Answer :
To find the standard deviation of the given data set: 1.46, 1.52, 1.57, 1.58, 1.64, 1.71, 1.79, 1.87, 1.89, 2.22, follow these steps:
Calculate the Mean:
The mean is the average of all the numbers. Add all the numbers together and then divide by the total count of numbers.
[tex]\text{Mean} = \frac{1.46 + 1.52 + 1.57 + 1.58 + 1.64 + 1.71 + 1.79 + 1.87 + 1.89 + 2.22}{10}[/tex]
[tex]\text{Mean} = \frac{17.25}{10} = 1.725[/tex]
Subtract the Mean and Square the Result:
For each number, subtract the mean, and then square the result to avoid negative values.
[tex]\begin{align*}
(1.46 - 1.725)^2 &= 0.070225, \\
(1.52 - 1.725)^2 &= 0.042025, \\
(1.57 - 1.725)^2 &= 0.024025, \\
(1.58 - 1.725)^2 &= 0.021025, \\
(1.64 - 1.725)^2 &= 0.007225, \\
(1.71 - 1.725)^2 &= 0.000225, \\
(1.79 - 1.725)^2 &= 0.004225, \\
(1.87 - 1.725)^2 &= 0.021225, \\
(1.89 - 1.725)^2 &= 0.027225, \\
(2.22 - 1.725)^2 &= 0.245025
\end{align*}[/tex]Calculate the Variance:
The variance is the average of these squared differences. Sum all the squared differences and divide by the number of elements.
[tex]\text{Variance} = \frac{0.070225 + 0.042025 + 0.024025 + 0.021025 + 0.007225 + 0.000225 + 0.004225 + 0.021225 + 0.027225 + 0.245025}{10}[/tex]
[tex]\text{Variance} = \frac{0.46245}{10} = 0.046245[/tex]
Calculate the Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\text{Standard Deviation} = \sqrt{0.046245} \approx 0.2151[/tex]
So, the standard deviation of the data set is approximately 0.2151.
