Master Elementary Statistics from Zero

Statistics is about summarizing data with a single representative value. Three such values are: Mean — the average of all values. Sum divided by count. Median — the middle value when data is arranged in order. Mode — the most frequently occurring value. For example, data: 5, 5, 7, 6, 5 Mean = (5+5+7+6+5)/5 = 28/5 = 5.6 Median = arrange → 5,5,5,6,7 → middle = 5 Mode = 5 (appears 3 times)

Key Formulas

Mean (Average)

Mean = Sum of all values / Number of values

Most common measure. Affected by extreme values (outliers).

Median (odd n)

Median = ((n+1)/2)th term after sorting

Example: n=9, median = 5th term after sorting.

Median (even n)

Median = Average of (n/2)th and (n/2+1)th terms after sorting

Example: n=8, median = average of 4th and 5th terms.

Empirical Relation

Mode = 3 × Median - 2 × Mean

Very important! Use when 2 of the 3 are given and you need the third.

⚡ Shortcut Tricks

Empirical formula: Mode = 3 × Median - 2 × Mean. If Mean=66 and Mode=60, Median = (Mean×2 + Mode)/3 = (132+60)/3 = 64.

Median is NOT affected by extreme values — use it when data has outliers. Mean IS affected.

A data set can have NO mode (all values unique), ONE mode, or MULTIPLE modes (bimodal, trimodal).

For even number of observations, median = average of two middle values after sorting.

Mean of combined groups: (n1×A1 + n2×A2) / (n1+n2). Weighted mean formula.

If all values increase by k, mean and median also increase by k. Mode increases by k too.

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