Variance and Standard Deviation
Variance and standard deviation is a Grade 11 Algebra 1 statistics skill from enVision Chapter 11 measuring how far data points typically sit from the mean. Standard deviation s = sqrt[sum(xi - x_bar)^2 / (n-1)] requires finding variance first (average squared deviation), then taking the square root to return to original units. For {80, 82, 78, 84, 81} with mean 81: squared deviations are 1, 1, 9, 9, 0 summing to 20; variance = 20/4 = 5; s = sqrt(5) = 2.24. Unlike IQR, standard deviation uses every data point. A standard deviation near zero means data is tightly clustered.
Key Concepts
Property Standard deviation ($s$ for a sample, $\sigma$ for a population) is the most powerful measure of spread. It calculates the typical, or "standard," distance that every single data point sits away from the mean.
The calculation requires finding the Variance first (the average of the squared differences from the mean), and then taking the square root to return to the original units.
Sample Standard Deviation Formula: $$s = \sqrt{\frac{\sum(x i \bar{x})^2}{n 1}}$$.
Common Questions
What is standard deviation?
The typical distance of data points from the mean. Calculated as the square root of variance: s = sqrt[sum(xi - x_bar)^2 / (n-1)].
How do you calculate variance?
Find the mean, subtract it from each value and square the result, sum all squared differences, then divide by n-1 for a sample.
Calculate the standard deviation of {80, 82, 78, 84, 81}.
Mean = 81. Squared deviations: 1, 1, 9, 9, 0. Sum = 20. Variance = 20/4 = 5. SD = sqrt(5) = 2.24.
Why do we square the deviations when calculating variance?
Squaring prevents negative deviations from canceling positive ones. Without squaring, the sum of deviations from the mean is always zero.
What does a standard deviation of 0 mean?
Every data point equals the mean. There is no variation in the dataset.
How does standard deviation differ from IQR?
IQR only measures the middle 50% of data. Standard deviation uses every data point and is more sensitive to outliers.