Mean, Median, Mode & z-scores

“What’s the typical income in this town?” — that one question already needs two answers, because the mean and the median can be very different numbers, and which one is honest depends on whether one billionaire moved in. Descriptive statistics is the small toolkit for that: a couple of ways to say “where is the centre” (mean, median, mode), a couple for “how spread out” (variance, standard deviation), and at the end a tiny rescaling — the z-score — that turns “I got 80 on this test” into “I was 2 SDs above the class mean.”

These are cheap marks on the exam, and the z-score in particular is the exact move ML preprocessing makes when it standardises features.

Measures of centre

• Mean μ = (Σ x) / n — the arithmetic average, the balance point of the data.
• Median — sort the values; the median is the middle one (average of the two middle values if n is even).
• Mode — the most frequently occurring value; a dataset can have one, several, or no mode, and it is the only centre that works for categorical data.
• Range — max − min, the crudest measure of spread.

For the dataset 2, 4, 4, 6, 9: mean = 25/5 = 5, median = 4 (middle of five sorted values), mode = 4 (it appears twice), range = 9 − 2 = 7.

Spread: variance and standard deviation

Variance measures the average squared distance from the mean. There are two versions, and the divisor is the whole exam trap:

• Population variance divides by n: σ² = (Σ (x − μ)²) / n.
• Sample variance divides by n − 1: s² = (Σ (x − x̄)²) / (n − 1).

That n − 1 is Bessel’s correction — using the sample mean (computed from the same data) slightly understates spread, so dividing by n − 1 instead of n corrects the bias when you only have a sample. Standard deviation is just the square root of variance (σ or s), back in the original units.

The z-score — standardization

The z-score z = (x − μ) / σ answers: how many standard deviations does a value x sit above (positive) or below (negative) the mean? It strips away the units and scale, which is exactly why standardization is a staple of ML preprocessing — it puts every feature on the same footing. A z-score of 0 is exactly average; +2 means two SDs above the mean.

How GATE asks this

Reliably a quick NAT: a value, a mean, and a standard deviation are given, and you compute the z-score to a few decimals — or you are given a small dataset and asked for its mean/median/variance. MCQs probe the concepts: which divisor is the sample variance (n − 1), or which measure of centre is robust to outliers (the median). z-score normalization appeared in GATE DA 2024.

Worked example — a real 2024 question

A data value of 106000 comes from a distribution with mean μ = 96000 and standard deviation σ = 21000. Find its z-score.

Plug straight into the formula:

z = (x − μ) / σ
  = (106000 − 96000) / 21000
  = 10000 / 21000
  ≈ 0.476

So z ≈ 0.476 — the value sits roughly half a standard deviation above the mean. This is a real GATE DA 2024 question, and 0.476 is the verified answer. Note how the large raw numbers collapse to a clean, unit-free score once standardized.

Quick check