Reading a Latency Histogram

A web server answers most requests in a blink, but a few take ages. A histogram shows that long right tail, and percentiles like p95 tell engineers how long the slowest users wait.

How it works

Histograms, cumulative frequency and percentiles

In one sentence

A frequency distribution groups data into classes. The relative frequency of a class is $f/n$ , the cumulative relative frequency $F(x)$ is the proportion of observations at most $x$ , and the $k$ th percentile is a value with $k\%$ of the data at or below it.

General formula

\text{rel. freq.} = \frac{\htmlData{sym=0}{{f}}}{\htmlData{sym=1}{{n}}}, \qquad {\htmlData{sym=2}{F(x)}} = \frac{\#\{{\htmlData{sym=3}{x_k}} \le x\}}{\htmlData{sym=1}{{n}}}, \qquad {\htmlData{sym=4}{p_{95}}} = F^{-1}(0.95)

The histogram shows where the requests pile up. The cumulative curve answers what fraction were done by a given time, and a percentile answers the reverse question.

What the symbols mean

$f$
Class frequency
How many requests fall in one bar's range of times.
$n$
Number of requests
All 60 response times.
$F(x)$
Cumulative relative frequency
The share of requests that finished within x milliseconds.
$x_k$
One response time
The time the kth request took, in milliseconds.
$p_{95}$
95th percentile
The time by which 95% of requests are done.

Group the 60 response times into classes of equal width. Each bar's height is its relative frequency $f/n$ : the share of requests in that class.
The cumulative relative frequency $F(x)$ adds up the bars from the left: the share of requests finished by time $x$ .
A percentile reads that curve backwards: $p_{95}$ is the time by which 95% of requests are done.
The long right tail drags the mean above the median, which is why engineers quote percentiles instead of averages.

Learn the math

Levels of Measurement and Frequency TablesStats · Lesson 3, Sampling and Data Graphs of DataStats · Lesson 5, Descriptive Statistics Quartiles, Box Plots and OutliersStats · Lesson 8, Descriptive Statistics Summarizing Process DataEng Stats · Lesson 2, Engineering Data and Variation