Standard Deviation

Section	Details
Definition	Standard deviation is a measure of the dispersion of values in a dataset, indicating how much a data point deviates from the mean.
Types	1. Population Standard Deviation 2. Sample Standard Deviation
Population Standard Deviation	Formula: σ = √(1/N ∑(xi – μ)²) Where: σ = Population standard deviation N = Number of data points xi = Each data point μ = Population mean
Sample Standard Deviation	Formula: s = √(1/(n-1) ∑(xi – x̄)²) Where: s = Sample standard deviation n = Number of data points in the sample x̄ = Sample mean
Excel Functions	– Population Standard Deviation: STDEV.P – Sample Standard Deviation: STDEV.S
Python Functions	Use numpy.std() function: – Population: Set ddof=0 – Sample: Set ddof=1
Concept Explanation	A standard deviation (σ) measures data dispersion relative to the mean. – Low standard deviation: Data points cluster around the mean. – High standard deviation: Data points are more spread out.
Visual Representation	A graphical curve showing high and low standard deviations.
Calculation Example	For a class with average height of 75 inches: – Data Points: 56, 65, 74, 75, 76, 77, 80, 81, 91 – Mean (μ): 75 inches
Steps to Calculate	1. Subtract mean from each data point 2. Square the results 3. Sum the squared results 4. Divide by total number of data points 5. Take the square root
Statistical Insights	– 68% of heights within 75 ± 9.3 inches (1 standard deviation) – 95% within 75 ± 18.6 inches (2 standard deviations) – 99.7% within 75 ± 27.9 inches (3 standard deviations)