What is the first error she made in computing the variance?
A. Emi failed to find the difference of 89-78 correctly.
B. Emi divided by [tex]N-1[/tex] instead of [tex]N[/tex].
C. Emi evaluated [tex](46-78)^2[/tex] as [tex]-(32)^2[/tex].
D. Emi forgot to take the square root of -135.6.
Answer (2)
The student incorrectly calculated ( 46 − 78 ) 2 as − ( 32 ) 2 instead of ( − 32 ) 2 .
Squaring any real number should result in a non-negative value.
The correct calculation should be ( − 32 ) 2 = 1024 .
Therefore, the first error is: Emi evaluated ( 46 − 78 ) 2 as − ( 32 ) 2 .
Explanation
Understanding the Variance Formula Let's carefully examine the provided variance calculation step by step to pinpoint the first error.
The formula for variance is: σ 2 = N ∑ i = 1 N ( x i − μ ) 2 where:
x i represents each individual data point,
μ is the mean of the data,
N is the number of data points.
Calculating the Differences The calculation starts correctly by finding the differences between each data point and the mean (78):
87 − 78 = 9 46 − 78 = − 32 90 − 78 = 12 78 − 78 = 0 89 − 78 = 11
So far, so good.
Identifying the Error Now, let's look at the squaring of these differences in the original calculation:
σ 2 = 5 ( 87 − 78 ) 2 + ( 46 − 78 ) 2 + ( 90 − 78 ) 2 + ( 78 − 78 ) 2 + ( 89 − 78 ) 2
The next line shows:
= 5 ( 9 ) 2 − ( 32 ) 2 + ( 12 ) 2 + ( 0 ) 2 + ( 11 ) 2
Here is the error! The student wrote − ( 32 ) 2 instead of ( − 32 ) 2 . Squaring any real number always results in a non-negative value.
The First Error The error is that Emi evaluated ( 46 − 78 ) 2 as − ( 32 ) 2 instead of ( − 32 ) 2 = 1024 . This is because the square of -32 should be positive 1024, not negative.
Examples
Understanding variance is crucial in many real-world scenarios. For example, in finance, it helps measure the risk associated with an investment portfolio. A higher variance indicates greater volatility, meaning the investment's returns can fluctuate significantly. In quality control, variance is used to ensure the consistency of manufactured products. By minimizing variance, companies can produce goods that meet specific standards, reducing defects and improving customer satisfaction. Calculating variance accurately is therefore essential for making informed decisions and maintaining quality across various fields.
The first error Emi made in computing the variance was evaluating ( 46 − 78 ) 2 as − ( 32 ) 2 instead of ( − 32 ) 2 . Squaring a negative number should always yield a positive result. This miscalculation is crucial as it affects the final variance result.
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