Frequency Analysis (Understanding Data Distribution)

Frequency analysis turns raw counts into insight: you see modes, skew, heavy hitters, and whether a constraint like “at most k distinct” is even possible before designing an algorithm.

Why this shows up in the real world

Log analytics pipelines aggregate events per key; A/B dashboards compare cohort frequencies; security spots rare spikes that indicate abuse.

Core idea (explained for students)

Build counts first (Counter, dict, or array for small alphabets), then derive derived stats: max count, sum of top-k, cumulative histograms, or bucket boundaries.

Try this in Python

from collections import Counter


def top_two(nums: list[int]) -> list[tuple[int, int]]:
    c = Counter(nums)
    return c.most_common(2)


print(top_two([1, 2, 2, 3, 3, 3]))

Common mistakes

• Counting without normalizing for empty input or Unicode normalization when keys are strings.
• Confusing frequency of values vs frequency of pairs (need 2D counts).

Key takeaways

• Plot mentally: sorted (value, count) pairs reveal bucket logic quickly.
• When alphabet is small (≤26 letters), a list of length 26 beats a dict for cache locality.