Class 11 Statistics Notes · GSEB

Types of Skewness

Types of Skewness — tell positive from negative by the tail, the average order and the quartile rule. GSEB Class 11 Statistics notes with a three-curve gallery, comparison table and solved sums.

Last updated: 5 Oct 2026

Notes

Two Types, Two Tails

“Measures of skewness indicates the tendency of the group to depart from symmetry.” — Connor

Positive skewness

The right tail is elongated — the more-variable values stretch to the right.

x̄ > M > M₀

(Q₃ − M) > (M − Q₁) · more variation right of the mode

Book example: number of deaths in a hospital

Negative skewness

The left tail is elongated — low values stretch far below the peak.

x̄ < M < M₀

(Q₃ − M) < (M − Q₁) · more variation left of the mode

Book example: price of an item vs number of customers — common in business & finance

The sandwich rule

The median always lies between the mean and the mode in a skewed distribution — whichever way it leans, M is never the outermost marker. That is why it survives both directions while the mean gets dragged.

Lab: Compare Three Curves

Three real book distributions side by side — negative, symmetric, positive. Click a curve to make it active; the order strip and quartile bars update live.

Click a curve to inspect it

OrderQ₃6>x̄4.62>M4>M₀3=Q₁3Positive — hospital deaths

Positive — hospital deathsx̄ > M > M₀x̄ 4.62 · M 4 · M₀ 3

Right tail elongated · x̄ > M > M₀

M − Q₁ (left distance)1

Q₃ − M (right distance)2

The right distance is larger — that is the side the tail drags toward.

Side by Side

Positive vs negative skewness
Aspect	Positive	Negative
Elongated tail	Right	Left
⭐ Order of averages	x̄ > M > M₀	x̄ < M < M₀
Quartile rule	(Q₃ − M) > (M − Q₁)	(Q₃ − M) < (M − Q₁)
More variation lies	Right of the mode	Left of the mode
Book example	Hospital deaths	Item prices, investors

Bowley’s quick sign test works without computing any measure: Q₃ + Q₁ > 2M ⇒ positive · < 2M ⇒ negative · = 2M ⇒ symmetric.

Worked Walkthrough — Deaths in a Hospital

Solved Example

Problem

Deaths in a hospital were recorded for 60 days: x = 1–11 days, f = 2, 8, 14, 10, 8, 6, 4, 3, 2, 2, 1. Test whether the distribution is positively skewed.

Solution

x̄ = 4.62 > M = 4 > M₀ = 3, and (Q₃ − M) 2 > (M − Q₁) 1 → positively skewed.

Think about it

A few long-stay patients pull the right tail — the mean death count looks worse than the typical day. Anyone quoting only the average would misjudge the hospital’s everyday reality. The same trap sinks founders who quote “average order value” when a handful of bulk buyers drag it up — the typical customer’s order is far smaller.

Spotting the Type at a Glance

Compare mean & mode

If the mode exceeds the mean, the distribution is negatively skewed. Mean above mode ⇒ positively skewed; mean = median ⇒ symmetric.

Three quick verdicts

Mean above mode ⇒ positive; mean = median ⇒ symmetric; mean below median ⇒ negative.

Highest average

In a negative skew the highest average is the mode; in a positive skew it is the mean.

Quartile distances

(Q₃ − M) < (M − Q₁) means negative skewness— the left span from the median is the wider one.

Worked Example Gallery — one per type

Both types, book-exact

2 problems

Full question

The frequency distribution of number of deaths in a hospital of a city during 60 days is as follows :

No. of deaths	1	2	3	4	5	6	7	8	9	10	11
No. of days	2	8	14	10	8	6	4	3	2	2	1

x	f	fx	cf
1	2	2	2
2	8	16	10
3	14	42	24
4	10	40	34
5	8	40	42
6	6	36	48
7	4	28	52
8	3	24	55
9	2	18	57
10	2	20	59
11	1	11	60
Total	n = 60	Σfx = 277

x̄ = 277/60 = 4.62 · M₀ = 3 (f = 14) · M = 30.5th = 4⇒ 4.62 > 4 > 3.

Q₁ = 3, Q₃ = 6 ⇒ (Q₃ − M) = 2 > (M − Q₁) = 1 ⇒ positively skewed.

Full question

Obtain the frequency curve and the measures of central tendency of the following frequency distribution and test whether the distribution is negatively skewed:

Price of an item (₹)	6	7	8	9	10	15	17	18	19	20	Total
No. of customers (thousand)	1	2	3	4	5	8	10	7	4	1	45

x̄ = 642/45 = ₹14.27 · M = 23rd observation = ₹15 · M₀ = value with maximum frequency 10 = ₹17.

Order: 14.27 < 15 < 17 ⇒ x̄ < M < M₀ — negatively skewed.

Q₁ = 10, Q₃ = 18 ⇒ (Q₃ − M) = 3 < (M − Q₁) = 5 ⇒ quartile rule confirms negative skewness.

Fewer customers at lower prices — the left tail of buyers stretches down, the pattern every discount retailer watches.

Quick Reference

	Positive	Negative
Tail	Right elongated	Left elongated
⭐ Order	x̄ > M > M₀	x̄ < M < M₀
Quartiles	(Q₃ − M) > (M − Q₁)	(Q₃ − M) < (M − Q₁)
More variation	Right of mode	Left of mode
Example	Hospital deaths	Prices, investors

· Median always lies between mean and mode — M is never the outermost marker.
· Mean above mode ⇒ positively skewed · mean = median ⇒ symmetric · mean below median ⇒ negatively skewed.
· The elongated tail shows where the more variation lies — never the direction the peak points.
· Mode = 3M − 2x̄ is the empirical bridge (Karl Pearson) — it powers both skewness methods ahead.
· In a negative skew the highest average is the mode; in a positive skew it is the mean.

Words marked with a dotted underline carry the question they usually appear in — tap them to reveal it.

Key Takeaways

Positive skew = elongated right tail with x̄ > M > M₀; negative skew = elongated left tail with x̄ < M < M₀ — the tail, not the peak, decides the type.
The quartile test never lies: (Q₃ − M) larger than (M − Q₁) means positive, smaller means negative — so if Q₃ + Q₁ ≠ 2M you already know the direction.
Median always sits between mean and mode, which is why it is the steady marker while the mean gets dragged toward the tail.
Hospital deaths skew positive, prices and investor counts skew negative — in your own life, look for the tail before trusting any average, or your inventory (or your study plan) gets built around a number almost nobody matches.
The order strip (x̄ vs M vs M₀) is the fastest exam shortcut — but confirm with the quartile rule when the numbers are close.