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Measures of Central Tendency

 

There are three primary measures of central tendency: the mean, the median,  and the mode. 

The mean:

  • Also called the average, this is taken by adding the scores together and dividing by the number of scores. 
  • Uses every score and tends to be stable in different samples but can be influenced by extreme scores and skewed distributions, and can only be used with interval and ratio data.

The median:

  • The middle score when scores are ordered by magnitude.
  • (1) Arrange the scores in increasing magnitude, (2) find the position of the middle score by counting the scores, adding one to that value, and dividing by two, (3) find the score at the position just calculated. NOTE: If even number of scores, take the average of the two middle scores.
  • Relatively unaffected by skewed distributions and extreme scores. Can be used with ordinal, interval, and ratio data.

The mode: 

  •  The score that occurs most frequently in a dataset
  •  To find: (1) Arrange data in sequential order, (2) count the number of times each score occurs, and (3) identify the score that occurs the most. 
  • Useful for nominal data but can sometimes have 2 values (I.e, bimodal distribution) or more (I.e., multimodal distribution). 

Skew

Before deciding on the measure of central tendency, it is important to examine the frequency distribution of your data. This is because the best measure of central tendency depends on the distribution, as well as the type of data. 

In a normal distribution, the frequency distribution is symmetrical and appears like a bell. It is also called a bell curve. However, not all the data is normally distributed and this impacts the accuracy of certain types of central tendency. 

The aspects that impact the frequency distribution include skew and kurtosis. 

Skew is the lack of symmetry in a distribution, where a cluster of scores appears on one side or the other. If scores are clustered towards the lower end, and the tail points to the higher end, this is a positive skew. If scores are clustered towards the higher end, and the tail points to the lower end, this is a negative skew.  The image to the left is an example of a positively and negatively skewed distribution. 

Kurtosis is the pointyness of the frequency distribution. If the distribution has heavy tails and is pointy, it has a positive kurtosis and called leptokurtic. If the tails are thin and it is relatively flat, it has a negative kurtosis and called platykurtic. Examples are to the right. 

Distributions should have no skew or excess kurtosis.  The further from zero the skewness or excess kurtosis is, the further the frequency distribution is from zero. 

There are statistical tests, such as the Kolmogorov-Smirnov test and the Shapiro-Wilk test, that determine whether the skewness or kurtosis is significantly different from zero. If a frequency distribution is found to have significant skew or kurtosis, that needs to be considered when choosing a measure of central tendency. 

Click here for a link about testing for normality.

Kurtosis

Identifying the Measure of Central Tendency

For interval or ratio data, if there is a non-normal or highly-skewed distribution, or if the data has extreme values, use the median. If the data is normally distributed and has interval or ratio data, use the mean. Ordinal data uses the median. Nominal or categorical variables use the mode.