When is median higher than mean

The mean is the generally understood "average", where the sum of the values is divided by the number of values sometimes referred to as the count of the values. How can we set up a set of values so that the median is higher than the mean? We can do it by taking a set of numbers and skewing the values to be very low below the median and just above the median. For instance, if I take a set of five numbers and set the middle value as 10, I can place the two lower values at 1 and 2 and the higher values at In fact, the mean will be lower than the median in any distribution where the values "fall off", or decrease from the middle value faster than they increase from the middle value.

How can a median be greater than the mean? Jan 29, Hello Friends, I recently started learning Data Science modules, of which started with Statistics basics. I came across a situation where i tried to draw Summary Statistics for all the employees time sheet entry in in hours.

I just took 1 week data and below is the summary statistics. Average Dev I could understand that this is a -vely skewed. The histogram for the data: 4 5 6 6 6 7 7 7 7 8 is not symmetrical.

A distribution of this type is called skewed to the left because it is pulled out to the left. The mathematical formula for skewness is:. The greater the deviation from zero indicates a greater degree of skewness. If the skewness is negative then the distribution is skewed left as in Figure. A positive measure of skewness indicates right skewness such as Figure. The mean is 6. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so.

The histogram for the data: 6 7 7 7 7 8 8 8 9 10 , is also not symmetrical. It is skewed to the right. The mean is 7. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most. To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode.

If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean. As with the mean, median and mode, and as we will see shortly, the variance, there are mathematical formulas that give us precise measures of these characteristics of the distribution of the data.

Again looking at the formula for skewness we see that this is a relationship between the mean of the data and the individual observations cubed. Formally the arithmetic mean is known as the first moment of the distribution. The second moment we will see is the variance, and skewness is the third moment. The variance measures the squared differences of the data from the mean and skewness measures the cubed differences of the data from the mean.

While a variance can never be a negative number, the measure of skewness can and this is how we determine if the data are skewed right of left. In many cases, the date of diagnosis is close to the time of reporting, i. However, the study group often also includes patients who have been suffering from the disease for many years. If we calculate the mean of the individual time spans since disease onset, such large values have an enormous impact, making the mean larger than the actual distribution of data would suggest.

Therefore, here the median gives a more realistic picture of the data. If they are not too different , use the mean for discussion of the data, because almost everybody is familiar with it. If both measures are considerably different, this indicates that the data are skewed i.

Stuck in the middle — mean vs. As an example, let us consider the following five measurements of systolic blood pressure mmHg : , , , , The median is defined as the value which is located in the middle, i.

Mean vs. So which one should we use?