The Effect of Outliers on Data - Sixth Grade Mathematics
An outlier is an extreme statistical observation that deviates substantially from the overall pattern of a distribution, exerting a disproportionate influence on certain summary statistics.
What is an Outlier?
In any collection of measurements, most observations tend to cluster within a predictable region. An outlier is a value that lies abnormally far away from this central cluster—either far higher or far lower than the rest of the data.
Visualizing an Outlier on a Dot Plot:
Data: 10, 11, 11, 12, 12, 12, 13, 14, 25
Cluster Outlier
*
* * *
* * * * * *
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Outliers can arise from multiple causes:
1. Measurement or recording errors (e.g., accidentally typing 25 instead of 2.5).
2. Genuine, exceptional phenomena (e.g., a professional athlete visiting a youth sports camp, or a billionaire moving into a rural town).
How Outliers Impact Measures of Center
Different statistical measures respond with radically different sensitivity when an outlier is introduced.
Demonstrating the Impact with Numbers:
Original Data Set (Quiz scores for 5 students):
78, 80, 82, 84, 86
Sum = 410. Count = 5.
Mean = 410 / 5 = 82.
Median = 82 (the exact middle score).
Now imagine a 6th student was absent and scored a 0 (Extreme Low Outlier):
New Data Set: 0, 78, 80, 82, 84, 86
New Sum = 410 + 0 = 410. New Count = 6.
New Mean = 410 / 6 = 68.33! (Dropped by nearly 14 points!)
New Median = (80 + 82) / 2 = 81. (Changed by only 1 point!)
Summary of Sensitivity:
- The Mean is HIGHLY SENSITIVE to outliers. Because every single number is summed,
an extreme value pulls (drags) the mean in its direction.
- The Median is RESISTANT (robust) to outliers. Because it depends strictly upon
ordering and central position, extreme endpoints barely affect it.
- The Mode is generally unaffected by single extreme values.
How Outliers Impact Measures of Spread
Outliers also distort measures of variability: The overall Range (Maximum - Minimum) is drastically increased by an outlier. In our example above, the original range was 86 - 78 = 8 points. With the outlier 0 added, the new range exploded to 86 - 0 = 86 points! Conversely, the Interquartile Range (IQR), which measures the spread of the middle 50% of the data, remains completely unaffected by extreme outliers.
Deciding When to Report Mean vs. Median
Because of outlier distortion: When a data distribution is symmetric without significant outliers, report the Mean as the center and Mean Absolute Deviation (or standard deviation) as spread. When a data distribution is skewed or contains outliers, report the Median as the center and the Interquartile Range (IQR) as spread.
Exercises
Exercise 1. Consider the data set: 14, 15, 15, 16, 18, 19, 45. (a) Identify the outlier; (b) Calculate the mean and median with the outlier included; (c) Calculate the mean and median with the outlier removed; (d) Explain which measure changed more when the outlier was removed.
Exercise 2. A real estate agent lists five homes in a neighborhood for $250,000, $260,000, $275,000, $290,000, and $2,500,000 (a luxury mansion). (a) Calculate the mean and median home price; (b) Which measure should a prospective homebuyer consider to understand the typical home price in that neighborhood?
Exercise 3. True or False: (a) Adding a high outlier to a data set always increases the mean; (b) Adding a high outlier always increases the median; (c) The range of a data set is resistant to outliers.
Solutions
Solution to Exercise 1. For (a), 45 is the outlier because it is far larger than the cluster between 14 and 19. For (b), with outlier: Sum = 14 + 15 + 15 + 16 + 18 + 19 + 45 = 142. Mean = 142 / 7 = 20.29. Median = 16 (the 4th value). For (c), without outlier: Sum = 97. Count = 6. Mean = 97 / 6 = 16.17. Median = (15 + 16)/2 = 15.5. For (d), the mean dropped significantly by 4.12 points (from 20.29 down to 16.17), while the median changed by only 0.5 points (from 16 to 15.5). The mean was much more sensitive to the outlier.
Solution to Exercise 2. For (a), Sum = 250,000 + 260,000 + 275,000 + 290,000 + 2,500,000 = $3,575,000. Mean = 3,575,000 / 5 = $715,000. Median = $275,000 (middle price). For (b), the homebuyer should rely on the median ($275,000). The mean ($715,000) is heavily distorted by the single $2.5 million mansion and paints a completely misleading picture, as four of the five houses cost under $300,000!
Solution to Exercise 3. For (a), True, adding a value greater than the current mean always increases the mean. For (b), False; an additional high value may shift the middle position by at most one rank or keep the median identical. For (c), False; range is the difference between maximum and minimum, making it directly and heavily altered by any outlier.