So what is frequency? In math and statistics, frequency is the number of times a value or event occurs in a given dataset or interval. In physics, frequency is the number of complete cycles a periodic phenomenon completes per unit time, measured in hertz (Hz).
The two definitions share a name and a flavour — both count occurrences over a defined window — but the formulas are different.
Quick Reference
Field | Value |
|---|---|
Math definition | The number of times a value occurs in a dataset |
Physics definition | The number of cycles per unit time of a periodic phenomenon |
Symbol | $f$ |
Math formula | $f = \text{count of occurrences}$ |
Physics formula | $f = \frac{1}{T}$, where $T$ is the period in seconds |
SI unit (physics) | Hertz (Hz), where $1,\text{Hz} = 1$ cycle per second |
Used in | Statistics, signal processing, physics, music, electronics |
What is Frequency?
In a dataset, frequency simply counts how many times each distinct value appears. If five students scored $7$ on a quiz and three scored $8$, the frequency of the score $7$ is $5$ and the frequency of $8$ is $3$. A frequency table lists every distinct value alongside its count.
In physics, frequency measures how often a wave or oscillation completes a cycle. A sine wave that goes through three complete cycles every second has a frequency of $3$ hertz, written $3,\text{Hz}$. Frequency and period are reciprocals — a high-frequency wave has a short period; a low-frequency wave has a long period.
Why Frequency Exists
The word frequency in mathematical use traces back to the early statisticians of the $1700$s — Abraham de Moivre, Pierre-Simon Laplace — who needed a precise word for how often a value showed up in a dataset. The need was urgent because probability theory was being built around exactly that concept. The hertz, the SI unit of physical frequency, was named for Heinrich Hertz (1857–1894, Germany), who first produced and detected radio waves and confirmed that James Clerk Maxwell's equations described real, measurable electromagnetic phenomena.
The Formula
The math formula is the simplest in this article: the frequency of a value is the count of how many times it appears.
$$f_i = \text{number of occurrences of value } x_i$$
The total frequency across all values equals the sample size $N$:
$$\sum f_i = N$$
The relative frequency (or probability) of a value is its frequency divided by the total:
$$\text{relative frequency of } x_i = \frac{f_i}{N}$$
In physics, frequency relates to period $T$ (the time for one cycle) and angular frequency $\omega$:
$$f = \frac{1}{T}, \quad \omega = 2\pi f$$
Variable Key
Symbol | Meaning | Unit |
|---|---|---|
$f$ | Frequency | (math) count or proportion; (physics) hertz (Hz) |
$f_i$ | Frequency of value $x_i$ in a dataset | count |
$N$ | Total number of observations | count |
$T$ | Period of a wave (time for one cycle) | seconds |
$\omega$ | Angular frequency | radians per second |
Worked Examples of Frequency
Example 1: Frequency table
A teacher records test scores from $10$ students: $7, 8, 8, 9, 7, 6, 8, 7, 9, 10$. Build the frequency table.
Score | Frequency |
|---|---|
$6$ | $1$ |
$7$ | $3$ |
$8$ | $3$ |
$9$ | $2$ |
$10$ | $1$ |
Total | $10$ |
Final answer: The most frequent score is a tie between $7$ and $8$, each with frequency $3$.
Example 2: Frequency in physics
A pendulum completes one full swing every $0.5$ seconds. What is its frequency?
$$f = \frac{1}{T} = \frac{1}{0.5} = 2,\text{Hz}$$
Final answer: The pendulum has a frequency of $2$ hertz.
Common Confusions of Frequency Formula
Frequency vs Relative Frequency
Frequency is a raw count (e.g., $5$ students). Relative frequency is the count divided by the total (e.g., $\frac{5}{20} = 0.25$ or $25%$). Both are useful; only relative frequency lets you compare across different sample sizes.
Frequency vs Probability
Once a sample is large enough, relative frequency closely approximates probability. They are not the same thing — probability is theoretical; relative frequency is empirical — but for large $N$ they converge.
Frequency vs Amplitude (Physics)
Frequency is how often a wave cycles; amplitude is how big the cycle is. A low-frequency sound is bass; a high-frequency sound is treble. Loudness corresponds to amplitude, not frequency.
Where Frequency Appears
Beyond the obvious — frequency tables in statistics, hertz in physics — the concept shows up in surprising places. Frequency analysis is used in cryptography (the Caesar cipher is broken by counting how often each letter appears in the ciphertext).
Music theory uses frequency to define pitch — middle C is $261.6,\text{Hz}$. Audio engineering uses frequency-domain analysis (Fourier transforms) to compress music files into MP3s. Heart-rate monitors compute beats per minute, which is a frequency.
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