Introduction to Statistics · Chapter 2 · Descriptive Statistics
Skewness and the Mean, Median, and Mode
A histogram's shape and its three centers are two readings of the same data — naming the shape lets you predict the order of the mean, the median, and the mode before you compute any of them.
bookSHelf · Introduction to Statistics · §2.6 · a self-paced section
Learning objectives — by the end of this section you will be able to
Objectives
- Identify a symmetrical distribution from its histogram or dot plot and explain what the mirror-image test checks §2.6.1
- Describe a distribution as skewed to the left or skewed to the right from the shape of its graph §2.6.2–2.6.3
- Predict the typical ordering of the mean, the median, and the mode for a symmetrical, a left-skewed, and a right-skewed distribution §2.6.4
- Interpret a graph of real data by naming its shape and saying what that shape implies about which measure of the center to quote §2.6.4
§2.6.1 — the mirror-image test
Symmetrical Distribution
Definition 2.6.1 — Symmetrical Distribution
A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left of the line and the shape to the right of the line are mirror images of each other.
Fold the histogram down that line — do the two halves match?
Insight Note — the mean is a balance point
Picture the histogram as blocks on a seesaw.
The mean is where you would put the pivot so it balances. Fold a symmetrical shape down the middle and the weight on each side is equal — so the pivot and the fold line land in the same place.
§2.6.1 — sixteen values, one tall bar
Unimodal Distribution
Definition 2.6.2 — Unimodal Distribution
A distribution is unimodal if it has exactly one mode — a single value (or interval) that occurs more often than any other.
4; 5; 6; 6; 6; 7; 7; 7; 7; 7; 7; 8; 8; 8; 9; 10 — each interval has width one, so a bar's height is just a count.
Figure 2.6.1: the tall bar at 7 is the fold line, and the bars on either side match.
§2.6.1 — dropping the fold line
Seeing the fold line
Definition 2.6.1: drop the fold line and every bar on the left has a partner of the same height the same distance to the right.
For these sixteen values the mean, the median, and the mode are each seven — the fold line is the balance point and the halfway point at once. The data set has a single tallest bar too, so it is unimodal: all three centers land together.
Your turn — build the frequency picture first
Try It Now 2.6.1
Try It Now 2.6.1 — Kyle's tutoring-center quiz
Kyle scores a 5-point quiz for the twelve students who came into the campus tutoring center this week: 1; 2; 2; 3; 3; 3; 3; 4; 4; 4; 5; 5. Is this distribution symmetrical? Support your answer with the mean and the median.
Not symmetrical — mean (3.25) > median (3)
Bar heights: 1, 2, 4, 3, 2. Folding at the tall bar (3) gives 1, 2 on the left but 3, 2 on the right — not mirror images. x=121+(2)(2)+(3)(4)+(4)(3)+(5)(2)=1239=3.25 and M=3. The mean sitting above the median tells you the shape leans slightly toward the high end — the mirror test and the numbers agree.
§2.6.2 — a thin tail toward the low values
Skewed to the Left
Definition 2.6.3 — Skewed to the Left (Negative Skew)
A distribution is skewed to the left (negatively skewed) if it has a longer, thinner tail extending toward the lower values, with the bulk of the data at the higher values.
4; 5; 6; 6; 6; 7; 7; 7; 7; 8 — mean 6.3, median 6.5, mode 7.
Figure 2.6.2: the right side looks chopped off; the tail stretches toward the low values.
Mean (6.3) < median (6.5) < mode (7). The mean has to be the balance point, so the low tail tugs it down; the median only counts positions, so it moves less.
Context Pause — the name points at the tail, not at the pile
The label names the direction the tail stretches — not where the bars are stacked.
Students reliably get this backwards, because the eye is drawn to the tall bars. Tail on the left means skewed left, no matter where the bars are stacked.
Your turn — name the skew, then order the centers
Try It Now 2.6.2
Try It Now 2.6.2 — Jun's homework tally
Jun grades a section of eleven students on how many of 11 assigned homework sets each turned in: 11; 11; 11; 11; 10; 10; 10; 9; 8; 6; 3. Sketch the shape in words, name the skew, and order the mean, the median, and the mode.
Skewed left — mean (9.1) < median (10) < mode (11)
Ordered: 3; 6; 8; 9; 10; 10; 10; 11; 11; 11; 11. The tall bars bunch at the top of the scale and a thin tail of single values runs down to 3, so the tail points left. x=113+6+8+9+(10)(3)+(11)(4)=11100≈9.1; M=10; mode =11 — exactly the ordering the shape predicts.
§2.6.3 — the mirror case, and the common one
Skewed to the Right
Definition 2.6.4 — Skewed to the Right (Positive Skew)
A distribution is skewed to the right (positively skewed) if it has a longer, thinner tail extending toward the higher values, with the bulk of the data at the lower values.
6; 7; 7; 7; 7; 8; 8; 8; 9; 10 — mean 7.7, median 7.5, mode 7.
Figure 2.6.3: the pile sits at the low end; the tail stretches toward the high values.
Right skew is the shape you meet most often in real data: many quantities have a hard floor and no ceiling — nobody earns a negative salary or waits a negative number of minutes — so a handful of large values can always stretch the right tail.
Your turn — predict before you compute
Try It Now 2.6.3
Try It Now 2.6.3 — Marisol's café wait times
Marisol times how many minutes each of nine customers waited for a table: 2; 3; 3; 3; 4; 4; 5; 9; 15. Name the skew and predict, before computing, whether the mean or the median will be larger. Then check.
Skewed right — mean (≈ 5.3) > median (4)
Seven of the nine waits sit between 2 and 5 minutes; two stragglers at 9 and 15 stretch the tail right. x=92+3+3+3+4+4+5+9+15=948≈5.3; M=4 (the 5th ordered value). The single 15-minute wait moves the mean by more than a minute and the median not at all.
§2.6.4 — the headline result of §2.6
The mean is affected by outliers that do not influence the median — so shape predicts order.
Skewed left → mean is often less than the median. Skewed right → mean is often greater than the median. Symmetric → mean and median are approximately equal. An extreme value enters the mean's sum at full strength; the median only counts its position in the order.
Naming the shape tells you which measure of the center to quote — quoting the wrong one is how a true set of numbers gets used to tell a false story.
Read “often” carefully — these are strong tendencies, not theorems. Problem 2.6.16 is a genuinely left-skewed data set whose mean and median are both six: equal mean and median is something symmetry guarantees, not something that guarantees symmetry.
Try it in rāSHio
Read the ordering off one panel
Open rāSHio, paste a list into File → Delimited List…, then choose Stats → Summary Statistics. The mean and the median come back side by side, so you can predict the shape before you ever draw the graph — then draw it and check.
Try it on the ten values from §2.6.2 and again on the ten from §2.6.3 — the two panels disagree in opposite directions.
Your turn — reading a stem-and-leaf plot
Try It Now 2.6.4 · Presidents' Ages
| Stem | Leaves |
|---|---|
| 4 | 6 9 |
| 5 | 3 6 7 7 7 8 |
| 6 | 0 0 3 3 4 4 5 6 7 7 7 8 |
| 7 | 0 1 1 2 3 4 7 8 8 9 |
| 8 | 0 1 3 5 8 |
| 9 | 0 0 3 3 |
Table 2.6.1: the ages former U.S. presidents died. Key: 8|0 means 80.
Try It Now 2.6.4 — part b of a three-source shape-reading exercise
Of 39 ages, name the shape and compare the mean to the median.
Skewed right — mean (≈ 70.1) > median (68)
The tallest row is the 60s (12 ages), with a thinner tail running up through the 80s and 90s — the peak sits left of center. M=68 (the 20th ordered value); x=392733≈70.1.
Worked example — three authors, three shapes
Example 2.6.1 · Comparing Three Authors' Letter Counts
Example 2.6.1 — a simple random sample of word-length counts
Terry (crime column, short words): 7; 9; 3; 3; 3; 4; 1; 3; 2; 2
Delgado (plain sentences): 3; 3; 3; 4; 1; 4; 3; 2; 3; 1
Raman (arts, longer vocabulary): 2; 3; 4; 4; 4; 6; 6; 6; 8; 3
Figure 2.6.8: Raman's dot plot — symmetrically shaped.
Make a dot plot for each author, then find the mean and the median. Is there a pattern between the shape and the measures of the center?
Example 2.6.1 — worked through
The mode sits at the peak; the mean gets pulled toward the tail
Means and medians
Terry: x=1037=3.7, M=3 — skewed right, mean > median.
Delgado: x=1027=2.7, M=3 — skewed left, mean < median.
Raman: x=1046=4.6, M=4 — symmetrical, mean ≈ median.
The pattern
Line each author up against the three rules: every one matches. The median stays closest to the tall bar (the mode); the mean is the one pulled out toward whichever tail the shape has.
Naming the skew is enough to predict the order of all three centers before computing any of them — which is the whole payoff of §2.6.
Key Terminology
Five terms from §2.6
- symmetrical distribution — a distribution whose histogram can be split by a vertical line into two halves that are mirror images of each other.
- unimodal — having exactly one mode, a single value or interval that occurs more often than any other.
- skewed to the left — a longer, thinner tail toward the lower values, with the data piled at the higher values; also negatively skewed.
- skewed to the right — a longer, thinner tail toward the higher values, with the data piled at the lower values; also positively skewed.
- skewness — the departure of a distribution's shape from symmetry, named for the direction its longer tail points.
§2.6 — conclusions
What §2.6 leaves you with
The core idea
Fold-test a histogram to call it symmetrical; otherwise name the direction its tail stretches. The mode marks the peak, the median only counts positions, and the mean is the balance point — so it is the one an extreme value drags the farthest.
The failure case
Equal mean and median does not prove symmetry — Problem 2.6.16 is a visibly left-skewed data set where both happen to land on six. Symmetry guarantees the equality; the equality does not guarantee symmetry.
Next: §2.7 — measures of spread, where the shape this section reads by eye gets a number attached to it.