I. Introduction
As students and professionals alike, we all have encountered various numerical measures. Understanding the values of these measures is vital, especially in fields like Data Science, where a slight inaccuracy can lead to a significant failure. One of these measures is the median, a measure of central tendency that provides crucial information about a given data set. In this article, we will explore everything you need to know about finding median, calculating it and its significance in various fields.
II. What is median?
Median is a numerical measure of central tendency in a data set. It is the value that splits a data set into two equal halves, such that half of the data points are below the median, and half are above it.
The significance of median in a data set lies in its ability to identify the central value around which the data is grouped, while eliminating the influence of extreme values.
A simple example where median is used in everyday life is calculating the median salary of a company. The median salary is the value that lies in the middle of all salaries when they are arranged in ascending or descending order.
III. Calculating the median
Calculating the median can be done in a few simple steps.
First, arrange the data in ascending or descending order, depending on preference.
Second, find the middle value of the data set. If there is an odd number of values, the median is the middle value. If there is an even number of data values, take the average of the two middle values.
Example 1: Suppose we have the following 7 data values arranged in ascending order: 5, 6, 7, 8, 9, 11, 12. To find the median, we can complete the following steps:
Step 1: Arrange the data in ascending order which gives us 5, 6, 7, 8, 9, 11, 12.
Step 2: Find the middle value
Since the data set contains an odd number of values, the median is the middle value, which is 8.
Example 2: Suppose we have the following 8 data values arranged in ascending order: 3, 5, 6, 8, 8, 9, 11, 12. To find the median, we can complete the following steps:
Step 1: Arrange the data in ascending order which gives us 3, 5, 6, 8, 8, 9, 11, 12.
Step 2: Find the middle value
Since the data set contains an even number of values, we need to calculate the average of the two central values which are 8 and 9. Thus, the median of this data set is (8+9)/2 = 8.5.
IV. Median vs Mean
Median and mean are both measures of central tendency that are used to describe a data set numerically. The primary difference between the two is that while median represents the middle value of a data set, mean gives us the arithmetic average of all the values in a data set.
Example: Consider a class with five students in which one student, David, got a zero and the rest scored 70, 75, 80, and 85. Here, the mean is 62, while the median is 75.
Median is often preferred over mean when we are dealing with extreme values or skewed data sets. Mean is more sensitive to variations in larger atypical values, whereas median is less sensitive.
V. Spotting a skewed distribution
A skewed distribution is a data set where the values cluster at one end of the distribution, while a few larger values pull the mean to the right or left. A skewed data set can negatively affect the calculation of the median, particularly if the skewness is severe.
To identify if a data set is skewed, we can use summary statistics like the mean, mode, and range. Another way is to create a boxplot or a histogram. Box plots provide a graphical representation of data based on five numbers: the minimum, maximum, median, and the first and third quartiles. A histogram, on the other hand, represents the distribution of the data using rectangular bars.
VI. The importance of median in real-life applications
Median is widely used in various professions and industries, including healthcare, economics, finance, and education. In healthcare, for example, median is used to determine the average length of hospital stays or the median income of different medical specialties. Economists use median to calculate household income, while financiers use median to calculate the median return on investments.
In many cases, median is preferred over mean because it eliminates the impact of extreme values and provides a more accurate representation of the central values around which data clusters.
VII. Variations of the median
There are variations of the median that are used in specific situations, including the weighted median and the median absolute deviation. The Weighted median is used when different data points in a data set need more or less treatment than the others, for example, when determining salaries in a company where a CEO salary carries more weight than that of a junior employee.
The Median Absolute Deviation (MAD) is a measure of the variation in a set of data. MAD is calculated by taking the median of the absolute values of the differences between each data point and the overall median. MAD is a useful tool in identifying outliers in the data set.
VIII. Practice problems
Here are some practice problems to help you sharpen your skills.
Problem 1: Determine the median of the following data set: 4, 6, 7, 8, 10, 12.
Problem 2: Calculate the median of the following even data set: 2, 3, 4, 7, 8, 10, 12, 14.
Problem 3: The salaries (in thousands of dollars) of ten employees in a company are: $32, $42, $56, $60, $80, $90, $100, $150, $160, $250. Find the median salary of the employees.
IX. Quizzes or Interactive Elements
If you are curious about finding the median, try out some of these quizzes now.
Quiz 1: Which of the following are measures of central tendency? (A) Range (B) Mean (C) Mode
Quiz 2: Consider the following data set: 7, 8, 10, 13, 14, 15. What is the median?
X. Conclusion
In this article, we have covered the definition, calculation, and significance of the median as a measure of central tendency. We have also distinguished between median and mean and their importance across various fields. We have provided tips to identify skewed data sets, described different variations of the median, and offered practice problems to improve your skills. Hopefully, you will apply the knowledge you have gained in your own studies or work.