Section 9.3

Estimating a Population Standard Deviation

Use the chi-square distribution to construct confidence intervals for population variance and standard deviation.

1

Chi-Square Distribution

The chi-square (χ²) distribution is used to estimate population variance and standard deviation.

Properties

Not symmetric — skewed right

Values are always non-negative (χ² ≥ 0)

Shape depends on degrees of freedom:

Critical Warning: Data MUST come from a normally distributed population. These methods are not robust to departures from normality.

Interactive Chi-Square (χ²) Distribution Explorer

Discover how chi-square is used for estimating population variance

Why Chi-Square for Variance?

When estimating population variance , the statistic follows a chi-square distribution with . Unlike the normal distribution, chi-square is right-skewed and only takes positive values — which makes sense since variance can never be negative!

2 (very skewed)50 (more symmetric)
0612182430χ² (df = 10)Mean = 10Normal Approx.
10
Mean (μ = df)
20
Variance (σ² = 2df)
8
Mode (df - 2)
0.894
Skewness
Moderate skew

Chi-Square Properties

  • ✓Only takes positive values (x ≥ 0)
  • ✓Right-skewed (especially for small df)
  • ✓Shape depends on degrees of freedom
  • ✓Approaches normal as df → ∞

Why It's Different

  • •Sum of squared standard normal variables
  • •Squaring makes all values positive
  • •More df = more terms = more symmetric
  • •Critical values are asymmetric
The Connection: If are independent standard normal random variables, then follows a chi-square distribution with degrees of freedom. This is why — it's essentially a sum of squared standardized values!
2

Confidence Interval for σ²

Unlike confidence intervals for means and proportions, these intervals are not symmetric and are not in the "point estimate ± margin of error" form.

Note: The larger chi-square critical value goes in the denominator of the lower bound.

3

Confidence Interval for σ

To find the confidence interval for the standard deviation σ, simply take the square root of the lower and upper bounds of the variance interval.

4

Variance & Standard Deviation Calculator

χ² Confidence Interval Calculator for σ² and σ

Important: This method requires the population to be normally distributed. Chi-square confidence intervals are not robust to departures from normality.

Chi-Square Critical Values (df = 24)

12.4010
39.3640
95% CI for Population Variance (σ²)
(10.7550, 34.1392)
Sample variance s² = 17.6400
95% CI for Population Std Dev (σ)
(3.2795, 5.8429)
Sample std dev s = 4.2000

Formulas Used

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