The median, a type of average in mathematics and statistics, represents the middle value in a dataset.
Despite its apparent simplicity, the meaning of the median involves some complexity. This article delves into the essence of what is the meaning of median in maths as well as how to calculate median across various applications.
An Introduction to Averages
In examining median meaning, it’s important to understand averages which serve as mathematical tools to grasp the essence of data.
To oversimplify, imagine determining if it's a hot day by measuring the temperature. However, as temperature fluctuates throughout the day, relying on a single measurement proves inadequate. Taking multiple measurements at intervals resolves this issue but doesn't provide a clear picture. By averaging these measurements, a comprehensive overview will emerge, and this median helps in the analysis and comparison of datasets.
In addition, averages also enable predictions and data interpolation.
Despite common complaints about the applicability of math in daily life, average temperatures, for instance, inform decisions regarding appropriate attire which can be particularly useful when weather forecasts are unavailable. In the same way, historical averages similarly aid in vacation planning. In fact, knowing how to calculate the median in most industries is usually a valuable tool.
While our series explores different types of averages, this article centres on the median meaning—a critical component in understanding data distribution.
What is the Median in Maths and How to Calculate it
The median, in essence, is straightforward —it represents the middle number within a dataset.
To compute it, simply arrange all values numerically and select the middle one. However, determining which value occupies the middle position varies based on the total number of values.
Thus, when it comes to how to calculate the median, there are two methods: one for datasets with an odd number of values, and another for those with an even number.
How to Calculate Median with an Odd Number of Values
Suppose we sought the median of the highest temperatures recorded for Johannesburg this week (at the time of writing) to determine the average temperature. We simply arrange the numbers in ascending order:
- The highs for Johannesburg this week are 29, 26, 30, 30, 29, 27, and 28.
- In ascending order, they are 26, 27, 28, 29, 29, 30, and 30.
- With 7 values, the fourth value stands as the precise middle (with 3 values on each side). Thus, the median high temperature for Johannesburg this week is 29.
By contrast, the mean (another form of average) of these numbers computes to 28.875, marginally lower than the median.
How to Calculate Median with Even Number of Values
In scenarios where the dataset contains an even number of values, determining the median requires a different approach as there's no precise middle value.
Instead, we compute the average of the two middle values. And this is where the question, what is the median in maths, gets slightly more complicated.
Consider augmenting our dataset with an additional day featuring a temperature of 22: 22, 29, 26, 30, 30, 29, 27, and 28. Arranging these values in ascending order yields: 22, 26, 27, 28, 29, 29, 30, and 30. Here, the fourth and fifth values occupy the middle position (with 3 values on each side).
These values are 28 and 29. When the middle values differ, we compute their mean.
Simply put, the mean is the midway point between two values, calculated by summing them and dividing by the number of values.
In this instance, the sum of the values is 28 + 29 = 57. As there are 2 values, 57 divided by 2 equals 28.5, and the median of this dataset is therefore 28.5.
The mean of this set of numbers would be 27.625, approximately one degree lower than the median.
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The Advantages of Using the Median
Now that you have an idea of median meaning and even how to calculate median, let’s consider the advantages of using it in maths.
In our weather example, we looked at the use of both the median and the mean to yield distinct averages. Despite "average" commonly referring to the mean, the median holds equal validity as a measure of central tendency. Each average possesses its own merits and demerits, guiding the choice of average based on the dataset and its context.
The median boasts several advantages, notably its robustness against outliers. Consider a scenario where a faulty thermometer erroneously reports a temperature of 50 degrees during the week. In this case, the dataset becomes 29, 26, 30, 50, 29, 27, and 28.
While the mean would yield an average of 31.2857, the median remains at 29. Instead, by using the median, the impact of outliers or extreme values is mitigated, resulting in less distortion by such data points.
This resilience to outliers makes the median preferred in financial contexts, such as average salaries or property values, where a few high-net-worth individuals can significantly skew the data. For instance, while many South Africans earn salaries in the tens of thousands, a single millionaire or billionaire's income drastically inflates the mean average. Similarly, property prices are heavily influenced by a few exceptionally expensive properties. Hence, reports on financial data tend to favour the median for a more representative portrayal.
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Disadvantages of Using the Median?
Understanding what is the meaning of median in maths and even knowing how to calculate median, does not mean it is without its drawbacks. In small datasets with unevenly distributed values, adding or removing a single value can significantly alter the median. Consider the following datasets:
- 10, 20, 30, 40, 50
- 10, 20, 30, 40, 50, 1 000
In the first dataset, the median is 30, while in the second, it's 35.
Despite the presence of a large outlier in the second dataset, the median remains relatively unaffected. Conversely, the mean for the first dataset is also 30, but for the second, it's 191.67. In this scenario, the median appears to be the preferable average as it's less influenced by the outlier.
In addition, adding another 1 000 to the second dataset alters the median to 40. However, including yet another 1,000 shifts the dataset to 10, 20, 30, 40, 50, 1,000, 1 000, 1 000, resulting in a median of 525. Despite the mean being only 268.75 due to its sensitivity to extreme values, the median is considerably more impacted by the inclusion of larger values.
In summary, the median is more susceptible to changes in small datasets, whereas the mean is typically more affected by extreme values in a dataset.
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When To Choose the Mean Over of the Median?
You could also use the mode as a form of average and this decision often boils down to choosing between the mean and the median for datasets. Understanding the strengths and weaknesses of each helps to make an informed choice.
Here is guide to help decide which one to use:
- If your data lacks extreme values, opt for the mean, as it remains unaffected by outliers. However, when dealing with extreme values, the median serves to mitigate their impact effectively.
- The median provides a less precise average, being restricted to one of the dataset's values or the mean of two, making it suitable for situations where precision isn't paramount. Conversely, for detailed numerical analysis requiring precision, the mean is preferable.
Although the mean is commonly used as the default average, it's important to acknowledge the utility of the median, especially in scenarios involving outliers.
To delve deeper into what is the median in maths, or any other type of average, consider seeking support from qualified maths and statistics tutors on the Superprof platform. With many tutors offering complimentary initial sessions, you can explore various options before selecting the ideal tutor to embark on your journey to improve and deepen your mathematical understanding.
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