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In below example, Is it possible to calculate new SD for one additional observation?

In a class of 25 students, 24 of them took an exam in class and 1 student took a make-up exam the following day. The professor graded the first batch of 24 exams and found an average score of 74 points with a standard deviation of 8.9 points. The student who took the make-up the following day scored 64 points on the exam.

New mean is 73.6, What is new SD? How to calculate it?

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I think this might be a near duplicate of this – Glen_b Jul 4 '14 at 9:16
up vote 5 down vote accepted

Yes, it's possible. Taken from this post, the updated mean and standard deviation (SD) can be calculated as follows: $$ \bar{X}_{n} = \frac{1}{n}\left(X_{n} + (n - 1)\bar{X}_{n-1}\right) $$ Where $n$ is the sample size (including the new observation), $X_{n}$ the value of the new observation $\bar{X}_{n-1}$ is the mean of the $n-1$ first observations. For the standard deviation, we have: $$ s_{n} = \sqrt{\frac{n-2}{n-1}s_{n-1}^{2}+\frac{1}{n}\left(X_{n} - \bar{X}_{n-1}\right)^{2}} $$ Where $s_{n-1}^{2}$ denotes the variance of the $n-1$ first observations.

Using the numbers $n=25, \bar{X}_{n-1}=74, s_{n-1}^{2}=8.9^{2}=79.21, X_{n}=64$, we get: $$ \begin{align*} \bar{X}_{n} &= 73.6 \\ s_{n} &= 8.939216 \end{align*} $$

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