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?
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: $$ \begin{align*} \bar{X}_{n} &= 73.6 \\ s_{n} &= 8.939216 \end{align*} $$
