Positional Averages: Median and Mode Management YouTube Lecture Handouts

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Positional Averages: Median and Mode Management


Median is basically the middle value of the observations after arranging them according to the size i.e.. , either in increasing and decreasing order.


For Ungrouped data:

N + th observation

For Grouped Data


  • I = Lower limit of median Class,
  • N = Number of observations,
  • Cf = cumulative frequency of class preceding the median class,
  • F = frequency of median class,
  • H = class size (assuming class size to be equal)

Related Positional Measures

  • The median divides the series into 2 equal parts. Similarly, there are certain other measures which divide the series into equal parts.
  • These are quartiles, Deciles, and percentiles. Quartiles divide the series into 4 equal parts, deciles into 10 parts whereas Percentiles divide it into 100 equal parts


  • Quartiles basically divide the series in 4 equal parts and each part contains equal observations. In this way there are three quartiles.
  • If a statistical series Is divided into 4 equal parts, then end value of each part is called a quartile.


  • Deciles: They divide the series into 10 equal parts and there are 9 deciles expressed as D1, D2 and each one is called a decile. Here D5 is the median.
  • Deciles are used in finance and economics to divide data into sets for the purpose of analysis.
  • For example, the data set of mutual fund portfolio or the data set of income-tax return filers can be divided for analyzing the top 10 per cent and so on.


  • The percentiles divide the entire series into 100 equal parts and are generally expressed as P.
  • For any series there are 99 percentiles because at the end of every part a percentile is formed.
  • Percentiles indicate the percentage of scores that fall below a particular value. They tell you where a score stands relative to other scores. For example, a person with an IQ of 120 is at the 91st percentile, which indicates that their IQ is higher than 91 percent of other scores.


  • It is denoted by Mo. The value which is repeated the most numbers is basically called the Mode.
  • We can have more than one mode i.e.. , a unimode, Bimode etc.

Ungrouped Data

Formula = most repeated value

Grouped Data

  • Where L = lower limit of the modal class,
  • H = size of the class interval (assuming all class sizes to be equal) ,
  • = frequency of the modal class,
  • = Frequency of the class preceding the modal class,
  • = Frequency of the succeeding the modal class,

Relations between Measures of Central Tendency

  • In symmetrical distribution:
    • Mean = Median = Mode
  • In a positively skewed distribution
    • Mean > Median > Mode
  • In negatively skewed distribution
    • Mean < Median < Mode
  • According to Karl Pearson the relation between three can be expressed as:
    • Mode = 3 Median-2 Mean


According to Karl Pearson, the relation between mean, median and mode is?

A. 2 mode-3 mean = median

B. Mode = 3 median-2 mean

C. Both A and B

D. None of the above

Answer: C

How many percentiles are there in a percentile?





Answer: 3