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Explain why the differences must be squared during the calculation of standard deviation and variance, and describe the problem that would occur if the student only calculated the average of the raw differences.
Case context: A student in an introductory psychology lab is working with a dataset of eight scores (, and ) with a mean of . While trying to compute the standard deviation, the student asks why they cannot simply skip the squaring step and find the average of the raw differences between each score and the mean, since squaring seems to make the numbers artificially large.
Question: Explain why the differences must be squared during the calculation of standard deviation and variance, and describe the problem that would occur if the student only calculated the average of the raw differences.
Sample answer: Squaring the difference between each score and the mean is necessary to make all the difference values positive. If the student only averages the raw differences, the positive differences (from scores above the mean, like and ) and negative differences (from scores below the mean, like and ) will cancel each other out. This sum will equal zero, which fails to show any of the actual variability present in the dataset.
Key points:
- Squaring makes all differences positive.
- Without squaring, positive and negative differences cancel each other out.
- The sum of raw differences from the mean always equals zero.
- Squaring is required to produce a non-zero measure of variability (variance and standard deviation).
Rubric: Response must explain that squaring converts negative differences to positive values, and describe how raw differences sum to zero due to positive and negative values canceling out, thus failing to represent variability.
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Research Methods in Psychology - 4th American Edition @ KPU
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