Calculating Standard Deviation (Small Data Set)

Find the standard deviation of: $4, 8, 6, 5, 7$

Find the mean: $\bar{x} = \frac{4 + 8 + 6 + 5 + 7}{5} = \frac{30}{5} = 6$ = Mean = $6$

Find each deviation from the mean: $4-6=-2$, $8-6=2$, $6-6=0$, $5-6=-1$, $7-6=1$ = Deviations: $-2, 2, 0, -1, 1$

Square each deviation: $(-2)^2=4$, $(2)^2=4$, $(0)^2=0$, $(-1)^2=1$, $(1)^2=1$ = Squares: $4, 4, 0, 1, 1$

Find the mean of squared deviations (variance): $\text{Variance} = \frac{4+4+0+1+1}{5} = \frac{10}{5} = 2$ = Variance = $2$

Take the square root to get standard deviation: $\sigma = \sqrt{2} \approx 1.41$ = $\sigma \approx 1.41$

Answer: The standard deviation is approximately $1.41$, meaning values typically differ from the mean by about $1.41$ units.

Comparing Two Data Sets

Class A test scores: $70, 72, 68, 71, 69$ (Mean = $70$) Class B test scores: $50, 90, 60, 80, 70$ (Mean = $70$) Which class has more consistent scores?

Calculate Class A deviations and squares: $(70-70)^2 + (72-70)^2 + (68-70)^2 + (71-70)^2 + (69-70)^2$ $= 0 + 4 + 4 + 1 + 1 = 10$ = Sum of squares = $10$

Calculate Class A standard deviation: $\sigma_A = \sqrt{\frac{10}{5}} = \sqrt{2} \approx 1.41$ = $\sigma_A \approx 1.41$

Calculate Class B deviations and squares: $(50-70)^2 + (90-70)^2 + (60-70)^2 + (80-70)^2 + (70-70)^2$ $= 400 + 400 + 100 + 100 + 0 = 1000$ = Sum of squares = $1000$

Calculate Class B standard deviation: $\sigma_B = \sqrt{\frac{1000}{5}} = \sqrt{200} \approx 14.14$ = $\sigma_B \approx 14.14$

Compare the results: $\sigma_A \approx 1.41$ vs $\sigma_B \approx 14.14$ = Class A is much more consistent

Answer: Class A has more consistent scores. Even though both classes have the same mean ($70$), Class A's standard deviation ($1.41$) is much smaller than Class B's ($14.14$).

Finding Variance from Standard Deviation

A data set has a standard deviation of $\sigma = 5$. What is the variance?

Recall the relationship: Variance = $(\sigma)^2$ Standard deviation = $\sqrt{\text{Variance}}$ = Variance is the square of standard deviation

Square the standard deviation: Variance = $5^2 = 25$ = Variance = $25$

Verify by reversing: $\sqrt{25} = 5 = \sigma$ (correct!) = Verification passed

Answer: The variance is $25$. Since standard deviation is the square root of variance, squaring the standard deviation gives you the variance.

Mistake: Forgetting to square the deviations

Why: Without squaring, positive and negative deviations would cancel out, giving a misleading result of zero.

Correct: Always square the deviations: $(x_i - \bar{x})^2$. This ensures all values are positive.

Mistake: Forgetting the final square root

Why: Without the square root, you have variance, not standard deviation. They measure the same thing but in different units.

Correct: Standard deviation = $\sqrt{\text{variance}}$. Always take the square root at the end.

Mistake: Confusing population and sample formulas

Why: Sample standard deviation divides by $(n-1)$ instead of $n$ to correct for bias when estimating from a sample.

Correct: Population: divide by $n$. Sample: divide by $(n-1)$. In this lesson, we use the population formula.

Quality Control in Manufacturing

Factories measure product dimensions to ensure consistency. Standard deviation shows how much variation exists.

A bolt factory aims for bolts of $10$ mm diameter. If the standard deviation is $0.1$ mm, most bolts are between $9.9$ and $10.1$ mm.

Investment Risk Analysis

Financial analysts use standard deviation to measure how much an investment's returns fluctuate (volatility).

Stock A has returns with standard deviation $5\%$. Stock B has standard deviation $20\%$. Stock B is riskier because its returns vary more.

Q: What is the difference between standard deviation and variance?

A: Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. They measure the same thing, but standard deviation is in the same units as your data, making it easier to interpret.

Q: Can standard deviation be negative?

A: No. Standard deviation is always zero or positive. Since we square all deviations (making them positive) and then take a square root, the result cannot be negative. A standard deviation of zero means all values are identical.

Q: Why do we square the deviations?

A: If we just added up the deviations, positive and negative values would cancel out (their sum equals zero). Squaring ensures all values become positive, so they accumulate instead of canceling.