Answer :
To find the standard deviation of the numbers 1.47, 1.52, 1.55, 1.57, 1.58, 1.73, 1.84, 1.92, 1.94, and 2.22, follow these steps:
Step 1: Calculate the Mean
The mean (average) is calculated by adding all the numbers together, then dividing by the total count of numbers.
Mean [tex]\bar{x} = \frac{1.47 + 1.52 + 1.55 + 1.57 + 1.58 + 1.73 + 1.84 + 1.92 + 1.94 + 2.22}{10}[/tex]
[tex]\bar{x} = \frac{17.34}{10} = 1.734[/tex]
Step 2: Calculate the Variance
Variance is the average of the squared differences from the mean.
[tex]\text{Variance} = \frac{(1.47 - 1.734)^2 + (1.52 - 1.734)^2 + \ldots + (2.22 - 1.734)^2}{10}[/tex]
First, calculate the squared differences for each number:
[tex](1.47 - 1.734)^2 = 0.069444[/tex]
[tex](1.52 - 1.734)^2 = 0.045136[/tex]
[tex](1.55 - 1.734)^2 = 0.033664[/tex]
[tex](1.57 - 1.734)^2 = 0.026824[/tex]
[tex](1.58 - 1.734)^2 = 0.023104[/tex]
[tex](1.73 - 1.734)^2 = 0.000016[/tex]
[tex](1.84 - 1.734)^2 = 0.011236[/tex]
[tex](1.92 - 1.734)^2 = 0.034596[/tex]
[tex](1.94 - 1.734)^2 = 0.042724[/tex]
[tex](2.22 - 1.734)^2 = 0.235684[/tex]
Sum these squared differences:
[tex]0.069444 + 0.045136 + 0.033664 + 0.026824 + 0.023104 + 0.000016 + 0.011236 + 0.034596 + 0.042724 + 0.235684 = 0.522428[/tex]
Now, divide by the number of numbers (10):
[tex]\text{Variance} = \frac{0.522428}{10} = 0.052243[/tex]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance.
[tex]\text{Standard Deviation} = \sqrt{0.052243} \approx 0.2285[/tex]
Therefore, the standard deviation of the given numbers is approximately 0.2285.
