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Yuri computes the mean and standard deviation for the sample data set {12, 14, 9, 21}. He finds the mean is 14. His steps for finding the standard deviation are below.

[tex]\[ s = \sqrt{\frac{(12-14)^2 + (14-14)^2 + (9-14)^2 + (21-14)^2}{4}} \][/tex]
[tex]\[ = \sqrt{\frac{(-2)^2 + (0)^2 + (-5)^2 + (7)^2}{4}} \][/tex]
[tex]\[ = \sqrt{\frac{4 + 0 + 25 + 49}{4}} \][/tex]
[tex]\[ = \sqrt{\frac{78}{4}} \][/tex]
[tex]\[ = \sqrt{19.5} \][/tex]

What is the first error he made in computing the standard deviation?



Answer :

GinnyAnswer
Yuri's first error occurred when he divided by the total number of data points [tex]\( n \)[/tex] instead of [tex]\( n - 1 \)[/tex]. This is a crucial step when calculating the sample standard deviation. For a sample, the denominator should be [tex]\( n - 1 \)[/tex] to account for the degrees of freedom in the estimation of the population standard deviation from the sample data.

Let's go through the steps correctly:

1. Calculate the mean:
[tex]\[ \text{mean} = \frac{12 + 14 + 9 + 21}{4} = \frac{56}{4} = 14 \][/tex]

2. Find the squared differences from the mean:
[tex]\[ (12 - 14)^2 = (-2)^2 = 4 \][/tex]
[tex]\[ (14 - 14)^2 = 0^2 = 0 \][/tex]
[tex]\[ (9 - 14)^2 = (-5)^2 = 25 \][/tex]
[tex]\[ (21 - 14)^2 = 7^2 = 49 \][/tex]

3. Sum the squared differences:
[tex]\[ 4 + 0 + 25 + 49 = 78 \][/tex]

4. Divide by [tex]\( n - 1 \)[/tex] (since [tex]\( n = 4 \)[/tex]):
[tex]\[ \frac{78}{4 - 1} = \frac{78}{3} = 26 \][/tex]

5. Take the square root to find the sample standard deviation:
[tex]\[ s = \sqrt{26} \][/tex]

Therefore, the correct sample standard deviation is:
[tex]\[ s = \sqrt{26} \][/tex]

The first error Yuri made was not dividing by [tex]\( n - 1 \)[/tex] (which is 3 in this case) when calculating the sample standard deviation. Instead, he incorrectly divided by [tex]\( n \)[/tex].

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