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I have a sample of grades from 1000 students. The average mark was 60 with a standard deviation of 3.

A year later I collected a sample of grades from 50 students sitting the same test. The average mark was 80 with a standard deviation of 4.

What is the best way to use these results to predict the expected average grade from another random sample of students (same time period as second sample)?

Should I use the first data set or the second, or some combination of the two?

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2 Answers 2

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Standard weighted averages use the square of the reciprocal of the deviation as their weight:

$$w_{1}=\frac{1}{3^2}=\frac{1}{9},w_{2}=\frac{1}{4^2}=\frac{1}{16}$$ The value then becomes $$\frac{\sum_{i} w_{i} x_{i}}{\sum_{i}w_i}=\frac{\frac{1}{9}60+\frac{1}{16}80}{\frac{1}{9}+\frac{1}{16}}=67.2$$

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You have enough data to say with rather high assurance that the two test results did not come from the same distribution. What would make you think the new sample comes from the same distribution as either of the other two ? Did you have a different professor last year? Did the students know the questions last year because it was the same test? Probably the best you can do is use the new data and ignore the other.

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  • $\begingroup$ For the sake of simplicity, I want to assume that all conditions are the same for the students, except the period of time is separated by 1 year. Now I agree the samples are statistically different, but the 1st sample had a much greater sample size, and thus might still have some weight on my future prediction, as the most recent sample could be an anomoly. $\endgroup$
    – Kenshin
    Jul 3, 2015 at 14:56
  • $\begingroup$ Well it depends if this is a critical thinking assignment or a simple statistics assignment. $\endgroup$
    – Feyre
    Jul 3, 2015 at 14:56
  • $\begingroup$ @Feyre, just a thought experiment I invented myself. I welcome the most technical answers. $\endgroup$
    – Kenshin
    Jul 3, 2015 at 14:57
  • $\begingroup$ @mew Sample size only means it's likely to have a slimmer standard deviation. If no other data is available to interpret the data, simple statistics is the best you're gonna get. $\endgroup$
    – Feyre
    Jul 3, 2015 at 15:00
  • $\begingroup$ @Feyre, does credibility theory apply here?en.wikipedia.org/wiki/Credibility_theory $\endgroup$
    – Kenshin
    Jul 3, 2015 at 15:02

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