Which of the following is a statistical question?

How many minutes do the bus rides of the students in this program take?. A statistical question is one that anticipates variability in the answers. Asking about the bus rides of many students produces a set of different values that form a distribution, so it is statistical. The other three questions each have exactly one answer — one ride this morning, one posted limit, one bag — and therefore anticipate no variability. The plural subject is usually the signal, but the defining test is whether the answers can differ.

A quality inspector records the following defect counts over five shifts: 3, 5, 9, 11, 12. What is the mean absolute deviation?

3.2 defects. First compute the mean: (3 + 5 + 9 + 11 + 12) ÷ 5 = 40 ÷ 5 = 8 defects. Next find each value's absolute distance from 8: 5, 3, 1, 3, and 4. Averaging those distances gives (5 + 3 + 1 + 3 + 4) ÷ 5 = 16 ÷ 5 = 3.2 defects. The distractor 8 is the mean rather than the deviation, and 9 is the range minus nothing meaningful — the range here is 12 - 3 = 9, which measures spread but is not the MAD.

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?

Skewed right; the median best represents a typical salary.. The bulk of the data sits at the lower end with a long tail extending toward the two very high executive salaries, which makes the distribution skewed right — skew is named for the direction of the tail, not the location of the hump. Because those extreme values pull the mean upward to a figure no ordinary employee earns, the median is the resistant and honest measure of a typical salary.

Statistical Questions, Variability, & Descri | Free Guide 2026

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.

TABE Mathematics

10.4 Statistical Questions, Variability, & Describing Distributions

Key Takeaways