## Measures of Central Tendency: Mode, Median, Mean, and Midrange

This section explores the measures of central tendency: mode, median, mean and midrange. It shows you the commands for computing these metrics using a spreadsheet program and it gives you the correct format for entering those commands into the spreadsheet program.

### Mode

The mode is the value that occurs most frequently in the data. Spreadsheet programs such as Microsoft Excel or OpenOffice.org Calc can determine the mode with the function MODE.

=MODE(data)

In the Fall of 2000 the statistics class gathered data on the number of siblings for each member of the class. One student was an only child and had no siblings. One student had 13 brothers and sisters. The complete data set is as follows:

1,2,2,2,2,2,3,3,4,4,4,5,5,5,7,8,9,10,12,12,13

The mode is 2 because 2 occurs more often than any other value. Where there is a tie there is no mode.

For the ages of students in that class

18,19,19,20,20,21,21,21,21,22,22,22,22,23,23,24,24,25,25,26

...there is no mode: there is a *tie* between 21 and 22, hence there no single must frequent value. Spreadsheets will, however, usually report a mode of 21 in this case. Spreadsheets often select the first mode in a multi-modal tie.

If all values appear only once, then there is no mode. Spreadsheets will display #N/A or #VALUE to indicate an error has occurred - there is no mode. *No mode* is NOT the same as a mode of zero. A mode of zero means that zero is the most
frequent data value. Do not put the number 0 (zero) for "no mode." An example of a mode of zero might be the number of children for students in statistics class.

### Median

The median is the central (or *middle*) value in a data set. If a number sits at the middle, then it is the median. If the middle is between two numbers, then the median is half way between the two middle numbers.

For the sibling data...

1,2,2,2,2,2,3,3,4,4,|4|,5,5,5,7,8,9,10,12,12,13

...the median is 4.

Note the data must be in order (sorted) before you can find the median. For the data 2, 4, 6, 8 the median is 5: (4+6)/2.

The median function in spreadsheets is MEDIAN.

=MEDIAN(data)

### Mean (average)

The mean, also called the arithmetic mean and also called the average, is calculated mathematically by adding the values and then dividing by the number of values (the sample size n).

If the mean is the mean of a population, then it is called the population mean μ. The letter μ is a Greek lower case "m" and is pronounced "mu."

If the mean is the mean of a sample, then it is the sample mean x. The symbol x is pronounced "x bar."

The sum of the data ∑ x can be determined using the function =SUM(data). The sample size n can be determined using =COUNT(data). Thus =SUM(data)/COUNT(data) will calculate the mean. There is also a single function that calculates the mean. The function that directly calculates the mean is AVERAGE

=AVERAGE(data)

**Resistant measures:** One that is not influenced by extremely high or extremely low data values. The median tends to be more resistant than mean.

#### Population mean and sample mean

If the mean is measured using the whole population then this would be the population mean. If the mean was calculated from a sample then the mean is the sample mean. Mathematically there is no difference in the way the population and sample mean are calculated.

### Midrange

The midrange is the midway point between the minimum and the maximum in a set of data.

To calculate the minimum and maximum values, spreadsheets use the minimum value function MIN and maximum value function MAX.

=MIN(data)

=MAX(data)

The MIN and MAX function can take a list of comma separated numbers or a range of cells in a spreadsheet. If the data is in cells A2 to A42, then the minimum and maximum can be found from:

=MIN(A2:A42)

=MAX(A2:A42)

The **midrange** can then be calculated from:

midrange = (maximum + minimum)/2

In a spreadsheet use the following formula:

=(MAX(data)+MIN(data))/2

Source: Dana Lee Ling, http://www.comfsm.fm/~dleeling/statistics/text5.html

This work is licensed under a Creative Commons Attribution 4.0 License.