Mathematical Properties of Median in Statistics

The recognition of the median is denoted as a very simple amount of central propensity. For the calculation of a median, the arrangement of the numbers is required by observing their order of minimum to the maximum value.

The middle value is recognized as the median when an odd number of observations are there. When we can see an even number of observations, the average of the two middle values is recognized as the median.

The result (value) which conquers the midpoint location amongst the explanations at the time of organizing them in ascending or descending order is known as the median. The median values always are neither beyond nor beneath, just stay in the middle.

The median is also named as the positional average or fiftieth percentile. The median’s place is reliant on the set of data that involves an odd or even number of scores.

The procedures for properties of the median in statistics are different for the even as well odd values.

Properties of Mean Median and Mode in Statistics

Let’s discuss the properties of mean median mode in statistics:

Mean

The average of the values can be believed as the Mean. Here, let’s just do the summation of all the amounts. Then we need to divide the summation by the total amount of numbers.

Let’s just take an example of a set of numbers and calculate their mean.

Example: 18, 13, 16, 13, 14, 13, 14, 13, 21

Ans: We know that, mean is the usual average, so firstly we need to add them up and after that, we will divide:

18 + 13 + 16 + 13 + 14 + 13 + 14 + 13 + 21 = 135

Then,

135 ÷ 9 = 15

Did you notice anything? The mean is not a value from the given list. The result is generally appearing like this. The fact is that sometimes results may appear as per the enlisted number and sometimes it may not.

Median

The central score for a set of numbers that has been organized in the order of extent is recognized as the median. We need to take an example to compute the median.

Example: Consider that we have the data below:

The primary purpose is; we need to reorganize that numbers into the order of amount (least number first):

In the table given above, 22 is painted in the bold mark as it is our median mark. It becomes the middle mark as there are 5 numbers before it and 5 numbers after it.

This procedure does its work satisfactorily while we have an odd set of numbers.

Do you ever wonder what will occur, when you have an even set of numbers?

Alright, you just need to take the central two numbers and calculate the average result. Thus, if we consider the example below, we will know the best.

Example:

As per the principle, we will reorganize that data into an increasing order;

We need to consider and take the numbers 21 and 22 and do their average.

(21 + 22) / 2 = 21.5

So this will be the median point.

Mode

The most common number in our information set is recognized as the mode.

We can represent on a histogram as the uppermost bar in a bar chart or histogram as per the picture given below. Thus, we assume the mode as the most prevalent choice sometimes. The illustration in the picture will show you about the mode:

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Properties of Mean Median and Mode in Psychological Statistics

Now, we are going to provide information regarding this aspect.

It will help you to recite and understand research articles for projects, containing charts, statistical analyses, and interpreting graphs.

Properties of the Median with Formula in Statistics

Let’s calculate the median for the individual series as given below:

The arrangement of data should be in ascending or descending order.

For the odd-sized sample

The median = value of ([n + 1]/2)th item.

Here, n = cumulative frequency

For the even-sized sample

The median = ½ [ value of (n / 2)th item + value of ([n/2] + 1)th item]

Here, n = cumulative frequency

Let’s calculate the median for discrete series which is given below:

The data must be organized in ascending or descending order.

The computation of cumulative frequencies is required.

Median = (n/2)th term,

n = cumulative frequency.

For a continuous distribution, the median formula is:

Here,

Frequency of the median class = f

The sum of all frequencies = N

The lower limit of the median class = l

The cumulative frequency of the group earlier to the median group = C

The width of the median class = i

FAQ (Frequently Asked Questions)

Q1. What is the Reason for the Usage of the Median in Statistics?

Ans: Despite using the mean, the median is a supplementary method to find the center of a numerical data set.

This is the reason that the median is the middle of the data set.

Q2. Elaborate the Dissimilarity Between Mean and Median.

Ans: The mean is recognized as the average of the data set. When in someplace, we combine all the numbers. Then we divide the result by the total amount of given numbers.

But, in the case of the "median," it is known as the central value of the list of numbers.

Q3. Mention the Answer about When the Mean is Greater or Smaller than the Median.

Ans: Yes, the distribution is positively skewed, if the mean is greater than the median. But when it is smaller than the median, the result of distribution is negatively skewed.

Q4. The Median of 10, 11, 12, 8, 14, 9, 6 is _________? Calculate the Fourth Term.

Ans: Let’s just organize the numbers in ascending order such as:

6, 8, 9, 10, 11, 12, 14

For the n is odd, the median will be

= ([n + 1] / 2)^{th} the term = Median

= ([7 + 1] / 2)^{th} term

= 4^{th} term

Therefore, the fourth term is 10.