Median Formula

The median formula in statistics refers to the formula used to determine the middle number in the given data set arranged in ascending order. According to the formula count, the number of items in the data set adds with one. Therefore, the results will be divided by two to derive the place of the median value, i.e., the number placed on the identified position will be the median value.

It is a tool to measure the center of a numerical data set. It summarizes large amounts of data into a single value. One can define it as the middle number of a group of numbers sorted in ascending order. In other words, a sample median formula gives the number that would have the same amount of numbers both above and below it in the specified data group. It is a commonly used measure of data sets in statistics and probability theory.

Median is a statistical measure that denotes the middle or center value within a dataset when it is arranged in either ascending or descending order. Unlike the mean, which happens to be the arithmetic average of a dataset, the median remains unaffected by extreme values or outliers in data.

However, to calculate the median, the dataset has to be first organized in ascending or descending order. If the number of observations is odd, the middle number is directly identified as the median. However, if the number of observations is an even number, the average of the two middle values is considered the median.

The significance of this statistical measure is that it can provide a representative measure of a dataset’s central location. This is particularly useful where the dataset has outliers or has extreme values like income distribution for instance. Even though the income gap between the richest percentile and the poorest percentile within a country is huge, the median as a tool can be perfect for determining the central income of the country.

It is useful in fields such as economics, healthcare, and even education. Wondering how? For instance, the median formula statistics applied in education helps identify the middle-performance level amongst hundreds, if not thousands, of students appearing for the same exam.

Let us understand the formula that shall help us understand the intricate details of a sample median formula:

Where,

‘n’ is the number of items in the data set, and ‘th’ signifies the (n)th number.

Now that we understand the basics and intricacies of the concept, let us understand the practicality of the concept through the examples below.

Example #1

List of numbers: 4, 10, 7, 15, 2. Calculate the median.

Solution: Let us arrange the numbers in ascending order.

In ascending order, the numbers are: 2,4,7,10,15

There are a total of 5 numbers. Median is (n+1)/2th value. Thus, the Median is (5+1)/2th value.

Median = 3^rd value.

The 3^rd value in list 2, 4, 7, 10, 15 is 7.

Thus, the median is 7.

Example #2

Suppose there are 10 employees in an organization, including the CEO. CEO Adam Smith believes that the salary drawn by the employees is high. Therefore, he wants to gauge the salary drawn by the group and hence make decisions.

Mentioned below is the salary given to the employees in the firm. Calculate the median salary. The salaries are $5,000, $6,000, $4,000, $7,000, $8,000, $7,500, $10,000, $12,000, $4,500, $10,00,000

Solution:

Let us first arrange the salaries in ascending order. Salaries in ascending order are:

$4,000, $4,500, $5,000, $6,000, $7,000, $7,500, $8,000, $10,000, $12,000, $10,00,000

Therefore, the calculation of the median will be as follows:

Since there are 10 items, the median is (10+1)/2 th item. Median = 5.5^th item.

Thus, the median is the average of the 5th and 6th items. For example, the 5th and 6th items are $7,000 and $ 7,500.

= ($7,000 + $7,500)/2 = $7,250.

Thus, the median salary of 10 employees is $7,250.

Example #3

Jeff Smith, the CEO of a manufacturing organization, needs to replace seven machines with new ones. However, he is worried about the cost incurred and calls the firm's Finance Manager to help him calculate the median cost of the seven new machines.

The Finance Manager suggested that one could purchase new machines if the median price of the machines is below $85,000. The costs are as follows: $75,000, $82,500, $60,000, $50,000, $1,00,000, $70,000, $90,000. Calculate the median cost of the machines. The costs are as follows: $75,000, $82,500, $60,000, $50,000, $1,00,000, $70,000, $90,000.

Solution: 

Arranging the costs in ascending order: $50,000, $60,000, $70,000, $75,000, $82,500, $90,000, $1,00,000.

Therefore, the calculation of the median will be as follows:

Since there are 7 items, the median is (7+1)/2nd item, i.e., 4th item. Therefore, the 4th item is $75,000.

Since the median is below $85,000, one can purchase the new machines.

The main advantage of the median over means is that it is not unduly affected by extreme values, which are very high and very low. Thus, it gives an individual a better idea of representative value. For instance, if the weights of 5 people in kg are 50, 55, 55, 60, and 150. Mean is (50+55+55+60+150)/5 = 74 kg. However, 74 kg is not a true representative value as most weights are in the 50 to 60 range. Let us calculate the median in such a case. It would be (5+1)/2th term = 3rd term. The third term is 55 kg, which is a median. Since most of the data is in the 50 to 60 range, 55 kg is a true representative value of the data.

We have to be careful in interpreting what the median means. For instance, when we say that the median weight is 55 kg, not everyone weighs 55 kg. Some may weigh more, and some may weigh less. However, 55 kg is a good indicator of the weight of 5 people.

In the real world, to understand data sets such as household income or household assets, which vary greatly, the mean may be skewed by a small number of very large values or small values. Thus, the median is used to suggest what should be the typical value.

Bill is the owner of a shoe store. He wants to know which size of shoe he should order. He asks 9 customers what size their shoes are. The results are 7, 6, 8, 8, 10, 6, 7, 9, and 6. Calculate the median to help Bill in his ordering decision.

Solution: We first have to arrange shoe sizes in ascending order.

These are: 6, 6, 6, 7, 7, 8, 8, 9, 10

Below is given data for calculating the median of a shoe store.

Therefore, the calculation of the median in excel will be as follows:

In Excel, one can use an inbuilt formula for the median to calculate the median of a group of numbers. Select a blank cell and type this =MEDIAN(B2: B10) (B2: B10 indicates the range you want to calculate the median from).

The median of the shoe store will be –

Let us understand the distinctions between median and mean through the comparison below.

Median

• Represents the middle value in a dataset when arranged in ascending or descending order.
• Not influenced by extreme values or outliers, making it a robust measure of central tendency.
• Particularly useful when the data is skewed, providing a more representative measure.
• Median is often preferred when describing income distribution, recovery times in healthcare, or educational scores.

Mean

• Calculated by summing all values in a dataset and dividing by the number of observations.
• Sensitive to extreme values and outliers, as they significantly impact the calculated average.
• Offers a measure of central tendency but may not accurately represent the typical value in the presence of extreme values.
• Mean is commonly used in scenarios where the data distribution is symmetrical and not heavily influenced by outliers.