Median

A median is a measure of central tendency which divides data into $2$ parts, separating the upper and lower half of the data by a value which is called the median value. Before calculating a median, we need to arrange all the values in an ascending order. After that, we can calculate the median for different types of series in different ways:

• Individual Series: Arrange all the observations in an ascending order. See whether the number of observations is an odd or even number. If the number is odd, use the formula $$\left(\frac{N+1}{2}\right)^\text{th}$$ observation, where $N$ is the number of observations. When the number of observations are even, the median is calculated by taking the average of the $$\left(\frac{N}{2}\right)^{\text{th}}$$ observation and the next observation.

• Discrete Series: Take the cumulative frequency for all the observations by successively adding the previous frequencies and then use the same formula as stated above in individual series.

• Continuous Series: The formula is

$$l+\frac{\frac{N}{2}-cf}{f}\times h,$$ where

• $l$ represents the lower limit of the median class interval;
• $N$ represents the total number of frequency;
• $cf$ represents the cumulative frequency of the class interval preceding the median class;
• $f$ represents the corresponding frequency of the median class interval;
• $h$ represents the width of the median class interval.

For more information on the types of statistical series, see Statistical Series.

What is the median of the following distribution: $$1,2,3,5,6,8,9?$$

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Since the number of observations is odd, use the formula $\left(\frac{N+1}{2}\right)^{\text{th}}$. Here $N=7$. Putting this in the formula, we get that the median is the $4^{\text{th}}$ observation which is $5$. So, the median is $5$. $_\square$

What is the median of the following distribution: $$1,2,3,5,6,8,9,11?$$

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Since the number of observations are even in number, we have to take the average of the $\left(\frac{N}{2}\right)^{\text{th}}$ observation and the next observation. Here, $N=8$. So, the average of the $4^{\text{th}}$ and $5^{\text{th}}$ observations is the median of this distribution. So, the median is $$\frac{5+6}{2}=5.5. \ _\square$$