# Positional Averages: Median and Mode Management YouTube Lecture Handouts

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## Median

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

Formula

For Ungrouped data:

N + th observation

For Grouped Data

Where

• 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

• 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

• 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.

#### Percentiles

• 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.

## Mode

• 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

## MCQs

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

How many percentiles are there in a percentile?

1.88

2.100

3.99

4.9