Calculate the Standard Deviation

Calculating Standard Deviation

Suppose we have the following dataset representing the test scores of 10 students:

$85, 90, 75, 88, 92, 78, 84, 91, 79, 87$

Step 1: Calculate the Mean

First, find the mean (average) of the dataset.

$Mean(μ)=85+90+75+88+92+78+84+91+79+8710\text{Mean} (\mu) = \frac{85 + 90 + 75 + 88 + 92 + 78 + 84 + 91 + 79 + 87}{10}$

$Mean(μ)=84910=84.9\text{Mean} (\mu) = \frac{849}{10} = 84.9$

Step 2: Calculate the Variance

Next, calculate the variance. Variance is the average of the squared differences from the mean.

Subtract the mean from each data point and square the result:

$85 – 84.9)^2 = 0.01$ $90 – 84.9)^2 = 26.01$ $75 – 84.9)^2 = 98.01$ $88 – 84.9)^2 = 9.61$ $92 – 84.9)^2 = 51.61$ $78 – 84.9)^2 = 47.61$ $84 – 84.9)^2 = 0.81$ $91 – 84.9)^2 = 37.21$ $79 – 84.9)^2 = 34.81$ $87 – 84.9)^2 = 4.41$

Find the average of these squared differences:

$Variance(σ2)=0.01+26.01+98.01+9.61+51.61+47.61+0.81+37.21+34.81+4.4110\text{Variance} (\sigma^2) = \frac{0.01 + 26.01 + 98.01 + 9.61 + 51.61 + 47.61 + 0.81 + 37.21 + 34.81 + 4.41}{10}$

$Variance(σ2)=310.110=31.01\text{Variance} (\sigma^2) = \frac{310.1}{10} = 31.01$

Step 3: Calculate the Standard Deviation

Finally, take the square root of the variance to get the standard deviation.

$Deviation(σ)=31.01≈5.57\text{Standard Deviation} (\sigma) = \sqrt{31.01} \approx 5.57$

Summary

For the dataset $85, 90, 75, 88, 92, 78, 84, 91, 79, 87$ , the mean is 84.9, the variance is 31.01, and the standard deviation is approximately 5.57.

The standard deviation provides a measure of the spread of the data points around the mean. A lower standard deviation indicates that the data points are closer to the mean, while a higher standard deviation indicates a wider spread.