Statistics Reference: Formula Sheet, Traps & Key Takeaways

The complete formula sheet, every number used in class, the exam traps, student questions answered, and the ten things worth memorising.

32. Complete Formula Sheet

Central Tendency

Mean            μ  =  Σx / N

Median          odd  n  →  the ((n + 1) / 2)-th value
                even n  →  average of the (n/2)-th and ((n/2) + 1)-th values

Mode            the most frequently occurring value

Dispersion

Range           =  Maximum − Minimum

Variance        σ²  =  Σ(xᵢ − μ)² / N            (population)
                s²  =  Σ(xᵢ − x̄)² / (n − 1)      (sample)

Std Deviation   σ   =  √Variance

Coeff. of Var   CV  =  (σ / mean) × 100 %

Quartiles, IQR & Outliers

Q1 position     =  ¼ (n + 1)
Q2 position     =  (n + 1) / 2                    ← the median
Q3 position     =  ¾ (n + 1)

  → if a position comes out as a decimal (e.g. 9.5),
    take BOTH neighbouring values and average them

IQR             =  Q3 − Q1

Lower limit     =  Q1 − 1.5 × IQR
Upper limit     =  Q3 + 1.5 × IQR

Outlier         any value  <  Lower limit   OR   >  Upper limit

Five-point summary  =  Minimum · Q1 · Median · Q3 · Maximum

Excel formulas used

=MAX(range)          =MIN(range)          =AVERAGE(range)
=SUM(range)          =C6-$C$37            (lock with $ before AND after the column letter)

Non-negotiable prerequisites

Sort the data ascending before computing any quartile
Check for outliers before trusting the mean or standard deviation
Report standard deviation, not variance

33. All Numbers Used in Class

Machine A / Machine B (Example 3)

Target specification: 10.0 mm diameter
Machine A (5 parts): 9.9, 10.0, 10.1, 10.0, 10.0        → mean 10.0, range 0.2
Machine B (5 parts): 9.5, 10.5, 9.6, 10.4, 10.0         → mean 10.0, range 1.0
Machine A (6 parts): 9.9, 10.0, 10.1, 10.0, 10.0, 11    → range 1.1
Machine B (6 parts): 9.5, 10.5, 9.6, 10.4, 10.0, 10.4   → range 1.0

Netflix A/B test

Users per group:  50,000 control  /  50,000 treatment
Control watch time:    52 min/day
Treatment watch time:  55 min/day
Observed lift:          3 min

Political survey

Population: 1 lakh (100,000)
Sample:     2,000 randomly picked
Confidence: 95%

AI bias / Equal Opportunity Score

Group 1: 100 people, avg probability 0.6
Group 2:  20 people, avg probability 0.5   (Asian)
Group 3:  30 people, avg probability 0.9
Range = 0.9 − 0.5 = 0.4   → too high → model rejected

PIN codes

600036  ·  600042        (sum 1200078 — meaningless)

Calories example (variance)

100, 150, 200, 250, 300     →  mean = 200 calories
Deviations: −100, −50, 0, +50, +100
Variance units: calories²   ·   SD units: calories

Temperature sensor Excel exercise (25 readings)

Max 90 · Min 72.6 · Range 17.4 · Mean 75.64
Σ(x − μ)² = 244.90
Population variance = 244.90 / 25 = 9.8
Population SD = √9.8 = 3.13
With the 90 replaced by 75:  SD ≈ 1.10

Second Excel exercise

Min 11.34
Σ(x − μ)² = 12.75
Population variance = 12.75 / 25 = 0.51
Population SD = √0.51 ≈ 0.71

Coefficient of variation

Product A: 10, 12, 14, 13, 11      → mean 12    · σ 1.414 · CV 11.78 %
Product B: 100, 102, 98, 105, 101  → mean 101.2 · σ 2.315 · CV  2.28 %

Quartile example A (with 1 and 100 added)

Q1 = 9 · Q2 = 18 · Q3 = 24 · IQR = 15
Lower limit = −13.5 · Upper limit = 46.5
1 is NOT an outlier · 100 IS an outlier · (−20 would be an outlier)

Quartile example B (11 values)

7, 8, 8, 9, 10, 10, 10, 11, 12, 14, 30
Q1 = 8 · Q2 = 10 · Q3 = 12 · IQR = 4
Lower limit = 2 · Upper limit = 18 · 30 is an outlier
(Slide's Excel-method values: Q1 = 8.5, Q3 = 11.5, IQR = 3, limits 4 and 16)

Restaurant wait times (18 tables)

5, 8, 10, 10, 12, 13, 15, 15, 16, 18, 19, 20, 22, 24, 25, 28, 32, 50
Q1 = 12 · Median = 17 · Q3 = 24 · IQR = 12
Lower limit = −6 · Upper limit = 42 · 50 is the outlier
(Alternative convention: limits −9.25 and 44.75 — same conclusion)

IoT range case studies

CNC temperature: 72, 75, 74, 78, 73, 76, 74, 90, 75, 74 °C
   Max 90 · Min 72 · Range 18 °C

Robotic arm cycle: 12.1, 12.3, 12.0, 12.4, 12.2, 15.8, 12.1, 12.3 s
   Max 15.8 · Min 12.0 · Range 3.8 s

Feature selection

F₁: mean 30, SD 0.1   → drop (near-constant)
F₂: mean 30, SD 5     → keep
Chennai house prices: ₹1,00,000 / 1,05,000 / 1,12,000 / 1,08,000 / 98,000  → no variation

34. Exam Traps

Trap	The right answer
PIN code (all digits, no decimal)	Nominal categorical, not discrete — arithmetic on it is meaningless and there’s no order
Gender stored as 1 / 0	Still binary categorical, not numerical
Two datasets with the same mean	The mean cannot distinguish them — compare dispersion (Machine A vs. B)
“Which machine is better” after adding one part	The range can flip — one value changed A from best to worst
Comparing spread across different scales	Use CV, not raw SD — Product A vs. B give opposite verdicts
Reporting variance to a stakeholder	Report standard deviation — variance has squared units nobody can interpret. “Don’t use variance much."
"Is IQR central tendency or variability?”	Variability — it measures a distance, like range
Unusually large SD on tight-looking data	Suspect an outlier — the 90 drove SD from 1.10 to 3.13
Computing quartiles on unsorted data	Sort ascending first
Quartile position comes out as a decimal	Take both neighbouring values and average them
Whisker limit falls outside the actual data	Either the computed limit or the actual minimum is acceptable in an exam
”Which feature should the model use — the one with less deviation?”	No — take the one with more deviation. Low SD = no information
Mode used on numerical data	Mode is for categorical variables

35. Student Questions Answered in Class

Question	Answer	§
“Why is IQR a measure of variability if it uses the median?”	It measures a distance, not a centre — same logic as range, but middle-in rather than extremes-only	§22
”Must the dataset be in ascending order?”	Yes, always	§21
”Is the upper limit Q3 + 1.3 IQR?“	1.5, not 1.3. Most of the industry uses 1.5	§23
”Can we go below 1.5 in this example?”	It depends on the data, but most statisticians worldwide accept 1.5	§23
”What is the purpose of the IQR calculation in application terms?”	Outlier detection — it’s the formula for identifying outliers	§22, §23
”Why do population and sample have different formulas?”	You can rarely measure a whole population; n − 1 corrects for sampling. Concept = degrees of freedom (deferred)	§18
”What is sample standard deviation at the bottom of the sheet?”	Ignored for now — covered in the next class	§18
”Is PIN code discrete since it’s numeric with no decimal?”	No — arithmetic on it is meaningless, so it’s nominal categorical	§5
”What if Q1/Q2/Q3 positions come out as decimals?”	Take both neighbouring values and average (n = 18 → 9.5 → 9th and 10th)	§21
”For the exam, is the right whisker answer the computed limit or the actual minimum?”	Either gets full marks	§25
”Which method should I use — box plot, SD, log?”	IQR → outliers. SD → spread. Don’t use variance much	§28
”We’re learning AI — where does statistics fit in?”	The data analysis part; plus cleaning, feature work, and model governance	§29, §31
”Which feature is better for model building?”	The one with more deviation (F₂), not less	§30
”Can we use this for IT job logs?”	Yes — find jobs beyond the pattern, then optimise them	§29

36. Deferred to Later Chapters

Explicitly postponed by the instructor:

Hypothesis testing — null hypothesis vs. alternate hypothesis; the machinery behind A/B testing. “There are multiple formulas, multiple statistics out there. We will cover in the next week.”
Degrees of freedom — the full explanation of why sample formulas use n − 1. “I don’t want to confuse you with n and n−1 on the first day itself.”
Sample variance and sample standard deviation — “Currently we’ll ignore it. We’ll cover in the next class.”
Linear regression
Measure of frequency distribution, histograms, pie charts
Equal opportunity score in full — “You will get it in future.”

37. Key Takeaways

The ten things worth memorising

Descriptive summarises what you have; inferential generalises from a sample to a population.
Mean uses all observations but breaks on outliers. Median survives outliers but ignores most of the data.
Mode is for categorical variables. Mean and median are for numerical.
Missing values: no outliers → fill with mean. Outliers present → fill with median. Symmetric → either.
The mean goes towards the tail. Right-skew: Mean > Median > Mode. Left-skew: Mean < Median < Mode. Symmetric: all equal.
Range = Max − Min. Simplest measure, but it uses only 2 points and one outlier destroys it.
Standard deviation = √Variance — and the only reason it exists is to restore the original units. Don’t report variance.
IQR = Q3 − Q1 = the middle 50%, which is exactly how it ignores outliers.
1.5 × IQR is the industry-standard outlier — and anomaly — detector.
Low dispersion → the central tendency is more reliable. But low dispersion in a feature → useless for machine learning.

The three sentences that carry the whole session

“Statistics will help you create new features which are more predictable. That is the power of statistics.”

“Don’t use variance much. We use standard deviation.”

“These calculations are very simple metrics, but they will help to take many decisions.”

Compiled from: the primary class transcript (0:00–1:14 of a 1:51 session), 56 timestamped slide screenshots (to 1:49), the auto-generated session summary, the variance/standard-deviation session with its two Excel exercises, the quartiles/IQR/box-plot session, and the box-plot practice session. Sections tagged [S] rest on slide content without accompanying audio.