The standard deviation of 30 items in a data set is 5/6. If every item in the data set is multiplied by 6, then the variance of the data set will be:
Understand the Problem
The question is asking for the variance of a data set after every item in the set has been multiplied by 6. Given that the standard deviation is known, we need to apply the property that if every item is multiplied by a constant, the variance is multiplied by the square of that constant.
Answer
$25$
Answer for screen readers
The variance of the data set after every item is multiplied by 6 will be $25$.
Steps to Solve
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Calculate the variance from standard deviation To find the variance, we square the standard deviation. Given that the standard deviation is $\frac{5}{6}$, the variance, $Var$, is calculated as: $$ Var = \left(\frac{5}{6}\right)^2 $$ $$ Var = \frac{25}{36} $$
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Apply the variance scaling rule When each item in the data set is multiplied by a constant, the variance is multiplied by the square of that constant. Here, the constant is 6, so we apply this rule: $$ \text{New Variance} = Var \times (6^2) $$
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Calculate the new variance Now substituting the values into the equation: $$ \text{New Variance} = \frac{25}{36} \times 36 $$ The $36$ cancels out: $$ \text{New Variance} = 25 $$
The variance of the data set after every item is multiplied by 6 will be $25$.
More Information
When individual data points in a data set are scaled by a constant factor, the variance increases by the square of that factor. In this case, multiplying by 6 resulted in the variance becoming $25$, illustrating how scaling affects the spread of data.
Tips
- Confusing the relationship between standard deviation and variance. Remember, variance is the square of the standard deviation.
- Forgetting to square the constant when scaling variance. Always remember to use the square of the constant applied to the data set.
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