Introduction of Central Tendency – Median
Measures of central tendency refer to a single value that helps to correctly describe a set of data through the identification of the central position within a said data set. There are three valid central tendency measures, namely – mean, median and mode. As per the measures of central tendency definition, these are collectively classed as summary statistics.
While mean, median and mode are all valid measures for central tendency, some of them are more appropriate and accurate than others.
Following is an elaboration on the median, as one of the most significant among various measures of central tendency.
Median – Definition and Calculation
Median is a middle score in a set of data arranged according to their magnitude. It is used to divide a collection of data into two parts, where one half includes values that are greater than or equal to the median; whereas the second half contains values that are lesser than or equal to the median value.
The median value, unlike other measures of central tendency formulas, is not influenced by skewed data or outliers.
Now, if the observations in a variable are ordered by value, then a median value is given by the value corresponding to the middle value in that said ordered list. That is, the median value corresponds to that of a cumulative percentage of 50%, and its position is given by – {(n+1)/2} Th value; where “n” is the number of values present in that dataset.
Following is an example of the Calculation of Median Value
The Data above has to be First Arranged as per their Magnitude (Lowest to Highest)
Here, the median value is 53, since it is the middle mark, and five values are lying before it and five values after it. However, this method for computing the median works if there is an odd number of data present in the set. What happens if the number of data present is even?
Take a look!
The Data is Arranged as per Magnitude (Lowest to Highest)
Here, we consider the fifth and sixth data from this sequence and their average is the median in this scenario. That is,
Median = (49+53)/2 = 51.
Properties of Median
Median is considered among the best measures of central tendency owing to its following properties –
Calculating a median does not depend on all the values of data present in a dataset.
It is the value given by the middle point of the data set, such that half of this data is present above it and the other half is situated below it.
Every array of data has a single median.
The value for a median remains stable in a grouping procedure.
Medians cannot be determined for interval, rational, and ordinal scales.
The measure for the median is accurate over the mean when the distribution of data is skewed.
Medians cannot be combined or weighed or in general, manipulated algebraically.
Merits and Demerits of Median
As far as merits and demerits of central tendency median are concerned, there are several of which to take note of.
Following are a few merits of Median
It is easily understandable and computable.
It is well defined as an average.
It is not influenced by extreme values in a data set and is also independent of the dispersion and range of data.
Median can be utilised in the case of computing frequency distribution with open-ended classes.
It can be plotted graphically with the help of an ogive curve.
It can be used as a proper average for qualitative data for which items are scored, instead of being measured.
In a few cases, a median is a better measure of average than a mean.
Following are some of the demerits of Median
Since the computation of a median requires data to be arranged in ascending or descending order of value, it can be time-consuming when a data volume is large.
It can be affected by fluctuations in sampling, more than that in the case of Arithmetic mean.
It only gives a positional average and does not consider the magnitude of data.
Since it does not consider all observations, it cannot be considered the ideal representation of the average.
In case of considerable variation among data, the median will not be able to represent the data efficiently.
Since it is a positional average, further algebraic treatment for the data is not possible. For instance, computing the combined median between two groups of data is not possible.
It neglects considering the extreme values.
Here are a few of the merits and demerits of measures of central tendency for the median.
To learn more about median merits and demerits, alongside its integration in the calculation of central tendency, you can refer to our online learning programs. We, at Vedantu, have also introduced online live classes, which can assist you in learning about the advantages and disadvantages of the measures of central tendency and other such topics, from experts in the field.
So, without further ado, install the Vedantu app and start learning today!
Learning the Median
Median can be understood as when in a sorted list of numbers, the median is the value that is present in the middle. To get the median value in a set of data, the data has to be sorted in the value order ranging from highest to lowest or lowest to highest. The median can also be used to estimate an average or mean, but it should not be taken as the actual mean.
If the amount of numbers is odd, then the median value is the number that is in the middle, with the same number of values above and below it.
If the list comprises an even number of values, then we identify the middle pair and add them together, then it can be divided by the value two to return the median value.
When there are outliers present in the series that could somewhere affect the average of the numbers, sometimes the median is used instead of using the mean. Outliers or exceptions have less impact on the sequence median rather than on the mean.
FAQs on Measures of Central Tendency: Understanding Median
1. How is the Median specifically defined as a measure of central tendency?
The Median is the middle value in a dataset when the values are arranged in ascending or descending order. It effectively divides the data into two equal halves, meaning 50% of the values are below the Median and 50% are above it. It's often referred to as a positional average.
2. What are the step-by-step methods to calculate the Median for different types of data sets?
To calculate the Median accurately for your revision, follow these essential steps:
- First, always arrange the dataset in either ascending or descending order.
- If the number of observations (n) is odd, the Median is simply the value of the [(n+1)/2]th observation.
- If the number of observations (n) is even, the Median is the average of the (n/2)th and [(n/2)+1]th observations.
- For grouped data, a specific formula is used: Median = L + [(N/2 - C.F.) / f] * h, where L is the lower limit of the median class, N is the total frequency, C.F. is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and h is the class interval size.
3. What are the key differences between Mean, Median, and Mode that I should remember for my revision?
For a quick revision of measures of central tendency, remember these key differences:
- The Mean is the arithmetic average, calculated by summing all values and dividing by their count. It is sensitive to extreme values.
- The Median is the middle value in an ordered dataset and is less affected by extreme values or outliers.
- The Mode is the most frequently occurring value in a dataset. It is unique as it's the only measure suitable for nominal data.
Each measure offers a distinct perspective on the central point of the data.
4. When should I use Mean, Median, or Mode in data analysis?
The appropriate measure of central tendency depends on the type of data you're working with and its distribution:
- Use the Mean for interval or ratio data when the data distribution is symmetrical and there are no significant outliers.
- Use the Median for ordinal, interval, or ratio data, especially when the dataset is skewed or contains outliers, as it provides a more robust and representative central value.
- Use the Mode for nominal (categorical) data to identify the most frequent category. It can also be applied to other data types if the goal is to find the most common value.
5. What are the primary measures of central tendency covered in this chapter?
In this chapter on Measures of Central Tendency, the three primary types you need to grasp are the Arithmetic Mean, the Median, and the Mode. Each of these measures helps in understanding the typical or central value of a given dataset, forming a crucial part of your Commerce syllabus.
6. Why is the Median often preferred over the Mean when dealing with skewed data or outliers?
The Median is frequently preferred over the Mean in datasets with skewed distributions or outliers because it is a positional measure. Unlike the Mean, which is influenced by the magnitude of every single value, including extreme ones, the Median simply finds the midpoint of the ordered data. This characteristic makes the Median far more robust and a better representative of the 'typical' value in a dataset that is not symmetrically distributed, as it effectively neutralizes the impact of unusually high or low observations.
7. How do extreme values impact the Mean, Median, and Mode, and why is this important for revision?
Understanding the impact of extreme values (outliers) is vital for choosing the correct measure during your Commerce revision:
- The Mean is highly sensitive to extreme values, which can significantly pull it towards the outlier, making it less representative of the majority of the data.
- The Median is largely unaffected by extreme values because its calculation only depends on the position of values in the sorted list, not their specific magnitudes.
- The Mode is generally not affected by extreme values, as it focuses only on the most frequent data point, unless the outlier itself becomes a new, unique mode.
This knowledge ensures you select the most appropriate measure for different data scenarios.
8. What are the essential formulas for Mean, Median, and Mode that I need to quickly recall during my Commerce exam revision?
For quick recall during your Commerce exam revision, here are the essential formulas for measures of central tendency:
- Arithmetic Mean (for ungrouped data): Sum of all observations (Σx) / Number of observations (n)
- Median (for odd 'n', ungrouped data): Value of the [(n+1)/2]th observation
- Median (for even 'n', ungrouped data): Average of the (n/2)th and [(n/2)+1]th observations
- Median (for grouped data): L + [(N/2 - C.F.) / f] * h
- Mode (for ungrouped data): The observation with the highest frequency.
- Mode (for grouped data): L + [(f1 - f0) / (2f1 - f0 - f2)] * h, where f1 is frequency of modal class, f0 is frequency of preceding class, f2 is frequency of succeeding class.
Mastering these formulas is crucial for solving numerical problems in your exams.
9. Are there any specific situations where the Mode becomes the most suitable measure of central tendency, and why?
Yes, the Mode is uniquely suitable in specific situations, particularly when dealing with nominal data. This refers to categorical data that cannot be ordered or quantified numerically (e.g., colours, types of vehicles, survey responses like 'yes/no'). In such cases, you cannot calculate a meaningful Mean or Median. The Mode simply identifies the most frequent category or value, which is the only relevant measure of central tendency for qualitative attributes. It's also useful when identifying the most popular item or frequent occurrence is the primary objective, even in numerical datasets.
10. What are some common errors students make while interpreting measures of central tendency, especially the Median?
During your revision of measures of central tendency, be mindful of these common errors:
- Failing to arrange the data in ascending or descending order before attempting to calculate the Median.
- Confusing the Median with the Mean, particularly when dealing with skewed data distributions, leading to incorrect conclusions about the dataset's center.
- Assuming that a dataset can only have one Mode, or that a Mode must always exist; datasets can be bimodal, multimodal, or have no mode.
- Incorrectly applying a measure (e.g., calculating the Mean for nominal data).
- Misinterpreting what the 'average' truly represents based on the measure used, especially overlooking the Median's property of dividing the data into two exactly equal halves, which is vital for understanding income distribution or similar scenarios.
Avoiding these pitfalls will significantly improve your understanding and performance.
