Comprehending how the central point of a dataset is calculated tends to be one of the first lessons taught in a statistics class. Even though the average is pretty much the most commonly used central measure, there are some cases where the median is more convenient to use.
In this article, we will look at the occasions when the median is a better measure of center than the mean, focusing on it through practical examples and real-life uses. In the end, you will have a grasp of how these measures function, their advantages and disadvantages, and when to apply them.
Mean vs. Median: The Basics
Mean And Median In Detail:
But before we get to that, we have to discuss when it is appropriate to use mean, and why the median is better. So first, let us look at what these two measures of central tendency are:
Mean (Arithmetic Average):
The mean is simply gotten by adding all figures together in a dataset and dividing by the number of figures.
It is determined by and considers all values in a dataset including outliers or extreme values.
Median:
The median is the central number or midpoint in a data set when arranged in order.
It has high resistance to outlier influence and normal variance, making it a highly accurate measure of central tendency.
When is the Median Preferable Over the Mean?
The median becomes the preferred choice over the mean in specific scenarios. Let’s discuss these cases in detail:
1. When the Data Set Contains Outliers
Extreme outliers are data that are grossly different from most other entries in a data set. Because they differ from other entries, their mean can easily be adjusted, resulting in a delta that is uncharacteristic of the given set.
Data Set: 10, 12, 14, 15, 100
Mean:
Median: 14
In this example, the mean is greatly affected by the outlier value (100), while the median center (14) is much more representative of the data.
2. When the Data Is Skewed
In a skewed distribution, the data points are not symmetrically distributed but concentrated more heavily on one side of the distribution.
- Right-Skewed (Positive Skew): Long tail on the right. Examples include income distributions.
- Left-Skewed (Negative Skew): Long tail on the left. Examples include test scores where most students perform well.
In both cases, the mean is dragged toward the tail, whereas the median remains at the center.
Example:
- In a right-skewed income distribution:
- Mean income: $75,000 (inflated by high earners)
- Median income: $50,000 (more representative of the typical person’s income)
3. For Ordinal Data
The median can work with ordinal data-whereby the ordering is the primary consideration with no interval data. The average cannot be determined for such data.
Illustration: Customer rating from 1 (very dissatisfied) to 5 (very satisfied)
Median: the inner satisfaction level
Mean: Leslie’s average does not amount to anything
4. Small Data Sets
In smaller data sets, especially those with extremes, the median sometimes represents the center more accurately.
Example:
Test Scores: 40, 45, 50, 90
Mean:
Median:
In this instance, the median makes more sense in terms of the normal score.
Strengths and Weaknesses of the Median and Mean
Advantages of the Median:
• Resistant to Outliers: Unaffected by extreme values.
• Applicable to Ordinal Data: Can represent ordered categories.
• Reflects Typical Values in Skewed Distributions: Offers a better sense of the “typical” value.
Disadvantages of the Median:
• Ignores Data Magnitude: This does not account for the magnitude of values.
• Less Useful in Symmetric Distributions: The mean is equally effective and easier to calculate when data is symmetric.
Advantages of the Mean:
• Uses All Data Points: Provides a more comprehensive measure.
• Ideal for Symmetrical Data: Perfect for normally distributed data.
Disadvantages of the Mean:
• Sensitive to Outliers: Can be heavily skewed by extreme values.
• Not Suitable for Ordinal Data: Requires numerical values with consistent intervals.
Comparing Mean and Median
Criteria | Mean | Median |
---|---|---|
Sensitivity to Outliers | Highly sensitive | Not affected |
Data Type | Numerical (interval/ratio) | Numerical and ordinal |
Skewed Data Handling | Skewed by extreme values | Robust against skewness |
Symmetrical Data | Suitable | Suitable |
Ease of Calculation | Easy for large, symmetrical sets | Easy for small and skewed sets |
Real-World Applications of the Median
- Income and Wealth Analysis
Incomes are often right-skewed due to a small upper class good at using the median.
Example: Median household income is commonly used in economic reports. - Property Prices
Real estate markets often feature outliers (e.g., luxury properties). The median house price is a more reliable metric than the mean. - Exam Scores
When evaluating student performance, outliers like very high or very low scores can distort the mean. The median offers a fairer assessment. - Survey Data
The median is ideal for summarizing results for ranked survey responses (e.g., satisfaction ratings).
Key Scenarios Where the Median is a Better Choice
- Skewed Distributions: In distributions like income, the median income is a better representation of the typical value, as the mean is often distorted by extreme values.
- Ordinal Data: When dealing with ranked data such as satisfaction ratings, the median provides a clearer picture of the center since the mean cannot be calculated meaningfully.
- Symmetrical Data: For a normal distribution, both the mean and median are equally valid, as the data is symmetrically distributed.
- Income Analysis: In studies like income inequality, the median offers a more accurate depiction of central tendencies, minimizing the impact of outliers.
FAQs About the Median vs. Mean
How does the mean differ from the median?
The mean is the average of all the data while the median is simply the middle data point of an ordered one. The median does not change because of an extreme value but the mean does.
What makes the median a preferred choice for skewed data?
In skewed distributions, extreme values can pull the mean toward the tail, making it less representative. The median remains centered and reflects the typical value.
Is it possible for both mean and median to be equal?
Yes, they tend to be equal in most symmetrical distribution cases like a normal one.
When do we apply the median?
Examples include income distributions, house prices, and survey satisfaction ratings—especially when data contains outliers or is skewed.
What is the influence of outliers on the mean and median?
Outliers can skew the mean greatly but not the median therefore the median is more reliable in such situations.
Conclusion: When to Use the Median Over the Mean
So, in conclusion, whether you decide to find the median or the mean is highly dependent on your data. The median is also preferred over the mean when there are outliers, ordinal data, or the data in general is skewed. On the other hand, the mean works well with symmetric and normally distributed data. However, it is important to note that the median is preferred in a lot of real-world cases. Knowing when to use each statistic will help you analyze data properly.