Question

roberto took a sample of 5 recent matches for his favorite soccer team. here is how many goals the team scored in each match: 0, 2, 1, 2, 5 roberto found the mean was \\(\bar{x} = 2\\) goals. he thinks the standard deviation is \\(s_x = \sqrt{\frac{(2 - 2)^2 + (1 - 2)^2 + (2 - 2)^2 + (5 - 2)^2}{4}}\\) what is the error in robertos standard deviation calculation? choose 1 answer:

Explanation:

Step1: Recall sample standard deviation formula

The formula for the sample standard deviation \( s_x \) is \( s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}} \), where \( n \) is the sample size, \( x_i \) are the data points, and \( \bar{x} \) is the sample mean. For a sample of size \( n = 5 \), we use \( n-1=4 \) in the denominator? Wait, no, wait. Wait, the data points are \( 0,2,1,2,5 \). Let's list all the deviations. The data points are \( x_1 = 0 \), \( x_2=2 \), \( x_3 = 1 \), \( x_4=2 \), \( x_5 = 5 \). The mean \( \bar{x}=2 \). So the deviations are \( (0 - 2)^2 \), \( (2 - 2)^2 \), \( (1 - 2)^2 \), \( (2 - 2)^2 \), \( (5 - 2)^2 \). Roberto's formula has only four terms in the numerator, but there are five data points. So he missed the term for \( x_1 = 0 \), which is \( (0 - 2)^2 \).

Step2: Identify the error

In the sample standard deviation calculation, we need to sum the squared deviations for all data points. The data set has 5 values: \( 0,2,1,2,5 \). So the sum in the numerator should be \( (0 - 2)^2+(2 - 2)^2+(1 - 2)^2+(2 - 2)^2+(5 - 2)^2 \), but Roberto's formula only includes four terms (he missed the \( (0 - 2)^2 \) term). Also, let's check the denominator: for a sample, the denominator is \( n - 1 \), where \( n = 5 \), so \( n-1 = 4 \), which is correct. But the numerator is missing one squared deviation (the one for \( x = 0 \)).

Answer:

Roberto missed the squared deviation for the data point \( 0 \) (i.e., the term \( (0 - 2)^2 \)) in the numerator of the standard deviation formula. The correct numerator should include the squared deviation for each of the 5 data points, but his formula only includes 4.