Access this content offline
Omniguides by NOX
Skip to the reading

4.6 Differences Between Two Sample Means

Compare Two Averages with a Fixed Order

Two library branches have known population checkout-time parameters in our teaching model. Branch A has mean 18 minutes and SD 6 minutes; branch B has mean 15 minutes and SD 4 minutes. Take independent SRSs of 36 visits from A’s 6,000 recorded visits and 64 from B’s 8,000 visits. Both populations are moderately skewed without extreme outliers. Define the statistic as A’s sample mean minus B’s.

For independent populations and independently sampled values,

μ(xˉ1−xˉ2)=μ1−μ2,σ(xˉ1−xˉ2)=σ12n1+σ22n2.\mu_{(\bar x_1-\bar x_2)}=\mu_1-\mu_2,\qquad \sigma_{(\bar x_1-\bar x_2)}=\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}.

The means subtract in the chosen order. The two variance contributions add under the square root because both independent sample averages fluctuate. The population SDs must be squared before dividing by their respective sample sizes. Neither subtracting SDs nor pooling sample sizes gives the required formula.

For these branches,

μD=18−15=3 minutes,σD≈6236+4264=1.25=1.1180 minutes.\mu_D=18-15=3\text{ minutes},\qquad \sigma_D\approx\sqrt{\frac{6^2}{36}+\frac{4^2}{64}}=\sqrt{1.25}=1.1180\text{ minutes}.

The approximation acknowledges the small without-replacement sampling fractions and omits a finite-population correction. If the difference were defined B minus A, its mean would be −3 minutes, while its SD would still be 1.1180 minutes.

Two Random Samples Need Two Population Checks

For sampling without replacement, obtain two independent random samples and check each n against its own N:

n1≤0.10N1,n2≤0.10N2.n_1\leq 0.10N_1,\qquad n_2\leq 0.10N_2.

Our samples are independent SRSs, with 36≤60036\leq 600 and 64≤80064\leq 800. These fractions support the usual independence approximation. Recording the same visitors under two conditions would instead create paired data; the independent two-population formula would not be justified just because there are two columns.

The known values 18,15,6 and 4 describe populations. Observed sample means and SDs are different quantities. When the parameters are unknown, estimating a difference and its standard error is an inference problem. Do not treat one observed sample summary as a known true population value.

Shape Depends on Both Populations

If both population distributions can be modeled as normal, the difference between their independent sample means has a normal model, including at small sample sizes. If the populations are not normal, the course uses n1≥30n_1\geq 30 and n2≥30n_2\geq 30 to support an approximately normal difference. A combined sample size of 60 does not replace checking both sizes. One group with 5 and the other with 55 does not meet this two-group large-sample condition.

Population models and independent samplingDifference-mean shape
Both normal, n₁ = 9 and n₂ = 16Normal under the supplied models
Both moderately skewed, n₁ = 40 and n₂ = 50Approximately normal is reasonable
Both non-normal, n₁ = 20 and n₂ = 80The two-group large-sample check is not satisfied

The library example has 36 and 64 observations, and the supplied population descriptions support a CLTEditorial qualification: The captured n ≥ 30 statement is not a universal guarantee. Under the CLT’s assumptions, the accuracy of a Normal approximation depends on the population and sample size. approximation. The broader CLT caution still applies: extreme skew or influential outliers may require larger samples. A threshold is a guide for evaluating a model, not permission to ignore the actual distributions.

Compare Randomly Assigned Treatments Carefully

In an experiment, random assignment of experimental units to two treatments supplies the randomization condition. Two population SRSs and their sampling-fraction checks are not required merely to compare the assigned treatments. Describe the experimental units, assignments and response, and retain any stated response-independence and distribution assumptions used for a probability model.

For example, randomly assigning 80 independently produced test batteries to two charging protocols, 40 per group, differs from sampling two town populations. Assignment supports a treatment comparison; it does not alone establish that those test batteries represent every battery sold. The normal/large-sample shape reasoning for averages still needs attention. Do not interpret the experimental randomization route as a guarantee that every response distribution is normal.

Worked Example: Could Branch A’s Sample Average Be Lower?

Our statistic is D=xˉA−xˉBD=\bar x_A-\bar x_B. “A’s sample average is no higher than B’s” means D≤0D\leq 0. Using the justified approximation,

P(D≤0)≈P(Z≤0−31.1180)=P(Z≤−2.6833)≈0.00365.P(D\leq 0)\approx P\left(Z\leq\frac{0-3}{1.1180}\right)=P(Z\leq -2.6833)\approx 0.00365.

Across repeated pairs of independent SRSs of 36 A visits and 64 B visits, the average difference is 3 minutes; those differences typically deviate from 3 by about 1.118 minutes. About 0.365% of the sample pairs would show A’s sample average at most B’s under the stated model. The population means remain 18 and 15; a rare reversed sample order does not reverse the known population means.

Try It Yourself

The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.

Original generated practice for the Fall 2026 course.

Compare Average Battery Lifetimes

Brand A’s defined production population has N_A=100,000 batteries with known mean lifetime 10 hours and known SD 2 hours. Brand B has N_B=200,000 batteries with known mean lifetime 9 hours and known SD 3 hours. Both lifetime distributions are moderately skewed with finite variance and no extreme outliers. Take independent SRSs without replacement of 64 A batteries and 100 B batteries. Define D=mean lifetime in the A sample minus mean lifetime in the B sample. Use the usual 10% approximation and a justified normal approximation.

Part a

Calculate D’s sampling-distribution mean and SD with complete substitutions. Explain the squared population SDs, separate sample sizes and chosen subtraction order. State what would change if D were reversed.

Part b

Justify the original sampling conditions and approximate normal shape for D. Include independent random samples, both numerical 10% checks, and both sample sizes and supplied population descriptions.

Part c

Evaluate these separate sampling models: (i) both populations normal with independent samples of 9 and 16; (ii) both populations moderately skewed with independent random samples of 40 and 50 and small sampling fractions; (iii) both populations non-normal with random samples of 25 and 50 and small sampling fractions. Explain the appropriate shape branch for each and why a large combined sample alone is insufficient in case (iii).

Part d

A separate experiment randomly assigns 80 independently produced test batteries, 40 to charging protocol C and 40 to protocol D, then measures lifetime in hours. The supplied protocol-specific response models are moderately skewed, finite-variance and without extreme outliers, with independent responses. Explain the experimental randomization requirement, whether two population SRSs or 10% population checks are needed, and whether the group sizes support a normal approximation for the mean difference. Distinguish treatment comparison from population representativeness.

Part e

For the original brand samples, estimate P(D≤0.5)P(D\leq 0.5). Show the cutoff and area direction. Interpret D’s mean, SD and this probability in hours for repeated pairs of the specified brand samples.

Worked answer and rubric

Glossary and deeper reading · 15 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Population

    A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.

    Read more: Population in the captured Statistics reading →

  • Population parameter

    A population parameter is a numerical feature of the population or probability model being studied. It is fixed for that defined population, although its value may be unknown. For example, the true proportion of all enrolled students who walk to school is a parameter.

    Read more: Population parameter in the captured Statistics reading →

  • Variance

    Variance is the square of the standard deviation. It measures spread in squared units, with the precise averaging rule depending on whether it describes a sample or a probability distribution. For example, a standard deviation of 3 centimeters corresponds to a variance of 9 square centimeters.

    Read more: Variance in the captured Statistics reading →

  • Statistical independence

    Statistical independence means that learning one outcome does not change the probability distribution of another. For two events, it implies that the probability of both occurring equals the product of their probabilities. For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.

    Read more: Statistical independence in the captured Statistics reading →

  • Mean

    The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.

    Read more: Mean in the captured Statistics reading →

  • Sampling without replacement

    Sampling without replacement does not return selected items before later selections. Selections from a finite population are dependent, although treating them as approximately independent can be reasonable when the sample is a small fraction of the population. For example, selecting students without replacement ensures that no student is chosen twice.

    Read more: Sampling without replacement in the captured Statistics reading →

  • Random sampling

    Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.

    Read more: Random sampling in the captured Statistics reading →

  • Sample

    A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.

    Read more: Sample in the captured Statistics reading →

  • Standard error

    A standard error describes the standard deviation of a statistic’s sampling distribution, usually estimated from data. It measures uncertainty in an estimate rather than the spread of individual observations. For example, the estimated standard error of a sample mean is the sample standard deviation divided by the square root of the sample size.

    Read more: Standard error in the captured Statistics reading →

  • Statistical inference

    Statistical inference uses data and a study design to draw conclusions beyond the observed values. Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions. For example, a random sample can estimate a school-wide preference without surveying every student.

    Read more: Statistical inference in the captured Statistics reading →

  • Central limit theorem

    Editorial qualification: The captured n ≥ 30 statement is not a universal guarantee. Under the CLT’s assumptions, the accuracy of a Normal approximation depends on the population and sample size.

    For independent identically distributed observations with finite variance, the central limit theorem explains why the standardized sample mean approaches a Normal distribution as sample size increases. There is no universal sample size that makes every population sufficiently close to Normal; skewness and extreme tails affect the approximation. For example, averaging many independent waiting times can produce a less skewed distribution of averages than the original waiting-time distribution.

    Read more: Central limit theorem in the captured Statistics reading →

  • Outlier

    An outlier is an observation unusually far from the overall pattern. The 1.5-IQR rule flags possible outliers below the first quartile minus 1.5 IQR or above the third quartile plus 1.5 IQR. For example, a very long commute may be flagged for investigation without automatically being a recording error.

    Read more: Outlier in the captured Statistics reading →

  • Experiment

    An experiment deliberately imposes treatments and measures responses. Random assignment, comparison, and replication help distinguish treatment effects from other variation. For example, assigning plants to fertilizer treatments is an experiment.

    Read more: Experiment in the captured Statistics reading →

  • Random assignment

    Random assignment uses chance to allocate experimental units to treatment conditions. It tends to balance other influences across groups and supports causal inference in a well-conducted experiment, without guaranteeing identical groups. For example, shuffled labels can assign plants to two fertilizer treatments.

    Read more: Random assignment in the captured Statistics reading →

  • Treatment

    A treatment is a specific condition imposed on an experimental unit. In a multifactor experiment, a treatment is a combination of factor levels. For example, low fertilizer with eight hours of light is one treatment combination.

    Read more: Treatment in the captured Statistics reading →

Search lessons, terms and problems