Inferences about One-Population Standard Deviation
Copyright © 2007–2022 by Stan Brown, BrownMath.com
Copyright © 2007–2022 by Stan Brown, BrownMath.com
Summary: In class you learn to estimate population means and test hypotheses about them. It can also be important to estimate or test variability — standard deviation or variance of a population. This page shows you how. All operations can be done in the accompanying Excel workbook (43 KB), or in the downloadable TI-83/84 program MATH200B part 5.
The tests on standard deviation or variance of a population require that the underlying population must be normal. They are not robust, meaning that even moderate departures from normality can invalidate your analysis. See MATH200A Program part 4 for procedures to test whether a population is normal by testing the sample.
Outliers are also unacceptable and must be ruled out. See MATH200A Program part 2 for an easy way to test for outliers.
For Excel users, Normality Check and Finding Outliers in Excel is an easy way to make both tests. A free workbook is included.
You already know how to test the mean of a population with a t test, or estimate a population mean using a t interval. Why would you want to do that for the standard deviation of a population?
The standard deviation measures variability. In many situations not just the average is important, but also the variability. Another way to look at it is that consistency is important: the variability must not be too great.
For example, suppose you are thinking about investing in one of two mutual funds. Both show an average annual growth of 3.8% in the past 20 years, but one has a standard deviation of 8.6% and the other has a standard deviation of 1.2%. Obviously you prefer the second one, because with the first one there’s quite a good chance that you’d have to take a loss if you need money suddenly.
Industrial processes, too, are monitored not only for average output but for variability within a specified tolerance. If the diameter of ball bearings produced varies too much, many of them won’t fit in their intended application. On the other hand, it costs more money to reduce variability, so you may want to make sure that the variability is not too low either.
Write your hypotheses in the usual way. For H0, you compare (=, ≤, or ≥) the population standard deviation σ to the claimed value σo. H1 contains ≠, >, or < as usual.
The test statistic is
χ² = (n−1) s² / σo² with df = n− 1
You perform a one-tailed test by computing the cumulative probability from 0 to the χ² (left tail) or from χ² to 10^99 or ∞ (right tail). For a two-tailed test, compute the cumulative probability and double it.
Example 1: A machine packs cereal into boxes, and you don’t want too much variation from box to box. You decide that a standard deviation of no more than five grams (about 1/6 ounce) is acceptable. To determine whether the machine is operating within specification, you randomly select 45 boxes. Here are the weights of the boxes, in grams:
Solution: First, use
Stats to find the sample standard deviation,
which is 6.42 g. Obviously this is greater than the target
standard deviation of 5 g, but is it enough greater that you can
say the machine is not operating correctly, or could it have come from
a population with standard deviation no more than 5 g?
Your hypotheses are
H0: σ = 5, the machine is within spec (some books would say H0: σ ≤ 5)
H1: σ > 5, the machine is not working right
No α was specified, but for an industrial process with no possibility of human injury α = 0.05 seems appropriate.
Next, check the requirements: is the sample normally distributed and free of outliers? You can do it with a TI calculator or with a statistics package. (Excel users, please see Normality Check and Finding Outliers in Excel.) On my TI-84, I used MATH200A part 4 to make a quantile plot to check for normality, and MATH200A part 2 to make a box-whisker plot to check for outliers. The results are shown at right, and demonstrate that the sample of box weights is normally distributed with no outliers.
|TI-89||Stats/List Editor: [|
Now compute the test statistic:
χ² = (n−1) s² / σo² = 44×6.42²/5² = 72.54
Next, compute the p-value. Use either the
accompanying Excel workbook or your TI
χ²cdf function; see
keystrokes at right for your model. (If you don’t have Excel or
a suitable calculator, you can probably find a table of the χ²
distribution at the back of your textbook.)
Use n−1 = 44 degrees
p-value = χ²cdf(Ans, 10^99, 44) = 0.0043
Conclusion: p-value < α; reject H0 and accept H1. The machine’s output is too variable: at the 0.05 level of significance the standard deviation is greater than 5 g.
Example 2: You have a random sample of size 20, with a standard deviation of 125. You have good reason to believe that the underlying population is normal. Is the population standard deviation different from 100, at the 0.05 significance level?
Solution: n = 20, s = 125, σo = 100, α = 0.05. Your hypotheses are
H0: σ = 100
H1: σ ≠ 100
This is a two-tailed test.
Compute the test statistic:
χ² = (n−1) s² / σo² = 19×125²/100² = 29.6875
Compute the p-value.
Since this is a two-tailed test, find the one-tailed p and double it.
(If the one-tailed p-value is >0.5, subtract from 1 and then
Either use the accompanying
Excel workbook or use your TI
χ²cdf function with degrees of
freedom n−1 = 19:
p = 2 × χ²cdf(Ans, 10^99, 19) = 0.1118
p-value > α; you fail to reject H0 and cannot reach a conclusion. The population standard deviation may be different from 100, or it may not.
You may have noticed that the test for σ actually uses the sample variance s² and the hypothetical population variance σo². Therefore, to make a test for variance you follow exactly the same procedure except that you already have the variance and you don’t square it to obtain the test statistic.
To estimate the standard deviation σ of a population at confidence level 1−α, the bounds are
√(n−1)s² / χ²crit(df,α/2) < σ < √(n−1)s² / χ²crit(df,1−α/2)
In the formula, χ²crit(df,rtail) is the critical χ² value that divides the curve with area (probability) rtail to the right and 1−rtail to the left. It’s an inverse χ² function, analogous to inverse t or inverse normal.
As usual when you’re dealing with standard deviations, df = n−1.
The regular χ²(df,teststat) means the
probability of getting a test statistic greater than teststat.
That probability is also the total area under the curve to the right
of that test statistic. You find this in a χ² table by looking
at the row and column for the test statistic and degrees of freedom,
and the probability is at their intersection in the body of the table.
On a calculator, it’s simply the
Excel has CHISQ.DIST.RT or CHIDIST.
By contrast, the inverse χ² function, χ²crit(df,rtail), means where—at what value of the test statistic—is the dividing line between area rtail to the right and area 1−rtail to the left. In terms of probability, it says: there’s some critical value of the test statistic, call it c, such that by random chance there’s an rtail probability of getting a test statistic > c, and it asks what that critical value c is. In a χ² table, you look along the row or column for the degrees of freedom until you find that rtail (or close to it), then look at the crossing row or column to find the critical value of the test statistic. See Finding χ²crit(df,rtail), below, for calculator and Excel options.
Caution: For standard deviation of a population, the confidence interval is not symmetric and the point estimate is not in the middle of the confidence interval. Therefore the confidence interval can be expressed only in endpoint form, not in s ± E form.
Example 3: Of several thousand students who took the same exam, 40 papers were selected randomly and statistics were computed. The standard deviation of the sample was 17 points. Estimate the standard deviation of the population, with 95% confidence. (Recall that test scores are normally distributed.)
Solution: 1−α = 0.95, so α = 0.05, α/2 = 0.025, and 1−α/2 = 0.975. df = n−1 =39. The confidence interval is
√37 × 17² / χ²crit(39,0.025) < σ < √37 × 17² / χ²crit(39,0.975)
How do we find the two required χ²crit values? There are several methods, laid out in Finding χ²crit(df,rtail), below. For now let’s just use the values: χ²(39,0.025) = 58.12006 and χ²(39,0.975) = 23.65432. Continuing with the calculation,
√39×17²/58.12006 < σ < √39×17²/23.65432
13.9 < σ < 21.8 with 95% confidence
Remark: The middle of the confidence interval is (13.9+21.8)/2 ≈ 17.9, but the point estimate was 17. The confidence interval is not symmetric: it extends 3.1 below and 4.8 above the point estimate.
χ²crit, also known as inverse χ², is not easy to compute, but you’re not necessarily reduced to looking up tables in a book. Here are several methods using various forms of technology:
χ²crit(df,rtail) ≈ ½ [zcrit(rtail) + √2·df−1]², for df ≥ 30
Example: Using the normal approximation,
χ²(39,0.025) ≈ ½ [z(0.025)+√2×39−1]²
z(rtail) is easily found on your calculator by invNorm(1−rtail). In this case, z(0.025) = invNorm(1−0.025) = 1.9600, so
χ²(39,0.025) ≈ (1.9600+√77)² / 2
χ²(39,0.025) ≈ 57.61934
That agrees quite well with the true value of 58.12006 given above. Using the same technique, χ²(39,0.975) ≈ 23.22212 and the 95% confidence interval on σ is 14.0 to 22.0, close to the interval we found above.
Variance is the square of standard deviation, so the confidence interval procedure is the same except that you don’t take square roots:
(n−1)s² / χ²crit(df,α/2) < σ² < (n−1)s² / χ²crit(df,1−α/2)
Example 4: Heights of US males aged 18–25 are normally distributed. You take a random sample of 100 from that population and find a variance of 7.3 in². (Remember that the units of variance are the square of the units of the original measurement.) Estimate the variance of the height of US males aged 18–25, with 95% confidence.
Solution: For a 95% confidence interval, 1−α = 0.95 and α/2 = 0.025. From the accompanying workbook we find
χ²crit(df,α/2) = χ²crit(99,0.025) = 128.4219887
χ²crit(df,1−α/2) = χ²crit(99,0.975) = 73.36108022
The endpoints of the interval are therefore
99 × 7.3 / 128.42199 < σ² < 99 × 7.3 / 73.36108
5.6275... < σ² < 9.8512...
We’re 95% confident that the variance in heights of US males aged 18–25 is between 5.6 and 9.9 in².
Inserted the subscript crit to indicate using an inverse function to find a critical value; see explanation in text.
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