Measures of Central Tendency

Definition of Measures of Central Tendency
A Measures of Central Tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location.

The mean, median and mode are all valid measures of central tendency but, under different conditions, some measures of central tendency become more appropriate to use than others.

• These are statistical terms that try to find the “center” of the numbers that are in a group of data
• This is a method of calculating
• The average of a set of data
• The average represents the center of the distribution
• These types of statistics are descriptive which means they seek to summarize the data

Principal of Measures of Central Tendency

• Mean
• Median
• Mode

The measures for a sample can differ from those for the whole population.

Here are the examples with solutions :

Mean

The sum of the data points or values in a set of data divided by the number of data points or values in the set (the average).

• Commonly called the average or the arithmetic mean
• Add up all of the values to find a total
• Divide the total by the number of values you added together
• The data have no extreme values (values that are much greater or much less than the rest of the data)

Advantages and Disadvantages of the Measures in Mean

• Defined algebraically
• Stable from sample to sample
• But usually does not actually occur in the data
• And heavily influenced by outliers

Formula

 

This is because in statistics, samples and populations have very different meanings and these differences are very important, even if in the case of the mean, they are calculated in the same way. 
To acknowledge that we are calculating the population mean and not the sample mean, we use the Greek lower case letter “mu”, denoted as  :

Example

Median

The middle number or value when the values are ranked in order in a set of data.

• The midpoint of a data set
• Put the numbers in numerical order and identifying the middle number
• If there are two middle values, then the median is the mean of these two middle values
• The data have extreme values
• There can only be one median

Advantages and Disadvantages of the Measures in Median

• Unaffected by extreme scores
• Usually its value actually occurs in the data
• But cannot be entered into equations, because there is no equation that defines it
• And not as stable from sample to sample, because dependent upon the number of scores in the sample

Example

Mode

The mode is the most frequently (common or popular) occurring value in a set of data.

• Count how many of each value appears
• The value that appears the most
• You can have more than one mode
• Data have many repeated numbers
• It is possible to have no mode

Advantages and Disadvantages of the Measures in Mode

• Typically a number that actually occurs in data set
• Has highest probability of occurrence
• Applicable to Nominal, as well as Ordinal, Interval and Ratio Scales
• Unaffected by extreme scores
• But not representative if multimodal with peaks far apart

Example

References :

• http://www.personal.kent.edu/~mtmoore1/Quant2CentralTendency.ppt
• https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php