# How to express error as a percentage?

I have data about how many unique users do a certain thing for each day of a month. I can average it, and i would like to display the variation in a intuitive format (such as % of something).

Is there a standard way of doing this?

I've found standard error, which is $\frac{\sigma}{\sqrt{n}}$, which is not particularly intuitive for users of the data.

EDIT: in response to the answers, it's for building out a analytics dashboard for use by the entire company (so many people probably don't understand standard deviation). In particular, we are doing A/B testing for various metrics over a period of say 1 month. We basically would average all the metric per day to give a number for that period, but there are variations day-to-day and would like to have some good way of expressing that.

It may be meaningful to just divide by the average. E.g. the average number is 1000 and the std is 200, so in a sense this means the actual number can vary by 20% from the baseline.

Also, if you could say who the user of the data are and what they are doing with it, it might be useful.

• hey i edited the question for a more specific use case for the data – FurtiveFelon Feb 28 '11 at 22:36
• What if the mean is 0? – rm999 Feb 28 '11 at 23:13
• If your data is such that the mean can be zero, then this is not a good solution. I.e. it is not meaningful to divide by the mean. This is even if the actual mean is not zero. E.g. you average something like the "x-position" of points. The mean here can be zero, and I'd argue dividing by the mean is not meaningful, even if in your particular sample the mean is not 0. – SheldonCooper Feb 28 '11 at 23:44

The standard error of the mean tells you how precise your estimate of the mean is; that doesn't seem to capture what you're trying to do. I would use either a) a histogram, if you care mostly about showing variation, or b) a line chart or area chart, if you want to say something about variation while also showing progression over time.

Assuming your distribution remotely resembles a normal curve, you could convert the standard errors into a more intuitive percentage value pretty easily. For example, if the distribution is pretty normal, approximately 95% of your population falls within +/-1.96*SE of the mean. Building from SheldonCooper's sample values, you could say, "The average was 1000 and about 95% of the population was between 600 and 1400." Likewise, about 70% of the population falls within +/- 1*SE, etc.

If your sample distribution tends to deviate from normal by a lot, don't despair, but try to provide more details so we can help.

• Hey i edited the question to provide more details, I don't know anything about the distribution of the data, though most likely normal if i have to guess. – FurtiveFelon Feb 28 '11 at 22:37
• Thanks for clarifying. If you're dealing with people's behavior on the Internet, the distribution is probably far from normal. In fact, you're likely to find very skewed distributions on almost any measure (that's the fun of the long tail!). I think you're instinct (mean + standard errors in some intuitive fashion) is great. The visualization suggestions others have made should be useful too. You might consider also presenting information about just how unequal the distribution is as well as data about "average" users at the bottom and top of the distribution too. – ashaw Mar 1 '11 at 3:08

As @rolando2 mentioned, the histogram might be a tool for displaying the variations; and also, as @ashaw stated, you might need to find where 95% percentage of number go to, then, you can probably just use box plot to generate the basic features of your dataset, and put the data to the dashboard that is shown in the company without the box plot. For box plot, you might check here:

http://en.wikipedia.org/wiki/Box_plot


Hope this helps, at least for inspiration...

$$\text{Percent Error} = \frac{\text{Error}}{\text{Theoretical value}}\times 100$$

• Would you mind providing more detailed explanation, especially with regard to the updated question (A/B testing)? – chl Jul 19 '12 at 8:49

The more intuitive form to express the Standard Error in terms of percent is called Relative Standard Error $(RSE)$ which is the expression of the Standard Error $(SE)$ as percent, this is the formula: $$RSE\%=\frac{SE}{\bar{x}}\times 100$$ where,

$RSE\%$ is the Relative Standard Error in percent
$SE$ is the Standard Error
$\bar{x}$ is the mean value of the sample

and,

$$SE=\frac{\sigma}{\sqrt{n}}$$

where, $\sigma$ is the sample standard deviation, and $n$ is the size (number of observations) of the sample.

I was dealing time before with the same problem because I had many variables with different units, and I found the $RSE$ as a good alternative to show my results. I hope it can be useful for you too. You can find additional information here: Australian Bureau of Statistics, and Investopedia.

Regards