Adding a new data point to a set of test scores will affect the mean (average) of that set. The mean is calculated by summing all the scores and then dividing by the number of scores.
Here’s how to understand the effect of adding a new data point:
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Calculating the Original Mean:
- Let’s say you have a set of test scores \(S\) with \(n\) scores. The mean \(\mu\) can be calculated as: \[ \mu = \frac{\text{Sum of scores in } S}{n} \]
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Adding the New Data Point:
- When you add a new score \(x\), the new set of test scores will have \(n + 1\) scores.
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New Mean Calculation:
- The new mean \(\mu'\) after adding the score \(x\) is calculated as: \[ \mu' = \frac{\text{Sum of scores in } S + x}{n + 1} \]
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Effect on the Mean:
- The new mean will be influenced by the value of \(x\) in relation to the original mean \(\mu\).
- If \(x\) is greater than \(\mu\), the new mean will increase.
- If \(x\) is less than \(\mu\), the new mean will decrease.
- If \(x\) is equal to \(\mu\), the new mean will remain unchanged.
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Magnitude of Change:
- The effect of \(x\) on the mean is also dependent on the size of the original set of scores. The more scores you have, the less influence a single new score will have on the mean.
In summary, adding a new data point either raises, lowers, or maintains the mean, depending on the value of the new score compared to the existing mean, and the impact diminishes as the size of the data set increases.