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BUSINESS STATISTICSβ€ΊUNIT 02β€ΊDISPERSION & SKEWNESS

Unit 02 β€” Measures of Variance and Skewness

Measures of variance (standard deviation for sample and population) and measure of skewness β€” plus the coefficient of variation, kurtosis and Chebyshev's theorem that your class taught alongside them.

SESSIONS: 2 TEXTBOOK: LEVIN Β§3.7–3.10, pp. 111–136 EXAM WEIGHT: THE OTHER HALF OF Q3 (a) β€” 5 MARKS Full PPT: Dispersion slides 8–10 Class notes 7
PRACTISE THIS UNIT

Every question shape in this unit is drilled on Unit 02 β€” How to Solve Anything: four topics, each with a "how to recognise this topic" card, a master method, and every exercise Levin sets solved with its page number. Topic 4 is the half of Q3 (a) that reads skewness and kurtosis.

β—† What the papers ask from this unit

This is the unit Q3 (a) really tests. Both papers hand you a table with SD, skewness and kurtosis for two service channels and ask, in the Final's words, "What implications does the difference in skewness and kurtosis have for consistency?" QP p. 2. The Re-Exam asks the same thing about "service consistency" QP p. 2.

The word consistency is the hinge. It means low standard deviation and high kurtosis. Say so explicitly. Both versions solved β†’

1. Why Dispersion Matters at All Core syllabus concept

1Understand the Concept

Two distributions can share a mean, a median, a mode and a skewness and still be completely different businesses to run. Your class notes make exactly this point with a sketch of three curves that have "same mean, mode, median; same skewness; different SD" Class notes p. 2.

Your class notes open the topic with the three tools Class notes p. 1:

3 Dispersion Tools
\[ \text{Range} \qquad \text{Variance } (\sigma^2, s^2) \qquad \text{Standard Deviation } (\sigma, s) \]

The deck adds the manager's reason for preferring the standard deviation: it is "a quantitative index of a distribution's spread or variability… used to calculate the likelihood (probability) of an event occurring" slide 8. That last clause is the bridge to Units 4–6: everything you do with z from here on is measuring a distance in standard deviations.

Sources
Lecture Sample Size deck β€” Measures of dispersion, why the SD slides 8–10
Textbook Levin Β§3.7 Dispersion: why it is important pp. 111–113

2. Variance and Standard Deviation β€” Sample vs Population Core syllabus concept

1The four formulas, and the one difference between them

Population Variance & SD
\[ \sigma^2 = \frac{\sum (x - \mu)^2}{N} \qquad \sigma = \sqrt{\sigma^2} \]
Sample Variance & SD
\[ s^2 = \frac{\sum (x - \bar{x})^2}{n - 1} \qquad s = \sqrt{s^2} \]
Grouped Data (Frequency)
\[ \sigma^2 = \frac{\sum f(x - \mu)^2}{\sum f} \qquad s^2 = \frac{\sum f(x - \bar{x})^2}{\sum f - 1} \]

THE ONLY DIFFERENCE: divide by N for a population, by (n βˆ’ 1) for a sample.

Dividing by n βˆ’ 1 is not an arbitrary convention. A sample's own mean sits, by construction, in the middle of that sample, so the deviations from it are systematically too small. Dividing by the smaller number n βˆ’ 1 inflates the answer just enough to make sΒ² an unbiased estimator of σ². That word unbiased returns in Unit 5 Sampling deck slide 29.

⚠ The mistake that costs a whole question

Deciding N vs n βˆ’ 1 by how big the dataset is. It has nothing to do with size. Ask only: is this every element I care about (population, Γ·N), or a subset I am generalising from (sample, Γ·nβˆ’1)? The Georgia hospital problem uses 200 randomly chosen patients β€” a sample β€” yet the class notes write ΞΌ and Οƒ because they treat those 200 as the complete dataset being described Class notes p. 4. If a question is ambiguous, say which you assumed in one line and carry on.

2Range β€” the crude one, and why it is still asked

Range
\[ \text{Range} = \text{Highest value} - \text{Lowest value} \]

Fast to compute, easy to explain, and hopeless in practice: it uses only two observations and one outlier destroys it. Levin Β§3.8 pp. 113–118 covers it plus the interfractile and interquartile ranges, which fix the outlier problem by ignoring the tails.

3Coefficient of variation β€” comparing apples with oranges

A standard deviation carries the units of the data, so an SD of β‚Ή500 and an SD of 500 grams cannot be compared. The coefficient of variation is a relative measure of dispersion β€” it strips the units out. Your class notes give it as Class notes p. 1:

Coefficient of Variation
\[ \text{CV} = \left(\frac{\sigma}{\bar{x}}\right) \times 100\% \]

Higher CV β†’ more variable RELATIVE to its own average. Lower CV β†’ more consistent.

ENPearson's coefficient, skewness & kurtosis Dr. Gajendra Purohit
Sources
Primary Levin Β§3.8–3.10 Ranges, average deviation measures, coefficient of variation pp. 113–136
Class CV formula and the three dispersion tools p. 1
Lecture Calculating a standard deviation, worked slides 9–10

3. Skewness β€” Departure from Symmetry Core syllabus concept

1Understand the Concept

Your class notes give the one-line definition to open any answer with Class notes p. 1:

Skewness β†’ departure from symmetry.

And the sign rule, exactly as written:

Skewness Sign Rule
\[ Sk = 0 \implies \text{Symmetric} \qquad Sk > 0 \implies \text{Positive (Right tail)} \qquad Sk < 0 \implies \text{Negative (Left tail)} \]

2Reading skewness from mean, median and mode

This is the specific skill Q3 (a) tests, because the question gives you all three and no data.

OrderingShapeWhat the long tail containsBusiness reading
Mean < Median ≀ Mode Negatively skewed (left tail) A minority of unusually low values, dragging the mean below the median. Most customers are satisfied; a small group had a bad experience and is pulling the average down.
Mean = Median = Mode Symmetric Nothing β€” the tails balance. Opinion is evenly spread around the centre.
Mean > Median β‰₯ Mode Positively skewed (right tail) A minority of unusually high values, dragging the mean above the median. Typical of incomes, waiting times and claim sizes β€” a few large values inflate the average.
⚠ The direction trap

"Negative skew" sounds like bad news and "positive skew" like good news. In a satisfaction survey it is the opposite: negative skew means most people are happy, with a small unhappy tail. The Final's Branch Counter has skewness βˆ’1.80 and the official key reads it as "the majority are highly satisfied" Key p. 3. Read the tail, not the sign.

3Pearson's coefficient of skewness β€” if you are asked to compute one

Neither paper asked for a number, but the syllabus says "measure of skewness" Syllabus p. 1, so know the two standard forms:

Karl Pearson 1st Skewness
\[ Sk = \frac{\text{Mean} - \text{Mode}}{\text{SD}} \]
Karl Pearson 2nd Skewness
\[ Sk = \frac{3(\text{Mean} - \text{Median})}{\text{SD}} \]
Bowley Quartile Skewness
\[ Sk = \frac{Q_3 + Q_1 - 2 Q_2}{Q_3 - Q_1} \]

All three carry the same sign convention and are unit-free for comparison across datasets.

ENExamples on Karl Pearson's coefficient of skewness Maths Platter by JC ENBowley's coefficient of skewness β€” solved examples Maths Platter by JC
Sources
Primary Class notes β€” skewness definition, tree and sign rule p. 1
Exam Final Q3 (a) p. 2 Β· official reading Key p. 3 Β· Re-Exam Key p. 2

4. Kurtosis β€” Peakedness, and What It Says About Consistency Core syllabus concept

1Understand the Concept

Your class notes define it as "a property of a data that talks about peakness / peakedness" Class notes p. 1, and the sketch on the next page shows three curves with the same mean, mode, median and skewness but different spread Class notes p. 2.

NameValueShapeWhat it means for a business
LeptokurticK > 3Tall and narrow β€” a sharp peak.Ratings are tightly clustered at one value. Consistent, predictable service.
MesokurticK = 3The normal curve's own peakedness.The benchmark. Neither unusually peaked nor unusually flat.
PlatykurticK < 3Flat and wide.Ratings are spread all over the scale. Inconsistent, unpredictable service.

Your notes' own explanation of the leptokurtic case, in your lecturer's words: "In leptokurtic, most of the data is peaked at the center… the frequency distribution is near the center and very few away" Class notes p. 2.

The sentence that answers Q3 (a) part b: "Low SD together with high kurtosis (leptokurtic) means the channel delivers a consistent experience; high SD with low kurtosis (platykurtic) means the experience varies widely from customer to customer." Both official keys say precisely this β€” Final Key p. 3, Re-Exam Key p. 2.
⚠ Two kurtosis conventions exist

The one your class uses compares K against 3. Excel's KURT() reports excess kurtosis, which subtracts 3 first β€” so Excel's benchmark is 0, and leptokurtic means > 0. Both papers quote values around 2 to 5 and label them platykurtic and leptokurtic, so they are using the K vs 3 convention. Use theirs.

ENExamples on Pearsonian's coefficient of kurtosis Maths Platter by JC ENChebyshev, skewness, kurtosis, percentiles in 6 minutes Ambarish Ganguly Academy

Practice B1–B6 β€” SD, CV, skewness reading and interpretation β†’

Sources
Primary Class notes β€” kurtosis, the three curves, leptokurtic explanation pp. 1–2
Exam Both papers' Q3 (a) tables quote kurtosis explicitly Final p. 2 Β· Re-Exam p. 2

5. Chebyshev's Theorem and the Empirical Rule Core syllabus concept

1Understand the Concept

Your class notes state Chebyshev's theorem in full Class notes p. 2: "no matter what the shape of the distribution is, almost 75% of the values would fall within Β±2 sigma from the mean and at least 89% of the values will lie within Β±3 sigma of the mean."

The power of it is the phrase "no matter what the shape". You do not need normality, symmetry or anything else β€” it holds for every distribution. The price is that the bounds are loose.

Chebyshev (Any Dist)
\[ \text{At least } 75\% \text{ inside } \mu \pm 2\sigma \qquad \text{At least } 89\% \text{ inside } \mu \pm 3\sigma \]
Empirical Rule (Normal)
\[ \mu \pm 1\sigma \approx 68\% \qquad \mu \pm 2\sigma \approx 95\% \qquad \mu \pm 3\sigma \approx 99\% \]

Both boxes are copied from your class notes p. 2 and p. 3. The general form of Chebyshev is at least (1 βˆ’ 1/kΒ²) of values lie within ΞΌ Β± kΟƒ; k = 2 gives 1 βˆ’ 1/4 = 75%, k = 3 gives 1 βˆ’ 1/9 = 88.9% β‰ˆ 89%.

2Worked example β€” continuing the Georgia hospital problem

From Unit 1 Concept 5: 200 patients, ΞΌ = 7.7 days, Οƒ = 4.6 days Class notes p. 4.

  1. Step 1 β€” Apply Chebyshev at k = 2 ΞΌ Β± 2Οƒ = 7.7 Β± (2 Γ— 4.6) = 7.7 Β± 9.2 = (βˆ’1.5 to 16.9)

    A stay cannot be negative, so the interval is read as 0 to 17 days.

  2. Step 2 β€” Convert the percentage into a count At least 75% of 200 = 200 Γ— 0.75 = 150 patients

    So at least 150 of the 200 patients stayed between 0 and 17 days.

  3. Step 3 β€” Now assume normality and compare

    If the distribution is treated as normal, the empirical rule gives around 95% inside ΞΌ Β± 2Οƒ Class notes p. 5:

    200 Γ— 0.95 = 190 patients
  4. Step 4 β€” Check it against the actual data

    Your notes then do the honest thing and count: "manually piles the frequency till 16–18 comes to 191, which is around 190." Adding the frequencies 18 + 90 + 44 + 21 + 9 + 9 = 191 patients stayed 18 days or fewer β€” the empirical rule's 190 was almost exactly right.

The point of that comparison: Chebyshev promised at least 150 and was correct but nearly useless. The normal assumption predicted 190 and was correct to within one patient. Assuming normality buys you precision β€” when it is justified. That trade-off is the whole reason Units 4–6 exist.
ENChebyshev's theorem The Organic Chemistry Tutor

Practice B5–B6 β€” Chebyshev vs empirical rule β†’

Sources
Primary Class notes β€” Chebyshev's theorem, the two boxes, Georgia hospital worked end to end pp. 2–5
Textbook Levin Β§3.9 Average deviation measures β€” standard deviation and Chebyshev pp. 119–130
Lecture "99% of its values are within Β±3 standard deviations from its mean" Sample Size deck slide 11