Real-world examples of hypothesis test for variance in statistics

Why examples of hypothesis test for variance matter in real work

Most people learning statistics get flooded with tests for the mean: z-tests, t-tests, ANOVA, and so on. But in many industries, the variance is the real star. Managers and regulators care about consistency, reliability, and risk. All of those are directly tied to variance.

A few everyday situations where examples of hypothesis test for variance are actually used:

• A manufacturer wants to know if a new machine produces parts with less variability in diameter.
• A hospital wants to check whether a new protocol stabilizes patient blood pressure readings.
• A financial analyst wants to see if a stock has become more volatile after a major policy announcement.

Each of these is an example of testing a claim about variance, not the mean. Below, we walk through several detailed examples of examples of hypothesis test for variance using the chi-square test (one variance) and the F-test (comparing two variances) in plain language and with real numbers.

Single-population chi-square examples of hypothesis test for variance

When you test whether one population’s variance equals some target value, you typically use the chi-square test for a single variance:

\[ \chi^2 = \frac{(n-1) s^2}{\sigma_0^2} \]

where:

• \( n \) is sample size
• \( s^2 \) is sample variance
• \( \sigma_0^2 \) is the hypothesized (target) variance

The test statistic follows a chi-square distribution with \( n-1 \) degrees of freedom under the null hypothesis.

Example 1: Manufacturing tolerance for bolt diameters

Imagine a plant that produces steel bolts. Engineering specs say bolt diameters should have a standard deviation of 0.02 inches (variance \( \sigma_0^2 = 0.0004 \)). The quality manager suspects the process has become more variable.

They take a sample of 30 bolts and find a sample standard deviation of 0.028 inches.

• \( n = 30 \)
• \( s = 0.028 \Rightarrow s^2 = 0.000784 \)
• \( \sigma_0^2 = 0.0004 \)

Set up the hypotheses for an upper-tail test (looking for larger variance):

• \( H_0: \sigma^2 = 0.0004 \)
• \( H_a: \sigma^2 > 0.0004 \)

Compute the chi-square statistic:

[
\chi^2 = \frac{(30-1)(0.000784)}{0.0004} = \frac{29 \times 0.000784}{0.0004} \approx 56.84
]

With \( df = 29 \), you compare this to the chi-square critical value at, say, \( \alpha = 0.05 \). From chi-square tables or software, \( \chi^2_{0.95, 29} \approx 42.56 \).

Since 56.84 > 42.56, you reject \( H_0 \). This is a textbook example of hypothesis test for variance: the data support the idea that bolt diameters are more variable than specified, which could trigger maintenance, recalibration, or even a production halt.

Example 2: Blood pressure stability in a hypertension clinic

Healthcare data often care about stability, not just average levels. Suppose a clinic introduces a new counseling program for patients with hypertension. The goal isn’t just to lower mean systolic blood pressure; they also want readings to be less erratic.

Assume past records show systolic blood pressure for a certain group has a standard deviation of 18 mmHg (variance 324). After the program, the clinic collects a sample of 40 patients and finds a sample variance of 250.

Set up a lower-tail test (testing for reduced variance):

• \( H_0: \sigma^2 = 324 \)
• \( H_a: \sigma^2 < 324 \)

Compute the test statistic:

• \( n = 40 \)
• \( s^2 = 250 \)

[
\chi^2 = \frac{(40-1) \times 250}{324} = \frac{39 \times 250}{324} \approx 30.09
]

With \( df = 39 \), you look at the lower critical value for \( \alpha = 0.05 \). From chi-square tables, \( \chi^2_{0.05, 39} \approx 24.99 \).

Because 30.09 is not less than 24.99, you fail to reject \( H_0 \). In plain language: this example of hypothesis test for variance suggests there isn’t enough evidence that blood pressure readings have become more stable yet.

If you want more context on why variance matters in blood pressure and cardiovascular risk, the National Heart, Lung, and Blood Institute (NHLBI) offers accessible background on blood pressure management.

Example 3: Climate variability in daily temperatures

Climate researchers don’t only track changes in average temperature; the variability of daily temperatures also matters for agriculture, energy demand, and health. Suppose a city’s historical daily high temperatures in July have a long-term standard deviation of 6 °F (variance 36). A scientist wants to know if recent years have become more variable.

They take a sample of 31 daily highs from a recent July and compute a sample variance of 52.

• \( n = 31 \)
• \( s^2 = 52 \)
• \( \sigma_0^2 = 36 \)

Hypotheses for an upper-tail test:

• \( H_0: \sigma^2 = 36 \)
• \( H_a: \sigma^2 > 36 \)

[
\chi^2 = \frac{(31-1) \times 52}{36} = \frac{30 \times 52}{36} \approx 43.33
]

With \( df = 30 \), \( \chi^2_{0.95, 30} \approx 43.77 \). Since 43.33 is **slightly below** 43.77, you would not reject \( H_0 \) at the 5% level. This example of hypothesis test for variance illustrates a borderline case: the sample looks more variable, but not quite enough to cross the conventional 5% threshold.

For a deeper dive into climate variability data and methods, the NOAA Climate.gov portal offers extensive datasets and methodology notes.

F-test examples of hypothesis test for variance (comparing two groups)

When you compare two variances, you typically use the F-test:

\[ F = \frac{s_1^2}{s_2^2} \]

where \( s_1^2 \) and \( s_2^2 \) are the sample variances from two independent groups. Under \( H_0: \sigma_1^2 = \sigma_2^2 \), this statistic follows an F distribution with \( df_1 = n_1 - 1 \) and \( df_2 = n_2 - 1 \).

Example 4: Old machine vs. new machine in a production line

A classic industrial engineering scenario: a company installs a new filling machine for cereal boxes and wants to know if the variance in fill weight is lower than that of the old machine.

• Old machine: sample of 25 boxes, standard deviation 1.8 oz (variance 3.24)
• New machine: sample of 20 boxes, standard deviation 1.2 oz (variance 1.44)

We want to test if the new machine has smaller variance. For the F-test, it’s standard to put the larger sample variance in the numerator so \( F \ge 1 \).

Let:

• \( s_1^2 = 3.24 \) (old machine)
• \( s_2^2 = 1.44 \) (new machine)

Hypotheses:

• \( H_0: \sigma_1^2 = \sigma_2^2 \)
• \( H_a: \sigma_1^2 > \sigma_2^2 \) (old machine more variable than new)

Compute:

\[ F = \frac{3.24}{1.44} = 2.25 \]

Degrees of freedom: \( df_1 = 24 \), \( df_2 = 19 \). At \( \alpha = 0.05 \), the upper critical F value \( F_{0.95, 24, 19} \) is around 2.12 (exact value depends on table or software).

Since 2.25 > 2.12, reject \( H_0 \). This example of hypothesis test for variance supports the claim that the old machine is more variable, and the new machine provides more consistent fills.

Example 5: Stock volatility before and after a policy change

Financial markets live and die on variance. Volatility is essentially variance in disguise. Suppose an analyst wants to compare the daily returns of a stock before and after a major Federal Reserve announcement.

• Period A (before): 60 trading days, sample variance of daily returns = 0.0009
• Period B (after): 60 trading days, sample variance = 0.0016

We want to know if volatility increased.

Let:

• \( s_1^2 = 0.0016 \) (after)
• \( s_2^2 = 0.0009 \) (before)

Hypotheses:

• \( H_0: \sigma_{\text{after}}^2 = \sigma_{\text{before}}^2 \)
• \( H_a: \sigma_{\text{after}}^2 > \sigma_{\text{before}}^2 \)

Compute the F statistic:

\[ F = \frac{0.0016}{0.0009} \approx 1.78 \]

Degrees of freedom: \( df_1 = 59 \), \( df_2 = 59 \). For \( \alpha = 0.05 \), \( F_{0.95, 59, 59} \) is around 1.54.

Since 1.78 > 1.54, you reject \( H_0 \) and conclude that variance (volatility) increased after the policy change. This is one of the best examples of hypothesis test for variance in finance because it directly informs risk management and portfolio allocation.

For more on volatility and risk measures, the Federal Reserve’s data portal and research pages offer timely series and analyses.

Example 6: Comparing variance in test scores between two teaching methods

Education research in 2024–2025 is heavily focused not just on average performance, but on equity and consistency across students. A school district tests a new online learning platform and wants to see whether it leads to less spread in standardized test scores compared with traditional instruction.

• Traditional class: 40 students, sample variance of math scores = 120
• Online platform: 35 students, sample variance = 80

We’re interested in whether the traditional class has greater variance.

Let:

• \( s_1^2 = 120 \) (traditional)
• \( s_2^2 = 80 \) (online)

Hypotheses:

• \( H_0: \sigma_1^2 = \sigma_2^2 \)
• \( H_a: \sigma_1^2 > \sigma_2^2 \)

\[ F = \frac{120}{80} = 1.5 \]

Degrees of freedom: \( df_1 = 39 \), \( df_2 = 34 \). For \( \alpha = 0.05 \), the upper critical F value is around 1.72.

Since 1.5 < 1.72, you fail to reject \( H_0 \). This example of hypothesis test for variance suggests that, with this sample, you can’t say the online platform produces more consistent scores, even though the raw variances differ.

For background on modern education research and assessment, the National Center for Education Statistics (NCES) is a reliable starting point.

Two-sided examples of hypothesis test for variance

So far, most examples of examples of hypothesis test for variance have been one-sided: “greater than” or “less than.” But sometimes you just want to know if variance is different in either direction.

Example 7: Pharmaceutical batch consistency

Drug manufacturing is tightly regulated. Suppose a pharmaceutical company produces a tablet that must have a specific active ingredient content with a target variance of 1.5 (mg²) across tablets. Regulators are concerned if the variance is too high or too low (too low might suggest over-control or issues with mixing assumptions).

A sample of 50 tablets from a new production line yields a sample variance of 2.1.

• \( n = 50 \)
• \( s^2 = 2.1 \)
• \( \sigma_0^2 = 1.5 \)

Two-sided hypotheses:

• \( H_0: \sigma^2 = 1.5 \)
• \( H_a: \sigma^2 \neq 1.5 \)

[
\chi^2 = \frac{(50-1) \times 2.1}{1.5} = \frac{49 \times 2.1}{1.5} \approx 68.6
]

\( df = 49 \). For a two-sided test at \( \alpha = 0.05 \), you look at both tails: \( \chi^2_{0.975, 49} \) and \( \chi^2_{0.025, 49} \). Approximate values:

• Upper critical: \( \chi^2_{0.975, 49} \approx 70.22 \)
• Lower critical: \( \chi^2_{0.025, 49} \approx 31.55 \)

Since 68.6 is between 31.55 and 70.22, you fail to reject \( H_0 \). This two-sided example of hypothesis test for variance shows how even a noticeable difference in sample variance may still be statistically compatible with the target variance when the sample is moderate.

For context on why variance control is so heavily monitored in pharmaceuticals, the FDA’s Drug Quality Sampling and Testing resources provide regulatory background.

Example 8: Comparing variance in wait times across two years

Customer experience teams in 2024–2025 often track not just average wait time but how unpredictable those waits are. Suppose a call center wants to know whether the variance in wait times changed from 2023 to 2024 after implementing an AI-based call routing system.

• 2023 sample: 100 calls, sample variance of wait times = 16 (minutes²)
• 2024 sample: 100 calls, sample variance = 10

We test whether the variances are different, not specifically higher or lower.

Let:

• \( s_1^2 = 16 \) (2023)
• \( s_2^2 = 10 \) (2024)

Hypotheses:

• \( H_0: \sigma_{2023}^2 = \sigma_{2024}^2 \)
• \( H_a: \sigma_{2023}^2 \neq \sigma_{2024}^2 \)

\[ F = \frac{16}{10} = 1.6 \]

Degrees of freedom: \( df_1 = 99 \), \( df_2 = 99 \). For a two-sided test at \( \alpha = 0.05 \), you use the upper critical value \( F_{0.975, 99, 99} \) and its reciprocal.

Approximate upper critical \( F_{0.975, 99, 99} \approx 1.39 \). The lower critical is about \( 1 / 1.39 \approx 0.72 \).

Since 1.6 > 1.39, the F statistic falls in the rejection region. This example of hypothesis test for variance suggests that variance in wait times did change between 2023 and 2024. The next step is interpretation: the 2024 variance is lower, so waits have become more predictable, which is usually great news for customers.

Interpreting examples of hypothesis test for variance in practice

Across these examples of examples of hypothesis test for variance, a few patterns show up:

• You almost always start with a null hypothesis that variances are equal (to a standard or between groups).
• The chi-square test handles a single variance vs. a target value.
• The F-test compares two independent sample variances.
• Results are highly sensitive to normality assumptions. If your data are very skewed or heavy-tailed, classical variance tests can mislead you.

In 2024 and beyond, analysts often pair these classical tests with:

• Levene’s test or Brown–Forsythe test for more robust comparisons of spread.
• Bootstrapping to estimate confidence intervals for variance without relying as heavily on distributional assumptions.

Still, the best examples of hypothesis test for variance—like those in manufacturing, healthcare, finance, and customer operations—remain grounded in these classic chi-square and F frameworks, especially when communicating with regulators, auditors, or other statisticians.

FAQ: Short answers about variance hypothesis tests

Q1. Can you give a simple example of hypothesis test for variance?
Yes. Suppose a process is designed so that the standard deviation of product weight is 2 grams. You sample 25 items, find a sample standard deviation of 3 grams, and test whether the variance is higher than the target using a chi-square test. That’s a straightforward example of hypothesis test for variance.

Q2. When should I use the chi-square test vs. the F-test for variance?
Use the chi-square test when you’re comparing one sample variance to a known or target variance. Use the F-test when you’re comparing two independent sample variances. Both assume normality of the underlying populations.

Q3. Are there real examples of variance tests in healthcare?
Yes. Hospitals might test whether a new medication schedule reduces the variance in blood pressure or glucose readings. Clinical trials also monitor variance in outcomes to check consistency across sites or subgroups, often alongside mean comparisons. Organizations like NIH frequently publish studies where stability and variability of patient outcomes are central.

Q4. What if my data are not normally distributed?
Classical chi-square and F-based examples of hypothesis test for variance rely on normality. If your data are skewed or heavy-tailed, consider transformations, nonparametric tests like Levene’s test, or resampling methods (bootstrapping) that don’t lean so hard on distributional assumptions.

Q5. Why do regulators care about variance, not just the mean?
Because variance tells you about reliability. A product with the correct mean but wild variability can still fail customers or patients. In pharmaceuticals, medical devices, or food production, too much variation can be a safety or quality red flag even when averages look fine.

These examples of examples of hypothesis test for variance are meant to give you realistic, data-focused templates you can adapt to your own work—whether you’re tuning a production line, analyzing market volatility, or checking how stable your latest experiment really is.