Measures of Central Tendency

• Measures of Central Tendency

  A single value that represents a data set, with the purpose of locating the center of a data set, commonly referred to as an average
• Mean


  • A set of data has only one mean
  • Applied for interval and ratio data
  • All values in the data set are included
  • Very useful in comparing two or more data sets
  • Affected by the extreme small or large values on a data set
  • Cannot be computed for the data in a frequency distribution with an open-ended class
• Arithmetic Mean (Mean)


  The only common measure in which all values plays an equal role meaning to determine its values you would need to consider all the values of any given data set
• Mean for Ungrouped Data


  Sum of all values / Number of values
• Population Mean


  Denoted by μ
• Sample Mean


  Denoted by X bar
• Mean for Grouped Data

  Sum of (Frequency x Midpoint) / Total Frequency
• Weighted Mean


  Useful when various classes or groups contribute differently to the total, found by multiplying each value by its corresponding weight and dividing by the sum of the weights
• Geometric Mean


  Used to determine the average percents, indexes, and relatives; to establish the average percent increase in production, sales, or other business transactions or economic series from one period of time to another
• Combined Mean


  The grand mean of all the values in all groups when two or more groups are combined
• Median


  The midpoint of the data array, appropriate measure of central tendency for data that are ordinal or above, but is more valuable in an ordinal type of data
• Properties of Median


  • It is unique, there is only one median for a set of data
  • It is found by arranging the set of data from lowest or highest (or highest to lowest) and getting the value of the middle observation
  • It is not affected by the extreme small or large values
  • It can be computed for an open-ended frequency distribution
  • It can be applied for ordinal, interval and ratio data
• Median for Ungrouped Data

  If n is odd, the median is the middle ranked

  If n is even, then the median is the average of the two middle ranked values
• Mode


  The value in a data set that appears most frequently
• Types of mode


  • Unimodal
  • Bimodal
  • Multimodal
  • No mode
• Properties of mode


  • It is found by locating the most frequently occurring value
  • It is the easiest average to compute
  • There can be more than one mode or even no mode in any given data set
  • It is not affected by the extreme small or large values
  • It can be applied for nominal, ordinal, interval and ratio data
• Midrange


  The average of the lowest and highest value in a data set
• Properties of midrange


  • It is easy to compute
  • It gives the midpoint
  • It is unique
  • It is affected by the extreme small or large values
  • It can be applied for interval and ratio data
• Effects of changing units on mean and median

  If a constant k is added to each observation, the mean increases by k and the median increases by k

  If each observation is multiplied by a constant h, the mean is multiplied by h and the median is multiplied by h
• The mode is the most frequently occurring score in a set of scores.
• The median is the middle number when data are arranged in ascending or descending order.
• The mean is the average value.