Derivation of formula for mean of a dataset after adding a new data pointI was reading an article that stated that if we know the mean x ¯ p r e v of a dataset with n datapoints, and that if we add a new data point x k to the dataset, then the new mean x ¯ n e w can be expressed by the formula: x ¯ n e w = 1 n + 1 ∑ i = 1 n + 1 x i = 1 n + 1 ( x k + ∑ i = 1 n x i )It is not obvious to me how this expression is true.This is my attempt at deriving the expression: x ¯ n e w = 1 n + 1 ∑ i = 1 n + 1 x i = 1 n + 1 ( x k + ∑ i = 1 n x i )Note that ∑ i = 1 n x i = n x ¯ p r e v x ¯ n e w = 1 n + 1 ( x k + n x ¯ p r e v )but is does not seem like this is getting me anywhere.

Question

Derivation of formula for mean of a dataset after adding a new data pointI was reading an article that stated that if we know the mean                     x        ¯                    p      r      e      v       of a dataset with n datapoints, and that if we add a new data point       x          k       to the dataset, then the new mean                     x        ¯                    n      e      w       can be expressed by the formula:                    x        ¯                    n      e      w        =      1          n      +      1            ∑          i      =      1              n      +      1                  x      i        =      1          n      +      1        (      x          k        +      ∑          i      =      1              n            x    i    )It is not obvious to me how this expression is true.This is my attempt at deriving the expression:                    x        ¯                    n      e      w        =      1          n      +      1            ∑          i      =      1              n      +      1                  x      i        =      1          n      +      1        (      x          k        +      ∑          i      =      1              n            x    i    )Note that       ∑          i      =      1              n            x    i    =  n                    x        ¯                    p      r      e      v                          x        ¯                    n      e      w        =      1          n      +      1        (      x          k        +  n                    x        ¯                    p      r      e      v        )but is does not seem like this is getting me anywhere.

Jovan Wong · Accepted Answer

x        ¯                    n      e      w        =      1          n      +      1        (      x          k        +  n                    x        ¯                    p      r      e      v        )  =                    x        ¯                    p      r      e      v        +      1          n      +      1        (      x          k        +  n                    x        ¯                    p      r      e      v        −  (  n  +  1  )                    x        ¯                    p      r      e      v        )

Derivation of formula for mean of a dataset after adding a new data point I was reading an article

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