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------------------------------------------------ What is the standard deviation of the following numbers: 1.47, 1.52, 1.55, 1.57, 1.58, 1.73, 1.84, 1.92, 1.94, 2.22?

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.