When to use mean or median?

Question

I am asked to put together some KPIs. I'm breaking down what the general lifespan of our subscribers is by industry.

There are some industries I feel outliers are impacting what the average would be, so I figure I should probably use median. But there are others where I think average might be a better course of action. What would be the best way to tell if I'm better off using average or median? Or if there's something else you think would be better, feel free to share.

In my Excel workbook, I actually did both average and median for each category. The issue is, for some of them, there is a WIDE gap in the difference.

For example, I have one category with a sample size of nearly 6000 where the average is 129, but the median is 162. I should also note that the sample size by industry can vary wildly. Some industries the same size is in the thousands while others might be as low as 100.

Here is a snapshot of the workbook:

Industry	Sample Size	Median Tenure	Average Tenure
Industry 1	5973	62	129
Industry 2	2090	90	155
Industry 3	1862	62	135
Industry 4	2491	161	111
Industry 5	2740	62	121

Using mean (and standard deviation) implies that the distribution is Normal. It might be that your data is far from being distributed according to the normal distribution. A plot, as suggested by @jlemaitre, would shed a light. — Vladislavs Dovgalecs, Nov 04 '15 at 22:47
Right, but the issue is, its a lot of different industries. Hundreds of them. So that would be rather tedious. — wizkids121, Nov 04 '15 at 22:57
Sorry, but @xeon's statement is quite incorrect. Mean and SD can be calculated for non-normal distributions and (some mathematical subtleties apart) are almost always well defined. It's a different point that mean and SD may not be the best way to summarize particular samples in practice, but much depends on purpose. Using the mean to summarize exponential and Poisson distributions is for example utterly standard. — Nick Cox, Nov 04 '15 at 23:10
@NickCox Now it is my turn to ask questions. Yes, mean and SD can be computed for just any sample. Another question whether those parameters describe the distribution (sufficient statistic?). My data can be exponentially distributed for example and I can compute mean and SD. Are those values useful? I don't think so. Perhaps I miss something. — Vladislavs Dovgalecs, Nov 04 '15 at 23:16
Look through any text on distributions and you will see mean and SD discussed routinely for non-normal distributions. The exponential is conventionally always parameterised in terms of either the mean or its reciprocal. The Poisson is always parameterised in terms of the mean and in fact its median is very awkward to work with. I agree naturally that with many skewed or long-tailed samples, indeed virtually all data, it is prudent to work with median as well as mean, and perhaps other summaries, but your statement that using mean and SD implies normality is false. — Nick Cox, Nov 04 '15 at 23:23
Picture's worth a thousand words. Plot the distributions by industry. — jlemaitre, Nov 04 '15 at 22:35
@xeon See my question, and the responses to it here Should the mean be used when the data are skewed. — Alexis, Mar 21 '16 at 04:59

score 3 · Answer 1 · answered Nov 04 '15 at 23:29

Honestly, the best case scenario is to present both mean and median. Doing so provides an accurate picture of your data, even if you don't include a picture.

If you can only provide one, remember that these are both measures of center. Which number better measures the center of your data? Generally, I would compute both and if there are many cases where the mean is far from the median, then report the median for every group. However, if there are not many cases where the two are far apart, then I would report the mean for every group.

score 2 · Answer 2 · answered Nov 04 '15 at 23:08

2

When the mean and the median are different, data distribution is said to be skewed. In the example you mentioned, the median (162) is larger than the mean (129), which means there are more observations towards larger values than towards smaller ones. In your context, this means that the lifespan of the subscribers in that industry tend to be larger, with a few low-valued observations pulling down the value of the mean.

This may help you.

answered Nov 04 '15 at 23:08

Samuel Barbosa

113

Thank you for that insight, but I still have a problem: How do I know which one I should be using? – wizkids121 Nov 04 '15 at 23:11
When the gap between them is very high, this means there are really few values influencing the mean. In that case, use the median. You could measure a percentage gap from the median by abs(median - mean) / median, for example. – Samuel Barbosa Nov 04 '15 at 23:12
3

You can get skewed distributions even if the mean and median are the same. – Nick Cox Nov 04 '15 at 23:13
Shoot I messed up my initial post. The median is actually 62, not 162. None the less, the percentage gap from the median is 1.079. What would that generally infer? – wizkids121 Nov 04 '15 at 23:16
@NickCox I know. Notice that I argued the other way around. – Samuel Barbosa Nov 04 '15 at 23:17
Shoot I messed up my initial post. The median is actually 62, not 162. None the less, the percentage gap from the median is 1.079. What would that generally infer? – wizkids121 Nov 04 '15 at 23:22
Good; but I don't want readers to jump to the wrong conclusion. – Nick Cox Nov 04 '15 at 23:25
@wizkids121 This means you have some observations deviating the mean from the median in aproximately 107,9% the value of the median. Do you have all the raw data? You would be way better off by using mean, median and SD. – Samuel Barbosa Nov 04 '15 at 23:28
Yes, I do have all the raw data, but how would SD come in to play in figuring out what the optimal way to use the data would be? – wizkids121 Nov 04 '15 at 23:33
@SamuelBarbosa Saying that you should only use the mean when the data aren't skewed is just an underhanded way of saying only use the median. – Alexis Mar 21 '16 at 05:02

When to use mean or median?

2 Answers2