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What is the mean and the median?

What does it indicate about a data set if the mean is lower than the median?

  • I just want some analysis on what sort of conclusions one can draw about a data set based on its mean and median values.

  • Answer:

    Think of a data set with three items in it. 9, 10, and 11. In this case the mean and the median are both 10. If you start increasing the highest number, 11, the mean jumps ahead of the median. Let's say you have 9,10, 1000. This is basically the "Bill Gates walks into a bar and make the average person a millionaire" affect. Now 10, and 11 constant and let's lower the 9, so let's say we have 0, 10, 11. Now the median is still 10 but the mean is going down. So in general (though not necessarily), if the median is lower then the mean, you have major outliers in the high end of the distribution, and if the mean is lower then median you have major outliers in the low end. In the case of income distribution for example, I don't think there's any country where the mean income is lower then the median, because the income distribution always looks like a very large amount of poor and middle class people with a very small super-rich minority, hence major outliers on the high end. A society where the median is higher then the mean would consist of a very large middle class, no extreme wealth, and a small and extremely poor minority.

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This might suggest that the data is skewed to the left. Loosely speaking, the entries lower than the median are farther away from the median than the entries higher than the median. In other words, the "left tail" is longer. See http://en.wikipedia.org/wiki/Nonparametric_skew for a measure of skewness using mean, median and standard deviation (note that this is not the most common way to define skewness). If the mean is lower than the median, then the nonparametric skew is negative. This is perhaps the strongest conclusion you can get from only comparing the mean and the median.

Ivan Li

Generally, not very much.  In small sample sizes, outliers can throw off means.  For example if your data set were test scores and some students received a null score for an absence, then that would lower the mean.  In this sense, if the data is independent and identically distributed, then the points may be skewed. http://en.wikipedia.org/wiki/Skewness

Ricky Kwok

In simple terms, it means that most values in the data set are higher than average. e.g, if the mean height of a group of people is 1.6m, and the median is 1.8m, then it means that most people in the group are above average, or >1.6m in the height. It also means by extension that the shorter people in the group are REALLY short - short enough to drag the mean down to 1.6 despite there being more people who are taller.

Hari Shankar

Let's make an example: We have a class with 28 students and for each student there is a given grade. (I do my calculations: from an assignment) We get: Mean = 55, SD = 18, Median = 56 1. If we have a median that is higher than the average it means there will be a huge distance between the good grades and the 'rubbish' grades. 2. If we have a lower average than median the 'rubbish' grades will be dominant and draw the average down 3. If they are close to one another (like this sample) then we have a very systemetically distribution

Frederik Leones

Jay Verkuilen

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