excluderho

2022-06-29

Derivation of formula for mean of a dataset after adding a new data point

I was reading an article that stated that if we know the mean ${\overline{x}}_{prev}$ of a dataset with n datapoints, and that if we add a new data point ${x}_{k}$ to the dataset, then the new mean ${\overline{x}}_{new}$ can be expressed by the formula:

${\overline{x}}_{new}=\frac{1}{n+1}\sum _{i=1}^{n+1}{x}_{i}=\frac{1}{n+1}({x}_{k}+\sum _{i=1}^{n}{x}_{i})$

It is not obvious to me how this expression is true.

This is my attempt at deriving the expression:

${\overline{x}}_{new}=\frac{1}{n+1}\sum _{i=1}^{n+1}{x}_{i}=\frac{1}{n+1}({x}_{k}+\sum _{i=1}^{n}{x}_{i})$

Note that $\sum _{i=1}^{n}{x}_{i}=n{\overline{x}}_{prev}$

${\overline{x}}_{new}=\frac{1}{n+1}({x}_{k}+n{\overline{x}}_{prev})$

but is does not seem like this is getting me anywhere.

I was reading an article that stated that if we know the mean ${\overline{x}}_{prev}$ of a dataset with n datapoints, and that if we add a new data point ${x}_{k}$ to the dataset, then the new mean ${\overline{x}}_{new}$ can be expressed by the formula:

${\overline{x}}_{new}=\frac{1}{n+1}\sum _{i=1}^{n+1}{x}_{i}=\frac{1}{n+1}({x}_{k}+\sum _{i=1}^{n}{x}_{i})$

It is not obvious to me how this expression is true.

This is my attempt at deriving the expression:

${\overline{x}}_{new}=\frac{1}{n+1}\sum _{i=1}^{n+1}{x}_{i}=\frac{1}{n+1}({x}_{k}+\sum _{i=1}^{n}{x}_{i})$

Note that $\sum _{i=1}^{n}{x}_{i}=n{\overline{x}}_{prev}$

${\overline{x}}_{new}=\frac{1}{n+1}({x}_{k}+n{\overline{x}}_{prev})$

but is does not seem like this is getting me anywhere.

Jovan Wong

Beginner2022-06-30Added 23 answers

${\overline{x}}_{new}=\frac{1}{n+1}({x}_{k}+n{\overline{x}}_{prev})={\overline{x}}_{prev}+\frac{1}{n+1}({x}_{k}+n{\overline{x}}_{prev}-(n+1){\overline{x}}_{prev})$

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