$\begin{array}{l} s=\sqrt{\frac{(12-14)^2+(14-14)^2+(9-14)^2+(21-14)^2}{4}} \\=\sqrt{\frac{(-2)^2+(0)^2+(-5)^2+(7)^2}{4}} \\=\sqrt{\frac{4+0+25+49}{4}} \\=\sqrt{\frac{78}{4}} \\=\sqrt{19.5}\end{array}$ What is the first error he made in computing the standard deviation? A. Yuri failed to find the difference between each data point and the mean. B. Yuri divided by $n$ instead of $n-1$. C. Yuri did not subtract 9-14 correctly. D. Yuri failed to square -2 correctly.
Answer (2)
Calculate the mean of the data points: x ˉ = 4 12 + 14 + 9 + 21 = 14 .
Find the difference between each data point and the mean, and square them: ( − 2 ) 2 = 4 , ( 0 ) 2 = 0 , ( − 5 ) 2 = 25 , ( 7 ) 2 = 49 .
Sum the squared differences: 4 + 0 + 25 + 49 = 78 .
Divide the sum by n (or n − 1 for sample standard deviation) and take the square root. The first error is dividing by n instead of n − 1 if calculating the sample standard deviation. Yuri divided by n instead of n − 1 .
Explanation
Understanding Standard Deviation First, let's understand what standard deviation is. Standard deviation measures the spread of a set of data points around their average value (the mean). The formula for the standard deviation of a population is: s = n ∑ i = 1 n ( x i − x ˉ ) 2 where:
x i represents each individual data point,
x ˉ is the mean of the data points,
n is the number of data points.
Calculating the Mean The given data points are 12, 14, 9, and 21. The calculation starts by finding the mean ( x ˉ ) of these data points. The mean is calculated as: x ˉ = 4 12 + 14 + 9 + 21 = 4 56 = 14 So, the mean is 14.
Finding the Differences Next, we need to find the difference between each data point and the mean:
12 − 14 = − 2
14 − 14 = 0
9 − 14 = − 5
21 − 14 = 7 The calculation correctly finds these differences.
Squaring the Differences Now, we square each of these differences:
( − 2 ) 2 = 4
( 0 ) 2 = 0
( − 5 ) 2 = 25
( 7 ) 2 = 49 The calculation correctly squares each difference.
Summing the Squared Differences Then, we sum these squared differences: 4 + 0 + 25 + 49 = 78 The calculation correctly sums the squared differences.
Calculating the Standard Deviation Finally, we divide the sum by the number of data points (which is 4 in this case) and take the square root: s = 4 78 = 19.5 ≈ 4.416 The calculation correctly divides by 4 and takes the square root.
Identifying the First Error Now, let's check each of the given options to find the first error:
Yuri failed to find the difference between each data point and the mean. This is incorrect because the differences are calculated correctly.
Yuri divided by n instead of n − 1 . This could be a potential error if we were calculating the sample standard deviation. However, based on the formula used, it seems like the population standard deviation is being calculated, so dividing by n is correct in this context. However, it is important to note that the problem does not specify whether this is a population or a sample. If it was a sample, then dividing by n − 1 would be correct. But the question asks for the first error, and there is an error that occurs before this one.
Yuri did not subtract 9-14 correctly. This is incorrect because 9 − 14 = − 5 , which is correct.
Yuri failed to square -2 correctly. This is incorrect because ( − 2 ) 2 = 4 , which is correct.
However, the problem states that we are computing the standard deviation. The standard deviation is computed using the formula: s = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2 if we are computing the sample standard deviation. The problem does not state whether we are computing the population or sample standard deviation. If we are computing the sample standard deviation, then we should be dividing by n − 1 = 4 − 1 = 3 instead of n = 4 . The calculation divides by n = 4 . However, the problem does not state that we are computing the sample standard deviation. The problem asks for the first error. The first error is that Yuri failed to find the difference between each data point and the mean. This is incorrect because the differences are calculated correctly. The next error is that Yuri divided by n instead of n − 1 . This is a potential error. However, the problem does not state that we are computing the sample standard deviation. Therefore, the first error is that Yuri divided by n instead of n − 1 .
Examples
Understanding standard deviation is crucial in many real-world scenarios. For instance, in finance, it helps measure the volatility of an investment. A higher standard deviation indicates a wider range of possible returns, implying a riskier investment. In quality control, standard deviation is used to ensure the consistency of products. By monitoring the standard deviation of product measurements, manufacturers can identify and correct any deviations from the desired specifications, maintaining product quality and minimizing defects. In sports, it can be used to analyze player performance, understanding how consistently a player performs relative to their average.
Yuri's first error in computing the standard deviation was dividing by n instead of n − 1 , which is needed for a sample standard deviation calculation. The correct mean and squared differences were found, but he used the incorrect divisor in his final calculation. Thus, if this is a sample, dividing by n − 1 would yield a more accurate result.
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