Mean, Median & Mode
What is the Mean, Median & Mode?
Mean, median, and mode are the three standard ways to describe a data set's "center" or typical value, and each responds differently to unusual data. The mean (xĢ = Ī£x/n) is the sum of all values divided by the count ā it's the most common measure, but it's sensitive to extreme outliers.
The median is the middle value once the data is sorted ā resistant to outliers, since it only depends on position, not magnitude. For an odd number of values, it's the single middle one; for an even number, it's the average of the two middle values. The mode is simply the value that appears most often ā a data set can have one mode, several, or none at all.
What Each Variable Means
When to Use It
- Mean ā best for roughly symmetric data without extreme outliers
- Median ā best when data has outliers that would skew the mean, like income or house prices
- Mode ā best for categorical data where averaging doesn't make sense, like a favorite color or shoe size
Step-by-Step Example
Problem: Find the mean, median, and mode of {4, 7, 2, 9, 3, 4}.
Add all values and divide by the count.
xĢ = (4+7+2+9+3+4)/6 = 29/6 ā 4.83Arrange from smallest to largest.
{2, 3, 4, 4, 7, 9}With 6 values (even), average the two middle ones.
Median = (4 + 4)/2 = 4Identify the most frequent value.
4 appears twice ā Mode = 4Common Mistakes
Mistake: Forgetting to sort the data before finding the median.
Fix: The median is defined by position in sorted order ā finding the "middle" of unsorted data gives a meaningless result.
Mistake: Averaging the two middle values only when the count is odd.
Fix: It's the reverse: average the two middle values when the count is even. With an odd count, there's a single true middle value and no averaging is needed.
Practice Questions
Find the mean of {10, 15, 20, 25, 30}.
Find the median of {8, 3, 5, 12, 1}.
Hint: Sort first: {1, 3, 5, 8, 12}.
Frequently Asked Questions
Can a data set have no mode?
Yes ā if every value appears exactly once (or all values appear the same number of times), there's no single most-frequent value, and the data set is described as having no mode.
Which measure should I use for a data set with an outlier?
The median ā it only depends on position in sorted order, so a single extreme value doesn't drag it far the way it would drag the mean.