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].
