Refresher - The Mean and the Median

The two most common measures of central tendency are the median and the mean, which can be illustrated with an example. Suppose we draw a sample of five students and measure their weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150 pounds.

 

The Median

• To find the median, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values. Thus, in the sample of five students, the median (after rearranging in increasing order) would be 130 pounds; since 130 pounds is the middle weight.

Basically, the median of a set of N numbers ${\displaystyle x_1,x_2,x_3,...,x_N}$ arranged in the order of magnitude is either:

The middle value if number of observations is odd.

Mean of the two middle values if number of observations is even.

Geometrically, the median is the value corresponding to the vertical line which divides a histogram into two parts having equal areas

When statisticians talk about the mean of a population, they use the Greek letter ${\displaystyle \mu }$ to refer to the mean score. When they talk about the mean of a sample, statisticians use the symbol ${\displaystyle \overline{x}}$ to refer to the mean score.

 

The Arithmetic Mean (Mean)

An Average is a value that is typical, or representative of a set of data. Several types of averages can be defined, the most common being the arithmetic mean, the median and the mode.

• The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations. Returning to the example of the five students, the mean weight would equal (100 + 100 + 130 + 140 + 150)/5 = 620/5 = 124 pounds. In the general case, the mean can be calculated, using one of the following equations:
  
  Population mean ${\displaystyle \mu =\frac{\Sigma x}{n}}$
  
  and sample mean is ${\displaystyle \overline{x}=\frac{\Sigma x}{n}}$
  
  where ${\displaystyle \Sigma x}$ is the sum of all the population observations, N is the number of population observations, ${\displaystyle \Sigma x}$ is the sum of all the sample observations, and n is the number of sample observations. In other words,
  
  The mean of a set of N numbers ${\displaystyle x_1,x_2,x_3,...,x_N}$ is denoted by and is defined to be:
  ${\displaystyle \overline{x}=\frac{x_1,x_2,x_3,...,x_N}{N}}$ or ${\displaystyle \overline{x}=\frac{\sum _{i-1}^N{x}^i}{N}}$ or simply ${\displaystyle \overline{x}=\frac{{\sum }^x}{N}}$