The test scores of a geometry class are given below:
[tex]$90,75,72,88,85$[/tex]
The teacher wants to find the variance for the class population. What is the value of the numerator of the calculation of the variance?
Variance:
[tex]$\sigma^2=\frac{\left(x_1-\mu\right)^2+\left(x_2-\mu\right)^2+\ldots+\left(x_N-\mu\right)^2}{N}$[/tex]
A. -160
B. -6
C. 16
D. 258
Answer (2)
Calculate the mean of the test scores: μ = 5 90 + 75 + 72 + 88 + 85 = 82 .
Subtract the mean from each score and square the result.
Sum the squared differences: 64 + 49 + 100 + 36 + 9 = 258 .
The numerator of the variance is 258 .
Explanation
Understanding the Problem We are given the test scores of a geometry class: 90 , 75 , 72 , 88 , 85 . The teacher wants to find the variance for the class population, and we need to find the numerator of the variance calculation. The formula for variance is given as: σ 2 = N ( x 1 − μ ) 2 + ( x 2 − μ ) 2 + … + ( x N − μ ) 2 where x i represents each individual score, μ is the mean of the scores, and N is the number of scores.
Calculating the Mean First, we need to calculate the mean ( μ ) of the test scores. The mean is the sum of all the scores divided by the number of scores. So, μ = 5 90 + 75 + 72 + 88 + 85 = 5 410 = 82
Finding the Differences from the Mean Next, we subtract the mean from each test score ( x i − μ ):
90 − 82 = 8
75 − 82 = − 7
72 − 82 = − 10
88 − 82 = 6
85 − 82 = 3
Squaring the Differences Now, we square each of the differences ( ( x i − μ ) 2 ):
8 2 = 64
( − 7 ) 2 = 49
( − 10 ) 2 = 100
6 2 = 36
3 2 = 9
Summing the Squared Differences Then, we sum the squared differences: i = 1 ∑ N ( x i − μ ) 2 = 64 + 49 + 100 + 36 + 9 = 258
Finding the Numerator of the Variance The numerator of the variance is the sum of the squared differences, which is 258.
Examples
Understanding variance is crucial in many real-world scenarios. For instance, in finance, it helps assess the risk associated with investments. A higher variance indicates greater volatility, meaning the investment's returns can fluctuate significantly. Similarly, in manufacturing, variance is used to monitor the consistency of product dimensions. By calculating the variance, manufacturers can identify and address any deviations from the desired specifications, ensuring product quality and minimizing defects. This concept extends to various fields, including sports analytics, environmental science, and education, where understanding data spread is essential for informed decision-making.
The numerator of the variance for the test scores is calculated by first finding the mean of the scores, then calculating the differences between each score and the mean, squaring those differences, and finally summing them up. The total sum of the squared differences is 258. Therefore, the answer is 258.
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