TutorChase logo
IB DP Maths AA SL Study Notes

4.1.2 Median

Definition

The median is the value that separates the higher half from the lower half of a dataset. It's a measure that gives us a central value, providing a clearer picture of the dataset's distribution. Specifically:

  • If there's an odd number of values in the dataset, the median is the middle number.
  • If there's an even number of values, the median is the average of the two middle numbers.

Calculation

Calculating the median involves a systematic approach:

1. Sort the Data: Before anything else, arrange the data in ascending order. This step ensures that the data is organised, making it easier to pinpoint the middle value(s).

2. Determine the Middle Value(s):

  • For an odd number of values, the median is the exact middle value.
  • For an even number of values, the median is the average of the two central values.

Example:

Consider the dataset: 7, 11, 9, 10, 12

  • When arranged in ascending order, it becomes: 7, 9, 10, 11, 12
  • Since there are 5 values (odd), the median is the third value: 10.

For a dataset like: 7, 11, 9, 10

  • In ascending order: 7, 9, 10, 11
  • With 4 values (even), the median is the average of the second and third values: (9 + 10) / 2 = 9.5

Properties of Median

The median boasts several distinctive properties:

1. Resilience to Outliers: One of the median's standout features is its resistance to outliers. Extreme values, whether incredibly high or low, don't sway the median like they do the mean. This resilience makes the median a robust measure of central tendency, particularly for skewed distributions.

2. Positional Average: Unlike the mean, which considers the magnitude of every value, the median is purely positional. It's determined by the order of values rather than their actual sizes.

3. Equal Halves: The median has a balancing act; it ensures that half of the values in the dataset lie below it, while the other half sit above.

Advantages of Using Median

The median isn't just another average; it has specific advantages that make it the go-to measure in certain scenarios:

  • Skewed Data: For datasets that lean heavily to one side (either left or right), the median offers a more accurate representation of the central value than the mean. It isn't pulled in the direction of the skew, ensuring a more balanced representation.
  • Ordinal Data: When dealing with ordinal data (data that can be ordered but not quantitatively measured), the median shines. It's more fitting than the mean, which might not make sense for such data types.
  • Open-ended Distributions: For data distributions that don't have a clear end (open-ended), the median can still be determined. The mean, in contrast, might be elusive.

Real-World Applications

The real estate sector frequently employs the median. Instead of the mean house price, reports often showcase the median house price. Why? Because the median isn't influenced by a few abnormally high or low prices. It offers a more genuine representation of the typical house price in an area.

For instance, in a neighbourhood with house prices of £250,000, £260,000, £265,000, £270,000, and £1,500,000, the mean would be heavily influenced by the £1,500,000 house. However, the median would stand at £265,000, painting a more realistic picture of the central price.

Practice Questions

  • A group of friends went out for dinner and spent the following amounts: £20, £25, £23, £22, £24, £26, £21. What's the median amount spent?

Solution: First, arrange the amounts in ascending order: £20, £21, £22, £23, £24, £25, £26. With 7 amounts (odd), the median is the fourth value: £23.

  • During a week, a shop recorded daily sales of: £500, £520, £510, £505, £515, £525, £530. What's the median sale?

Solution: In ascending order, the sales are: £500, £505, £510, £515, £520, £525, £530. With 7 days (odd), the median sale is the fourth value: £515.

In essence, the median is more than just an average. It's a powerful tool in data analysis, offering insights that other measures might miss. Whether you're dealing with skewed data, ordinal data, or open-ended distributions, the median stands out as a reliable measure of central tendency.

FAQ

In more complex statistical studies, the median serves as a robust measure for central tendency, especially in distributions that are not symmetrical. It's used in various fields, from economics to social sciences, often in income studies, where economists are interested in the middle income in a population. The median is also crucial in medical research, particularly in survival analysis, where the median survival time is a critical measure. These applications underscore the median's importance in providing a realistic picture of a dataset, unaffected by extreme values.

If every number in a dataset increases (or decreases) by a certain amount, the median will increase (or decrease) by the same amount. This is because the position of the median in the ordered list doesn't change; only the values themselves do. For example, if we add 5 to every number in a dataset, the new median will be the original median plus 5. This property shows the consistency of the median as a measure of central tendency, maintaining relative positions and differences within the data.

The median is versatile but not universally applicable. It is perfect for ordinal data (data that can be put in order but not necessarily equidistant from each other, like movie ratings) and interval data (data with meaningful differences between values, but no true zero, like temperatures). However, for nominal data (data that cannot be ordered, like categories or names), the median is meaningless because there's no logical way to determine a middle value when values can't be logically ordered.

The median is often favoured over the mean in datasets that are skewed or contain outliers. This is because the mean takes into account the actual values in a dataset, making it highly sensitive to extreme values which can distort the real picture of a dataset's central tendency. The median, however, solely depends on the order of the data, not the actual values, making it resistant to such distortions. Therefore, in a distribution with abnormally high or low values, the median provides a more accurate reflection of the central point.

Practice Questions

In a survey, the ages of 10 participants were recorded as follows: 24, 26, 25, 23, 24, 27, 26, 25, 24, 28. Calculate the median age of the participants.

To determine the median age, we first arrange the ages in ascending order: 23, 24, 24, 24, 25, 25, 26, 26, 27, 28. Since there are 10 ages (an even number), the median is the average of the fifth and sixth values. Averaging 25 and 25 gives 25. Therefore, the median age of the participants is 25 years.

A teacher recorded the marks of 9 students in a maths test out of 20. The marks were: 15, 12, 18, 16, 14, 13, 17, 15, 19. Calculate the median mark.

To find the median, we first need to arrange the marks in ascending order: 12, 13, 14, 15, 15, 16, 17, 18, 19. Since there are 9 marks (an odd number), the median is the fifth value. Therefore, the median mark is 15.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

Hire a tutor

Please fill out the form and we'll find a tutor for you.

1/2 About yourself
Still have questions?
Let's get in touch.