What do I do if there is an error in the scores I collected?
Many times when collecting data, we may make errors that force us to adjust the scores. It is crucial to understand how these modifications affect measures of central tendency and dispersion.
How does a shift affect measures of central tendency?
Imagine that when grading exams, you discover an error in two questions that gives each student four additional points. In this case, you would need to add those four points to each original score. This involves recalculating:
- Mean: You notice that the new mean is increased by the same magnitude as the offset. So, if your original mean was 68.6, it will now be 72.6.
- Median: It will also increase by four units, from 67 to 71.
- Mode: It is affected in the same way, going from 66 to 70.
This shows us that the shift affects all these measures equally. When you move all your data in a consistent manner, the central tendencies adjust uniformly.
Are the range and IQR also affected?
Although the minimum and maximum of the data set change, the difference between them, the range, remains the same. If the minimum and maximum have the same offset added to them, the distance between them remains the same, so the range and interquartile range (IQR) are not affected. This is crucial when dealing with aggregate measurements, as you only need to adjust the central measures without worrying about recalculating the dispersion indices.
What happens if I multiply my data by a scalar?
Shifting is not the only type of transformation that can happen to your data; you can also scale it, i.e. multiply it by a constant number. Suppose you decide to double your original scores.
How are central tendencies and dispersion affected?
Multiplying by a scalar affects both measures of central tendency and dispersion:
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Mean, Median, and Mode: they are all multiplied by the same factor. If your mean score was 68.6, multiplying by two makes it 137.2.
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Range and IQR: These values now also expand, as the relative distances between values increase. The dispersion grows proportionally to the scalar.
Thus, this type of transformation explicitly affects how we perceive variability in the data set.
What happens if I remove or add elements to the data set?
Modifying the number of elements in your dataset can result in significant impacts:
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Mean and Median: These are sensitive to the number of elements. By deleting a value, the mean and median may change, having now to divide by a new number of elements.
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Range: Would maintain its value if the item removed or added is neither the minimum nor the maximum.
These adjustments require a re-evaluation of the main metrics, allowing us to maintain an accurate representation of the data.
Practical Tip
To better understand how these changes operate, I recommend practicing with your own data. Try shifting your values or multiplying them by a scalar and observe how the measurements change. This practice not only facilitates the understanding of these mathematical concepts, but also strengthens your ability to solve complex problems. Keep exploring and expanding your knowledge in statistics!
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