Statistics I

• For inference to be valid, the sample must be representative of the population: randomly chosen and unbiased.
• A claim about every organisation in a city based on one organisation's data is an over-generalisation: the sample doesn't represent the population.

Categorical vs numerical

• Categorical (qualitative)

  What it is:

  Labels or groups

  Examples:

  Gender, city, blood group, T-shirt size, PIN code, roll number

• Numerical, discrete

  What it is:

  Countable values with gaps

  Examples:

  Children in a family, cars sold, goals scored

• Numerical, continuous

  What it is:

  Any value in a range (measured)

  Examples:

  Height, weight, time, temperature

Scales: N-O-I-R, each adds one property

• Nominal

  Adds:

  Names only, no order

  Allowed:

  = ≠

  Examples:

  Gender, religion, election symbol, city, jersey number

  Centre:

  Mode

• Ordinal

  Adds:

  Order (gaps unequal)

  Allowed:

  < >

  Examples:

  Ratings, ranks, grades, sizes S/M/L/XL, education level

  Centre:

  Mode, median

• Interval

  Adds:

  Equal gaps, no true zero

  Allowed:

  + −

  Examples:

  Temperature in °C/°F, calendar year, time of day

  Centre:

  Mode, median, mean

• Ratio

  Adds:

  True zero (0 = none)

  Allowed:

  + − × ÷

  Examples:

  Height, weight, income, age, sales, number of votes

  Centre:

  All

Relative frequency

fn

All relative frequencies add to 1.

Percentage

fn × 100

Pie-chart angle

fn × 360°

Count from a slice

n × angle360° or n × percentage

Mode

the category with the highest frequency

Median (ordinal only)

the middle category after ordering

Charts

Class width

upper limit − lower limit

Class mark (midpoint)

lower + upper2

Mean of grouped data

Σ f_ix_iΣ f_i

x_i = class midpoint

• Class [10, 20) includes 10 and excludes 20; a value of exactly 20 goes in the next class.
• Histogram: bars touch (no gaps); bar area ∝ frequency. Bar charts are for categories, histograms for numbers.
• Stem-and-leaf: stem 2 | leaf 3 means 23. Number of leaves = n. You keep every actual value, so counts, median and mean can be read straight off.
• Shape: symmetric (mean ≈ median); right-skewed, long tail to the right (mean > median); left-skewed (mean < median).

Mean

x = Σ x_in

Median (sorted data)

n odd: value at position n + 12 · n even: average of positions n2 and n2 + 1

Mode

most frequent value (there may be none, or several)

Combined mean

n₁x₁ + n₂x₂n₁ + n₂

Add one value x

new mean = nx + xn + 1

Fix a wrong value

new mean = x + correct − wrongn

y = ax + b

y = ax + b

Median, mode, quartiles and percentiles transform the same way.

Range

max − min

Population variance

σ² = Σ(x_i − μ)²n = Σ x_i²n − μ²

Sample variance

s² = Σ(x_i − x)²n − 1 = Σ x_i² − nx²n − 1

Standard deviation

√variance

Same units as the data.

Sample ↔ population

s² = nn − 1 σ²

y = ax + b

Var(y) = a² Var(x), SD(y) = |a| SD(x)

Recover Σx²

sample: (n − 1)s² + nx² · population: n(σ² + μ²)

What changes under y = ax + b

• Mean, median, mode, quartiles

  Adding b:

  + b

  Multiplying by a:

  × a

• Range, IQR

  Adding b:

  no change

  Multiplying by a:

  × |a|

• Variance

  Adding b:

  no change

  Multiplying by a:

  × a²

• Standard deviation

  Adding b:

  no change

  Multiplying by a:

  × |a|

IITM's percentile method (the answer keys use it)

Step 1

sort the data in ascending order

Step 2: index

i = P100 × n

Step 3

i not whole ⇒ round up, take that value · i whole ⇒ average the values at positions i and i + 1

Quartiles

Q₁ = P₂₅, Q₂ = P₅₀ = median, Q₃ = P₇₅

IQR

Q₃ − Q₁

Five-number summary

min, Q₁, median, Q₃, max

This is what a box plot draws.

Outliers

below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR

Joint relative frequency

cellgrand total

Row relative frequency

cellrow total

Column relative frequency

cellcolumn total

Marginal totals

row totals and column totals; each set adds to the grand total

• Contingency (two-way) table: one variable on the rows, the other on the columns, counts in the cells, totals in the margins.
• Association: compare row percentages across the rows. If they are (nearly) the same in every row, there is no association.
• A stacked (or 100% stacked) bar chart shows those row percentages visually.

Sample covariance

Σ(x_i − x)(y_i − y)n − 1 = Σ x_iy_i − nxyn − 1

Population covariance

Σ(x_i − μ_x)(y_i − μ_y)n

Scaling

cov(ax + b, cy + d) = ac · cov(x, y)

With itself

cov(x, x) = Var(x)

• Sign gives direction: positive (rise together), negative (one rises as the other falls), near 0 (no linear relation).
• Size depends on the units, so covariance can't measure strength. That's what correlation is for.
• Reading a scatter plot: direction, form (straight or curved), strength (how tightly points hug a line), outliers.

Pearson r

r = cov(x, y)s_x s_y

Use sample values throughout, or population values throughout. Never mix.

Computational form

r = nΣxy − ΣxΣy√[nΣx² − (Σx)²][nΣy² − (Σy)²]

Range

−1 ≤ r ≤ 1

Change of origin and scale

r(ax + b, cy + d) = r(x, y) if ac > 0, −r(x, y) if ac < 0

Reading r

When

one variable is dichotomous (0/1: pass/fail, male/female), the other numerical

With sample SD

r_pb = Y₁ − Y₀s_Y × √n₁n₀n(n − 1)

With population SD

r_pb = Y₁ − Y₀σ_Y × √n₁n₀n²

Same answer as the sample version.

Symbols

Y₁, Y₀: group means · n₁, n₀: group sizes · n = n₁ + n₀ · s_Y, σ_Y: SD of all Y values