What Changes When One Observation Is Removed?
In What Outliers Mean in Context, you learned that an outlier is a value that stands apart, not automatically a mistake. One way to understand how much an unusual observation affects a description is to recalculate summaries with and without it. This is a sensitivity check: it compares the summaries under two different data sets. It does not, by itself, justify deleting the observation.
Here we focus on three measures of center and spread: the mean, the median, and the sample standard deviation \(s\). Recalculate each measure using only the retained observations. Then compare the size of the changes. Since all three measures use the same units as the original data, you can compare their absolute changes directly. A positive or negative change gives the direction; the absolute change tells you how much the measure moved.
Recalculate From the Data That Remain
The mean is the sum of the observations divided by their count. After deleting a value, both the sum and the number of observations change. Recalculate the mean rather than trying to adjust it by simply subtracting the deleted value.
The median is the middle value after ordering the retained observations. If there is an even number of observations, take the average of the two middle values. Deleting one observation can change which values occupy the middle positions, so find the median again from the shortened, ordered list.
The sample standard deviation \(s\) describes the typical distance of observations from their mean. As in Describing Spread in Context, use the sample standard deviation formula for sample data. After deleting a point, recalculate both the mean and the distances from that new mean, as well as the sample size in the denominator.
The standard deviation is sensitive to unusually distant values: it uses squared distances from the mean, so large distances contribute substantially to the total. The mean is also sensitive because every observation contributes to its sum. The median depends on order and middle position rather than on how far an extreme value lies from the rest. It may therefore change less, although it is not guaranteed to stay the same.
When you report which measure changed most, compare the absolute differences between the original and revised summaries. For example, if a mean decreases by 3 units and a standard deviation decreases by 5 units, the standard deviation changed more. Do not compare only the signs of the changes, and do not compare percentages unless the question asks for them.
A Recalculation Routine
Keep the two versions of the data separate as you work. This makes it easier to catch a common error: removing the point from the mean calculation but accidentally leaving it in the standard deviation calculation, or vice versa.
Name the value being examined and the variable’s units. Treat it as a flagged or unusual observation unless there is evidence establishing why it should be excluded.
Use all observations to find the mean, median, and sample standard deviation.
Make a new list containing only the retained observations. Recalculate all three summaries from this list.
Subtract the original summary from the revised summary, take the absolute value, and report which of the three changes is largest, with units and context.
Worked Examples
Worked Example: A High Repair Time
A fictional repair desk records service times, in minutes, of 8, 9, 9, 10, 10, 11, 12, and 31. The 31-minute observation has been flagged as unusually high. Calculate the mean, median, and sample standard deviation with and without that observation, then report which changed most.
Original data. There are \(n=8\) observations with sum \(100\), so the mean is \(100/8=12.5\) minutes. The two middle observations are 10 and 10, giving a median of 10 minutes. The sum of squared deviations from 12.5 is \(402\), so
After removing 31. The retained observations have sum \(69\) and count \(7\), so their mean is \(69/7\approx 9.857\) minutes. Their middle observation is 10, so their median is 10 minutes. The sum of their squared deviations from the revised mean is \(76/7\approx 10.857\). Thus
The mean changes by \(|9.857-12.5|\approx 2.643\) minutes. The median changes by \(|10-10|=0\) minutes. The sample standard deviation changes by \(|1.345-7.578|\approx 6.233\) minutes. The standard deviation changed most among these three summaries.
Conclusion. In this fictional set of repair times, deleting the flagged 31-minute value lowers the mean and sharply reduces the sample standard deviation, while the median remains 10 minutes. This comparison shows how the value affects the summaries; it does not establish that the observation should be excluded from a description of all service times.
Worked Example: A Low Delivery Time
A fictional courier team records delivery times, in minutes, of 2, 18, 19, 20, 21, 22, 23, 24, and 25. The 2-minute time is flagged as unusually low. Find the three summaries before and after removing it, and identify which changes most.
Original data. The sum is \(174\), and there are \(9\) observations, so the mean is \(174/9\approx 19.333\) minutes. The fifth value in the ordered list is 21, so the median is 21 minutes. The sum of the squared observations is \(3740\). Using the computational form of the squared-deviation sum gives
After removing 2. The remaining values sum to \(172\), and their count is \(8\), so the mean is \(172/8=21.5\) minutes. Their two middle values are 21 and 22, giving a median of \((21+22)/2=21.5\) minutes. The sum of their squares is \(3736\), so
The mean changes by \(|21.5-19.333|\approx 2.167\) minutes. The median changes by \(|21.5-21|=0.5\) minutes. The sample standard deviation changes by \(|2.330-6.856|\approx 4.526\) minutes. The standard deviation changed most.
Conclusion. Removing the unusually low time raises the mean and median and lowers the sample standard deviation. Here, the standard deviation has the largest absolute change. The result is a comparison of summaries, not evidence that the 2-minute delivery was inaccurate.
Worked Example: An Outlying Equipment Reading
A fictional sensor records readings, in degrees Celsius, of 4, 5, 5, 6, 7, and 40. The 40-degree reading is flagged as unusually high. Recalculate the mean, median, and sample standard deviation with and without the reading.
Original data. The readings sum to \(67\), so the mean is \(67/6\approx 11.167\) degrees. The two middle observations are 5 and 6, so the median is \(5.5\) degrees. The sum of the squared readings is \(1751\), and the sum of squared deviations is
After removing 40. The five retained readings sum to \(27\), giving a mean of \(27/5=5.4\) degrees. Their middle value is 5, so their median is 5 degrees. Their squared readings sum to \(151\), and their sum of squared deviations is \(151-27^2/5=26/5\). Therefore,
The mean changes by \(|5.4-11.167|\approx 5.767\) degrees. The median changes by \(|5-5.5|=0.5\) degrees. The sample standard deviation changes by \(|1.140-14.162|\approx 13.022\) degrees. The standard deviation changed most.
Conclusion. For these readings, the 40-degree observation has a substantial effect on the mean and especially on the sample standard deviation, while the median moves only slightly. If the sensor reading is accurate and relevant, it may still belong in the data; investigate its source and the purpose of the description before deciding what to report.
Common Mistakes and AP Exam Tips
- Assuming an outlier must be deleted. A flag indicates that a value stands apart; it does not show that the value is wrong. A careful answer calls this a comparison “with the flagged observation removed,” rather than silently treating the shorter list as the only valid data set.
- Updating the mean incorrectly. After deletion, both the total and sample size change. Recalculate using the retained values and their count; do not just subtract the deleted value from the old mean.
- Leaving the old mean in the standard deviation formula. The sample standard deviation is based on distances from the mean of the data currently being summarized. Find the revised mean before calculating the revised standard deviation.
- Using the wrong denominator. For a sample standard deviation, use \(n-1\) for the data set being summarized. If one observation was removed, the revised denominator is the revised sample size minus one.
- Reusing the old median. Reorder the retained values and find their middle position or positions again. Deletion can change the number of observations and therefore which values are in the middle.
- Reporting only “the standard deviation changed.” State the two values, the absolute change, and the units. For full-credit communication, identify the data set and say which measure changed most in context.
- Comparing signed changes instead of magnitudes. One statistic may rise while another falls. Compare absolute differences to answer which changed most, and use the direction separately to describe what happened.
Check Your Understanding
For each question, use the observations that remain after deletion and report changes in the same units as the data.
- A fictional set of daily temperatures is 12, 13, 13, 14, 15, and 30 degrees. After removing 30, what are the original and revised means?
- For the same temperatures, find the original and revised medians. Explain why you must recalculate the median after deletion.
- Why does the sample standard deviation use a different denominator after one observation is removed?
- If the mean changes by 2 minutes, the median by 0 minutes, and the sample standard deviation by 4 minutes, which changed most? What should “changed most” mean here?
- A flagged reading is confirmed to be accurate and relevant. Does a large change in the standard deviation alone justify deleting it? Explain.