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If the existing mean is 84 . The 6 is added to each observation and then it is divided by 9 then how to find the new mean

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    The best way is taking a look at a text book or wiki. – 2017-02-26
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    What is divided by $9$ ? Each obvervsation or the sum ? – 2017-02-26
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    $(84 \times 9+54)÷9$ if I have understood correctly. – 2017-02-26
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    The question is at best poorly worded. I'm assuming there were $9$ numbers because it says to $÷9$ – 2017-02-26
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    The sum is divided by 9 @peter – 2017-02-26

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You have $X_1, X_2, \dots, X_n,$ and you transform to $Y_i = (X_i + 6)/9 = \frac{1}{9}X_i + \frac{2}{3}.$ Then

$$\bar Y = \frac{1}{n}\sum_{i=1}^n Y_i = \frac{1}{n}\sum_{i=1}^n \left(\frac{1}{9}X_i + \frac{2}{3}\right) = \frac{1}{n}\frac{1}{9}\sum_{i=1}^n X_i + \frac{1}{n} \left(n\frac{2}{3}\right) =\frac{1}{9}\bar X + \frac{2}{3}.$$

But you are given that $\bar X = 84,$ so you can find the numerical value of $\bar Y.$

As you can see, in general: if $Y_i = aX_i + b,$ then $\bar Y = a\bar X + b.$ Maybe that relationship is mentioned in your book.

Note: Later in your course you may use random variables and their 'expected values' (or 'means'). A similar relationship holds for random variables: If $X$ is a random variable with expected value $E(X)$ and $Y = aX + b,$ then $E(Y) = aE(X) + b.$