Statistical analysis is based on the concept that the mean, or average value of a set of data points is equal to the median. This is known in statistics as the normal distribution. The most common types of distributions are mean-on-target (MOT), median-on-target (MED), and mixed distributions (MLR). A normal distribution produces regular distribution, so the data does not lie anywhere else except at the median.
Using a sample size of n, the data is normally distributed in a normal (bell-shaped) curve with the mean being equal to the population’s probability distribution, but with no deviation of one side from the mean. When the sample size is sufficiently large to generate a normal distribution, then the median is the value in the middle of this distribution. For example, the mean of a normal distribution is 100. Therefore, the median would be the value at 50.
When the sample size is small, there is an uneven distribution of the data. This is known as a biased distribution because it is not normal, and it may have a high or low mean, which means that a particular value is likely to occur more often than average.
When the sample size is large enough to generate a normal distribution, then the mean and the median are more likely to lie somewhere near each other. In this case, the median is the value just below the mean, while the mean is the value at the very top of the distribution. In this situation, the median will be close to zero, but above the mean.
The number of values that lie between the median and the mean, or between the middle of the sample size and the mean of the sample size, will be less than 1.5%. If the sample size is large enough to produce a normal distribution, then the mean is likely to be greater than or equal to the median, and the median lies between the mean.
The number of values that lie within the range of the mean and median will be greater than the sample size. These values will fall outside the range of the mean and the range of the median, but are outside the range of the mean of the data. For example, if the sample size is five and the mean is 50, then the mean is likely to be between two and three standard deviations away from the mean. However, when the sample size is five, and the mean is five, then the mean is likely to lie between three and four standard deviations from the mean.
With a larger sample size, the distribution of the data will lie closer to the mean of the sample size. This is called a normal distribution with the mean being smaller than the median.
With a smaller sample size, the range of the data will lie closer to the median than to the mean. This is called a normal distribution with the mean being larger than the median.
Because of these differences in distributions, the median and the mean do not always coincide, even for a large sample size. A relatively small sample can still lie close to one of the mean values, while the mean is farther from the mean. For example, the sample size can be large enough so that the mean is equal to the median or even slightly higher, while still remaining too small to show any sign of a mean line on the curve.
A person who uses a median to predict the value of something is using a normal distribution, where there is a normal distribution of the mean and the range. Using a median to predict something else is more difficult, where there is an irregular distribution of the mean and the range.