6.4 Confidence Intervals for Variance and
Standard Deviation
• Key Concepts:
– Point Estimates for the Population Variance
and Standard Deviation
– Chi-Square Distribution
– Building and Interpreting Confidence Intervals
for the Population Variance and Standard
Deviation
6.4 Confidence Intervals for Variance and
Standard Deviation
• How do we estimate the population variance or
the population standard deviation using sample
data?
– The variation we see in the sample will be our best
guess.
• the sample variance, s2, is used to estimate σ2
• the sample standard deviation, s, is used to estimate σ
• To build confidence intervals for σ2 and σ, we
start with the sampling distribution of a modified
version of s2.
6.4 Confidence Intervals for Variance and
Standard Deviation
• If we find all possible samples of size n from a
normal population of size N and then record the
value of
n  1 2

 
s
2
2

2

for each sample, it can be shown that
follows
a chi-square distribution with n – 1 degrees of
freedom.
6.4 Confidence Intervals for Variance and
Standard Deviation
• Properties of the chi-square distribution:
– All chi-square vales are greater than or equal to zero.
– The shape of a chi-square curve is determined by the
number of degrees of freedom.
– The area below a chi-square curve is 1.
– All chi-square curves are positively skewed.
• Practice working with chi-square curves
#4 p. 334
#6
6.4 Confidence Intervals for Variance and
Standard Deviation
• How do we build confidence intervals using this
information?
We can start with:
n  1 2

2
 
 s  R
2
2
L

and use algebra to get to:
 n  1  s 2   2   n  1  s 2
 R2
 L2
6.4 Confidence Intervals for Variance and
Standard Deviation
• Fortunately, we can use the previous result for
both confidence intervals.
– To build a confidence interval for the population
variance, we use:
 n  1  s 2   2   n  1  s 2
 R2
 L2
– To build a confidence interval for the population
standard deviation, we use:
 n  1  s 2     n  1  s 2
2
2
R
L
6.4 Confidence Intervals for Variance and
Standard Deviation
• Guidelines for constructing these confidence
intervals are provided on page 332.
– Remember the population must be normal for
us to apply these techniques.
– When building our confidence intervals, we
need the chi-square curve with n – 1 degrees
of freedom.
• Practice:
#14 p. 334 (Cough Syrup)
#16 p. 334 (Washers)
#23 p. 335 (Waiting Times)