Last month, 5 customers of Macland Automotive bought the extended service plan with their purchase of a new car. All of their credit scores are given below. Calculate the standard deviation of this population data. Round your answer to 2 decimal places, if necessary.

\begin{tabular}{|c|}
\hline[tex]$x$[/tex] \\
\hline 697.6 \\
\hline 780.5 \\
\hline 746.8 \\
\hline 676.5 \\
\hline 689.8 \\
\hline
\end{tabular}

Standard deviation [tex]$=$[/tex] [tex]$\square$[/tex]



Answer :

To calculate the standard deviation of the given credit scores, we will follow a series of steps:

1. List the credit scores:
[tex]\[ 697.6, 780.5, 746.8, 676.5, 689.8 \][/tex]

2. Calculate the mean (average) of the credit scores:

The mean [tex]\(\mu\)[/tex] of a set of values is given by:

[tex]\[ \mu = \frac{\sum_{i=1}^{n} x_i}{n} \][/tex]

Here, [tex]\(n = 5\)[/tex] (the number of customers) and the scores [tex]\(x_i\)[/tex] are the credit scores provided.

[tex]\[ \mu = \frac{697.6 + 780.5 + 746.8 + 676.5 + 689.8}{5} = 718.24 \][/tex]

3. Calculate the deviations from the mean for each credit score:

Each deviation is calculated as [tex]\(x_i - \mu\)[/tex], where [tex]\(x_i\)[/tex] is each credit score.

[tex]\[ 697.6 - 718.24 = -20.64 \][/tex]
[tex]\[ 780.5 - 718.24 = 62.26 \][/tex]
[tex]\[ 746.8 - 718.24 = 28.56 \][/tex]
[tex]\[ 676.5 - 718.24 = -41.74 \][/tex]
[tex]\[ 689.8 - 718.24 = -28.44 \][/tex]

4. Square each deviation:

[tex]\[ (-20.64)^2 = 426.0096 \][/tex]
[tex]\[ (62.26)^2 = 3876.3076 \][/tex]
[tex]\[ (28.56)^2 = 815.6736 \][/tex]
[tex]\[ (-41.74)^2 = 1742.2276 \][/tex]
[tex]\[ (-28.44)^2 = 808.8336 \][/tex]

5. Calculate the variance ([tex]\(\sigma^2\)[/tex]) by taking the mean of these squared deviations:

[tex]\[ \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n} \][/tex]

[tex]\[ \sigma^2 = \frac{426.0096 + 3876.3076 + 815.6736 + 1742.2276 + 808.8336}{5} = 1533.8104 \][/tex]

6. Calculate the standard deviation ([tex]\(\sigma\)[/tex]), which is the square root of the variance:

[tex]\[ \sigma = \sqrt{\sigma^2} = \sqrt{1533.8104} \approx 39.16 \][/tex]

Therefore, the standard deviation of the credit scores is approximately [tex]\( \boxed{39.16} \)[/tex].