### Maths Median

##### What is Median?

The median is a statistical measure that represents the middle value of a dataset when assorted in the order from smallest to largest. It is a measure of central tendency, along with the mean and mode.

##### Median Definition

**Median** is a statistical measure that represents the middle value of a dataset when assorted in the order from smallest to largest. It is a measure of central tendency, along with the mean and mode.

##### Calculating the Median

To calculate the median, follow these steps:

**Arrange the data in ascending order.**This means putting the smallest value first and the largest value last.**If there is an odd number of data points, the median is the middle value.**For example, if the data set is {1, 3, 5, 7, 9}, the median is 5.**If there is an even number of data points, the median is the average of the two middle values.**For example, if the data set is {1, 3, 5, 7}, the median is (5+7)/2 = 6.

##### Properties of the Median

The median has several properties that make it a useful measure of central tendency:

**The median is not affected by outliers.**This means that a few extremely large or small values will not affect the median.**The median is a more robust measure of central tendency than the mean.**This means that the median is less likely to be affected by changes in the data.**The median can be used with both continuous and discrete data.**This makes it a versatile measure of central tendency.

The median is a useful measure of central tendency that is not affected by outliers and is more robust than the mean. It can be used with both continuous and discrete data and has a variety of applications.

##### Median Example

Median is the middle value of a given set of numbers when assorted in the order from smallest to largest. If there is an even number of values, then the median is the average of two middle values.

##### Example 1: Finding the Median of a Set of Numbers

Let’s consider a set of numbers: 2, 4, 6, 8, 10.

**Step 1: Assort the numbers in the order from smallest to largest.**

2, 4, 6, 8, 10

**Step 2: Determine the middle value.**

Since there are five numbers in the set, the middle value is the third one.

**Step 3: The median is the middle value.**

Therefore, the median of the given set of numbers is 6.

##### Example 2: Finding the Median of an Even Number of Values

Let’s consider a set of numbers: 1, 3, 5, 7, 9, 11.

**Step 1: Assort the numbers in the order from smallest to largest.**

1, 3, 5, 7, 9, 11

**Step 2: Determine the two middle values.**

Since there are six numbers in the set, the two middle values are the third and fourth ones.

**Step 3: Calculate the average of the two middle values.**

The average of 5 and 7 is (5 + 7) / 2 = 6.

**Step 4: The median is the average of the two middle values.**

Therefore, the median of the given set of numbers is 6.

##### Median Formula

The median is the middle value in a set of data when the data is arranged in order from smallest to largest. If there is an even number of data points, the median is the average of the two middle values.

##### Median Formula for Ungrouped Data

The median is similar to a type of average, that we utilised to locate the central value. Ungrouped data means the data which is not grouped or arranged properly. To work with such a group of data the below points are to be followed:

- We will start with arranging the data in increasing or decreasing order.
- Next thing is to calculate the total number of observations(denoted by n).
- Lastly, check if the number of observations estimated is even or odd. Depending on them we will apply the formula.

Check out this article on Variance and Standard Deviation.

##### Median Formula When n is Odd

The formula when the given set of numbers is odd or when the total number of observations tends to be odd is:

$$\text{Median} = \left(\frac{n+1}{2}\right)^\text{th term}$$

##### Median Formula When n is Even

An opposite case to the above condition, if the total number of observations tends to be even, then the formula is:

$$\text{Median} = \frac{\left(\frac{n}{2}\right)^\text{th term} + \left(\frac{n}{2}+1\right)^\text{th term}}{2}$$

##### Median Formula for Grouped Data

To compute the median of a grouped or continuous frequency distribution, we follow the below steps:

- For the total number of observations(n) obtain the class size(h), followed by dividing the data into separate classes.
- Next, compute the cumulative frequency per class and locate the class where the median would fall(i.e. n/2).
- Get the value of the l-lower limit of the median class and the value of the c- cumulative frequency of the class that precedes the median class.
- End up the calculation by substituting the values in the below formula.

$$\text{Median} = l + \frac{\left(\frac{n}{2}\right) - c}{f} \times h$$

Where ‘l’ is the lower limit of the median class, ‘f’ denotes the frequency of the median class, ‘h’ is the width of the median class, ‘c’ denotes the cumulative frequency of the class preceding the median class.

##### Application of Median Formula

The median is a statistical measure that represents the middle value of a dataset when assorted in the order from smallest to largest. It is often used to describe the typical value of a dataset that may have outliers or extreme values. The median is not affected by outliers, making it a robust measure of central tendency.

Here are some applications of the median formula:

**1. Finding the middle value of a dataset:**
The most straightforward application of the median formula is to find the middle value of a dataset. This can be useful for quickly identifying the typical value of a dataset without having to calculate the mean or other measures of central tendency.

**2. Comparing datasets:**
The median can be used to compare different datasets to identify similarities and differences. For example, the median income of two different countries can be compared to determine which country has a higher standard of living.

**3. Identifying outliers:**
The median can be used to identify outliers in a dataset. Outliers are values that are significantly different from the rest of the data. By identifying outliers, you can gain insights into the underlying distribution of the data and potential sources of error.

**4. Making predictions:**
The median can be used to make predictions about future values. For example, the median sales price of homes in a particular area can be used to predict the future sales price of a similar home in the same area.

**5. Statistical analysis:**
The median is used in various statistical analyses, such as regression analysis and hypothesis testing. It can also be used to calculate other statistical measures, such as the interquartile range and the coefficient of variation.

**6. Data visualization:**
The median can be used to create data visualizations, such as box plots and histograms. These visualizations can help to understand the distribution of data and identify patterns and trends.

**7. Quality control:**
The median can be used for quality control purposes to identify errors or inconsistencies in data. For example, the median weight of a product can be monitored to ensure that it meets the specified standards.

**8. Decision-making:**
The median can be used to support decision-making processes. For example, the median salary of a particular job role can be used to determine the appropriate salary range for a new hire.

**9. Risk management:**
The median can be used to assess and manage risks. For example, the median loss amount for a particular type of insurance policy can be used to calculate the appropriate premium rates.

**10. Research and analysis:**
The median is widely used in research and analysis across various fields, including economics, finance, healthcare, social sciences, and natural sciences. It provides valuable insights into the central tendency and distribution of data, helping researchers and analysts draw meaningful conclusions.

In summary, the median formula has numerous applications in various fields and disciplines. It is a versatile statistical measure that provides valuable insights into the typical value, distribution, and characteristics of data.

##### Median FAQs

##### What is the median?

The median is the middle value in a set of data when the data is arranged in order from smallest to largest. If there is an even number of data points, the median is the average of the two middle values.

##### How do you find the median?

To find the median, follow these steps:

- Arrange the data in order from smallest to largest.
- If there is an odd number of data points, the median is the middle value.
- If there is an even number of data points, the median is the average of the two middle values.

##### What is the difference between the mean and the median?

The mean is the average of all the values in a set of data. The median is the middle value in a set of data when the data is arranged in order from smallest to largest.

The mean can be affected by outliers, which are extreme values that are much larger or smaller than the rest of the data. The median is not affected by outliers.

##### When should I use the median?

The median is a good measure of central tendency when there are outliers in the data. The mean is a good measure of central tendency when the data is normally distributed.

##### Examples of the median

Here are some examples of the median:

- The median of the set of numbers 1, 2, 3, 4, 5 is 3.
- The median of the set of numbers 1, 2, 3, 4, 5, 6 is 3.5.
- The median of the set of numbers 1, 2, 3, 4, 5, 6, 7 is 4.

##### Conclusion

The median is a useful measure of central tendency that is not affected by outliers. It is often used in place of the mean when there are outliers in the data.