The Statistics of Web Performance Analysis

• Philip Tellis

• .com
• philip@lognormal.com
• @bluesmoon
• geek paranoid speedfreak
• http://bluesmoon.info/

Boston #WebPerf Meetup / 2012-08-14 The Statistics of Web Performance Analysis 1

I’m a Web Speedfreak


We measure real user website performance


This talk is about the Statistics we learned while building it


The Statistics of Web Performance Analysis

Philip Tellis / philip@lognormal.com

Boston #WebPerf Meetup / 2012-08-14


0
Numbers


Accurately measure page performance∗


Be unintrusive

If you try to measure something accurately, you will change
something related
– Heisenberg’s uncertainty principle


And one number to rule them all


What do we measure?

• Network Throughput
• Network Latency
• User perceived page load time


We measure real user data


Which is noisy


1
Statistics - 1


Disclaimer

I am not a statistician


1-1 Random Sampling


Population

All possible users of your system


Sample

Representative subset of the population


Bad sample

Sometimes it’s not


How to randomize?

http://xkcd.com/221/


How to randomize?

• Pick 10% of users at random and always test them

OR

• For each user, decide at random if they should be tested

http://tech.bluesmoon.info/2010/01/statistics-of-performance-measurement.html


Select 10% of users - I

if($sessionid % 10 === 0) {
// instrument code for measurement
}

• Once a user enters the measurement bucket, they stay
there until they log out
• Fixed set of users, so tests may be more consistent
• Error in the sample results in positive feedback


Select 10% of users - II

if(rand() < 0.1 * getrandmax()) {
// instrument code for measurement
}

• For every request, a user has a 10% chance of being
tested
• Gets rid of positive feedback errors, but sample size !=
10% of population


How big a sample is representative?

Select n such that
σ
1.96 √n ≤ 5%µ


1-2 Margin of Error


Standard Deviation

• Standard deviation tells you the spread of the curve
• The narrower the curve, the more conﬁdent you can be


MoE at 95% conﬁdence

σ
±1.96 √n


MoE & Sample size

There is an inverse square root correlation between sample size
and margin of error


1-3 Central Tendency


One number

• Mean (Arithmetic)
• Good for symmetric curves
• Affected by outliers

Mean(10, 11, 12, 11, 109) = 30


One number

• Median
• Middle value measures central tendency well
• Not trivial to pull out of a DB

Median(10, 11, 12, 11, 109) = 11


One number

• Mode
• Not often used
• Multi-modal distributions suggest problems

Mode(10, 11, 12, 11, 109) = 11


Other numbers

• A percentile point in the distribution: 95th , 98.5th or 99th
• Used to find out the worst user experience
• Makes more sense if you filter data first

P95th (10, 11, 12, 11, 109) = 12


Other means

• Geometric mean
• Good if your data is exponential in nature
(with the tail on the right)

GMean(10, 11, 12, 11, 109) = 16.68


Wait... how did I get that?

N
ΠN xi — could lead to overﬂow
i=1

ΣN loge (xi )
i=1
N
e — computationally simpler


Other means

And there is also the Harmonic mean, but forget about that


...though consequently

We have other margins of error
• Geometric margin of error
• Uses geometric standard deviation
• Median margin of error
• Uses ranges of actual values from data set
• Stick to the arithmetic MoE
– simpler to calculate, simpler to read and not incorrect


2
Statistics - 2


2-1 Distributions


Let’s look at some real charts


Sparse Distribution


Log-normal distribution


Bimodal distribution


What does all of this mean?


Distributions

• Sparse distribution suggests that you don’t have enough
data points
• Log-normal distribution is typical
• Bi-modal distribution suggests two (or more) distributions
combined


In practice, a bi-modal distribution is not uncommon


Hint: Does your site do a lot of back-end caching?


2-2 Filtering


Outliers

• Out of range data points
• Nothing you can ﬁx here
• There’s even a book about
them


DNS problems can cause outliers

• 2 or 3 DNS servers for an ISP
• 30 second timeout if ﬁrst fails
• ... 30 second increase in page load time
• Maybe measure both and ﬁx what you can
• http://nms.lcs.mit.edu/papers/dns-ton2002.pdf


Band-pass ﬁltering


Band-pass ﬁltering

• Strip everything outside a reasonable range
• Bandwidth range: 4kbps - 4Gbps
• Page load time: 50ms - 120s
• You may need to relook at the ranges all the time


IQR ﬁltering


IQR ﬁltering

Here, we derive the range from the data


Further Reading

lognormal.com/blog/2012/08/13/analysing-performance-data/


Summary

• Choose a reasonable sample size and sampling factor
• Tune sample size for minimal margin of error
• Decide based on your data whether to use mode, median
or one of the means
• Figure out whether your data is Normal, Log-Normal or
something else
• Filter out anomalous outliers


• Philip Tellis

• .com
• philip@lognormal.com
• @bluesmoon
• geek paranoid speedfreak
• http://bluesmoon.info/


Thank you


Photo credits

• http://www.flickr.com/photos/leoffreitas/332360959/ by leoffreitas
• http://www.flickr.com/photos/cobalt/56500295/ by cobalt123
• http://www.flickr.com/photos/sophistechate/4264466015/ by Lisa
Brewster


List of ﬁgures

• http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg
• http://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svg
• http://en.wikipedia.org/wiki/File:KilroySchematic.svg
• http://en.wikipedia.org/wiki/File:Boxplot_vs_PDF.png


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