. Always pair the two.
Measuring Spread: Mean Absolute Deviation
Mean absolute deviation (MAD) is the average distance of each value from the mean. Bigger MAD means more spread out.
Procedure:
- Compute the mean.
- Find each value's distance from the mean (always positive — that is what "absolute" means).
- Average those distances.
Worked example. Daily service calls: 8, 10, 12, 14, 16.
| Value | Distance from mean of 12 |
|---|---|
| 8 | 4 |
| 10 | 2 |
| 12 | 0 |
| 14 | 2 |
| 16 | 4 |
Mean $= 60 \div 5 = 12$. MAD $= (4 + 2 + 0 + 2 + 4) \div 5 = 12 \div 5 = \mathbf{2.4}$ calls.
Interpretation: on a typical day, the number of service calls is about 2.4 away from the average of 12.
Compare a second technician: 2, 7, 12, 17, 22. Same mean of 12, but distances are 10, 5, 0, 5, 10, giving MAD $= 30 \div 5 = 6$. Same center, very different consistency — and that is precisely what MAD exists to reveal.
Describing Shape
| Shape | Picture | Mean vs. median | Real-world example |
|---|---|---|---|
| Symmetric | balanced around the middle | mean ≈ median | heights of adults |
| Skewed right (positive) | long tail toward high values | mean > median | household income; a few very high values |
| Skewed left (negative) | long tail toward low values | mean < median | scores on an easy test; a few very low values |
| Bimodal | two separate peaks | neither summarizes well | commute times with two shift start-times |
| Uniform | roughly flat | mean ≈ median | rolls of a fair die |
Naming the skew. The direction is named for the tail, not the hump. A distribution with most values on the left and a few stragglers far to the right is skewed right, even though the data "sits" on the left. This is the most-missed vocabulary item in the topic.
Shape drives which center you report
| Shape | Best measure of center | Reason |
|---|---|---|
| Symmetric, no outliers | mean | uses every value; nothing distorts it |
| Skewed, or outliers present | median | resistant to extreme values |
| Categorical data | mode | the only measure that applies |
A distribution of salaries with one owner earning ten times the staff wage is skewed right; the median is the honest summary, and the mean is the number a recruiter quotes when they want the figure to look larger.
Summarizing a Distribution in Context
Standard 6.SP.5 asks candidates to summarize a data set by reporting the number of observations, what was measured and in what units, the center, the spread, and any striking overall pattern — all with reference to the context.
Full worked summary. Hours of training completed by 11 new hires: 4, 6, 6, 7, 8, 8, 8, 9, 10, 12, 30.
| Element | Value |
|---|---|
| Number of observations | 11 new hires |
| What was measured | training hours completed, in hours |
| Center (median) | 8 hours |
| Center (mean) | $108 \div 11 \approx 9.8$ hours |
| Spread (range) | $30 - 4 = 26$ hours |
| Spread (IQR) | $Q_3 = 10$, $Q_1 = 6$, so IQR $= 4$ hours |
| Shape | skewed right, with a clear outlier at 30 |
| Best summary | median of 8 hours, because the single 30-hour value pulls the mean up to 9.8 |
In context: ten of the eleven new hires completed between 4 and 12 hours of training, clustering around 8 hours, while one hire completed 30 hours — worth investigating separately rather than averaging away.
Reading Variability From a Display
You do not always need to compute. A dot plot or histogram shows spread directly:
- Marks clustered tightly above a few values → small spread, small MAD.
- Marks spread evenly across the whole axis → large spread, large MAD.
- A gap with isolated marks far to one side → an outlier and a skew.
TABE frequently shows two dot plots side by side and asks which data set is more consistent. The answer is the one whose marks are packed into a narrower stretch of the number line — no arithmetic required.
Which of the following is a statistical question?
A quality inspector records the following defect counts over five shifts: 3, 5, 9, 11, 12. What is the mean absolute deviation?
A distribution of annual salaries at a small firm has most values between $38,000 and $52,000, with two executives earning over $300,000. How should this distribution be described, and which measure of center best represents a typical salary?