Step-by-Step Median Calculation with Real-Life Examples
In the simplest terms, the median can be defined as the middle value of a given list of data. This value is selected when the list of data is arranged in proper order. One can decide to arrange the numbers in ascending order or descending order. For example, If a student is provided with a list of data in the order of 2, 3, and 4. From this list, the student is asked to find out the median number. The data list is already arranged in proper order, so it is easy to state that 3 is the median.
Apart from this, in Mathematics, the median can also be defined as a type of process that can be used to find the centre value of any set of data. Because of this reason, many people define the median as a measure of central tendency whenever they are asked to explain the answer to the question of what is the median number.
Readers should also be familiar with the fact that apart from the median, there are two other mathematical central tendencies. These measures of central tendencies are mean and mode.
Mean can be defined as the ratio of the sum of all observations or values and the total number of observations. Mode, on the other hand, is the value that is repeated the most in a given data set.
When it comes to geometry, a median is defined as the centre point of a polygon. Students follow several steps when they are trying to learn how to find out the median of a polygon. Also, the median of a triangle is the line segment that joins the vertex of the triangle to the centre of the opposite sides. We can also say that a median bisects the sides of a triangle.
In statistics, the median is a value that lies halfway between the given data set. A median is also the number that is separated by the higher half of a data sample from the lower half of the sample. This means that the value of the median will be different for different types of distributions.
Let’s consider an example to make this topic clearer. Let’s assume that a student has to find the value of the median from a given data set. The data set is 3, 3, 5, 9, 11. In this data distribution, the value of the median is 5. A similar method would work for as long as the total number of observations come up to be an odd number.
But what should one do if the total number of observations in a data set are even? In that case, the student should add both the digits that fall in the middle and, then divide the answer with two. For example, if one has to find the median of a data set: 2, 4, 5, 8, 9, 10, 12, 14, 16, 19, 20, 22. In this data set, both 10 and 12 fall in the middle.
So, we need to 10 + 12 = 22
22 / 2 = 11
This means that 11 is the medial of even numbers provided in the data set that was mentioned above.
The Median Formula
Till now, we have looked at the basic process of finding out the value of median in a simple data set. The next step is to understand the median formula even. Readers must remember that the formula for calculating median for even and an odd number of observations is different. This is why it is important to first identify the total number of observations in a data set before applying the even median formula.
Case 1: Odd Number of Observations
If the total number of observations in a data set is odd, then the following formula for calculating median can be applied.
Median = {(n + 1) / 2}th term
In this median sums formula, n is the total number of observations. This formula works for calculating the median of odd numbers.
Case 2: Even Number of Observations
If the total number of observations in a data set comes up to be an even number, then the median even formula should be followed. The median of even numbers formula is mentioned below.
Median = [(n / 2)th term + {(n / 2) + 1th] / 2
In this median formula for even numbers, n is the total number of observations.
How Can You Calculate the Value of Median of Data?
Calculating the value of the median of a complex set of data is not easy. However, one can follow a simple process to arrive at the final answer. First, one needs to begin by arranging the observations in an ascending or descending order. This is done as the middle value of the set of observations will provide the median of numbers.
However, there are still two scenarios that can arise in this case before arriving at the final answer. The total number of observations can either be odd or even. A different formula is used in both cases.
If one wants to find the value of the median of a set of numbers that are odd, then the formula mentioned below can be used.
Median = (n + 1 / 2)th observation
If one wants to know how to find the median number of a set of data that has an even number of observations, then the formula that is mentioned below is used.
Median = Mean of (n / 2)th and [(n / 2) + 1]th observations
The Graphical Representation of Median
Till now, we have answered questions like what is a median and what formulas are required for the median. It is now time to gain better insight into this topic by going over the graphical representation of the median.
There are three main central tendencies. These three central tendencies are mean, median, and mode. All three values are different and can be plotted differently on a graph. If you wish to learn how these values are plotted, then you should look at the image that is attached below. Students also often have to solve questions in which they have to graphically represent the values of median, mean, or mode. This image will help students answer those types of questions.
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Fun Facts about Median and Other Measures of Central Tendency
Did you know that mean is the most commonly used measure of central tendency? Mean can be defined as the average of all the observations in the data set. This measure of central tendency is applicable in the case of both continuous and discrete data.
Mean can also be defined as the sum of all the values of observations in data divided by the total number of observations. On the other hand, the mode is the most frequently occurring value in the data set. All these three values of central tendencies help in providing better insight into a data set and are of great importance in the subject of statistics.
FAQs on Median in Maths: Meaning, Formula, and How to Find It
1. What is the median in Maths? Explain with an example.
In mathematics, the median represents the middle value in a dataset that has been arranged in order of magnitude (either ascending or descending). It is a measure of central tendency that pinpoints the exact centre of the data. For example, to find the median of the dataset {5, 2, 8, 1, 9}, you first arrange it in ascending order: {1, 2, 5, 8, 9}. The middle number is 5, so the median is 5.
2. What is the most important first step before finding the median?
The most crucial first step is to arrange the dataset in order. You must list all the numbers from the dataset in either ascending (smallest to largest) or descending (largest to smallest) order. If this step is skipped, the middle value you pick will be incorrect and will not represent the true median of the data.
3. How do you find the median for a dataset with an even number of observations?
When a dataset has an even number of observations, there are two middle numbers. To find the median, you must calculate the average of these two middle numbers. For instance, in the ordered dataset {2, 4, 6, 8, 10, 12}, the two middle numbers are 6 and 8. The median is calculated as (6 + 8) / 2 = 7.
4. What is the difference between mean, median, and mode?
Mean, median, and mode are all measures of central tendency, but they describe the center of a dataset in different ways:
- The Mean is the 'average' value, found by summing all numbers and dividing by the count of numbers.
- The Median is the 'middle' value of a dataset that has been arranged in order.
- The Mode is the 'most frequent' value that appears in the dataset.
5. Why is the median sometimes a better measure of central tendency than the mean?
The median is a better measure than the mean when a dataset contains outliers (extremely high or low values) or is skewed. Outliers can significantly distort the mean, making it unrepresentative of the typical value. For example, when calculating the average income of a neighbourhood, a few billionaire residents would drastically increase the mean income. The median income, however, would provide a more realistic value for the typical resident's earnings by ignoring these extreme outliers.
6. How is the concept of median used in real-life situations?
The median is widely used in many fields to provide a more accurate representation of data. Key examples include:
- Economics: To report the 'median household income' or 'median house price', which prevents a few extremely high values from skewing the data.
- Education: To find the median test score, which helps teachers understand the performance of the typical student, separating them from the highest and lowest scorers.
- Statistics: Government agencies use it to report population statistics like median age, which gives a better sense of the population's age distribution.
7. How do you find the median for grouped data as per the CBSE syllabus?
For grouped data, the median is estimated using a specific formula. The steps are:
1. Calculate the cumulative frequency (cf) for each class.
2. Find the total number of observations (n) and calculate n/2.
3. Identify the median class, which is the class interval where the cumulative frequency is just greater than or equal to n/2.
4. Apply the formula: Median = l + [((n/2) – cf) / f] × h, where 'l' is the lower limit of the median class, 'cf' is the cumulative frequency of the class preceding it, 'f' is its frequency, and 'h' is the class size.
8. In the formula for the median of grouped data, what do 'l', 'cf', and 'f' represent conceptually?
Each variable in the grouped data median formula has a specific meaning:
- l (Lower Limit): This is the starting point of the median class. It acts as a base value from which we add a fraction to pinpoint the exact median.
- cf (Cumulative Frequency): This is the cumulative frequency of the class before the median class. It tells us how many data points are accumulated up to the start of the median class.
- f (Frequency): This is the frequency of the median class itself. It indicates how many data points are contained within that specific interval and influences how far into the class we need to go to find the median value.
