Question

For which sample (i or ii) would you expect the SD of heights to be larger? Or, would they be about the same? (a) (i) A sample of 10 women ages \(18-24,\) or (ii) a sample of 100 women ages \(18-24\). (b) (i) A sample of 20 male college basketball players, or (ii) a sample of 20 college-age men. (c) (i) A sample of 15 professional male jockeys, or (ii) a sample of 15 professional male biologists.

Short answer

Part A: SDs would be about the same. Part B: 20 college-age men (sample ii) would have a larger SD. Part C: 15 professional male biologists (sample ii) would have a larger SD.

Step-by-step solution

Comparing Sample Size and Population Variability - Part A

For sample (i) with 10 women ages 18-24 and sample (ii) with 100 women of the same age group, if we assume the populations from which these samples are drawn are roughly the same, then the variability in heights (SD) should be about the same. A larger sample size does not necessarily mean a larger standard deviation; it's the variability within the actual population that matters.

Diverse Population vs. Specific Subgroup - Part B

In sample (i), we have a specific subgroup of males who are college basketball players who are likely to be taller on average with less variance in their height. Sample (ii) is a more general population of college-age men who will have a more variable range of heights. Therefore, we would expect sample (ii) to have a larger SD of heights.

Professional Attributes Related to Height - Part C

Sample (i)'s population of professional male jockeys is selected based on a profession where there is typically a restriction on height and weight, leading to less variability. Sample (ii) of professional male biologists is not selected based on physical attributes related to height, so we would expect a greater variability in heights in this group. Sample (ii) would be expected to have a larger SD.

Key concepts

Population Variability

When examining data, population variability, is a measure of how much the individuals within that entire population differ from one another. To understand it, imagine a small town where almost everyone has a similar job, lifestyle, and income. The variability in lifestyles in this town would be low because there are fewer differences among its residents.

• In the context of height, if a population consists of adult females aged 18-24, we expect them to have various heights, but within a certain range. If the population is diversified with differences in genetic and environmental factors, there will be more variability.

Sample Size

Sample size plays a critical role in statistics as it impacts the reliability of the conclusions we draw. The larger the sample size, the more confident we can be that our sample accurately reflects the population.

• However, an important misconception is that a larger sample size would automatically result in a larger standard deviation (SD). This isn't necessarily true; the sample size influences the precision of our estimates of the population parameters, such as the mean or SD, rather than the values themselves.
• A larger sample may provide a more accurate estimate of the population SD, but it doesn't inherently change the spread of the data.

Standard Deviation

The standard deviation (SD) is a statistical measure that tells us how spread out the numbers in a data set are. In simpler terms, it's a way of seeing how much variation exists from the average (mean).

• A low SD indicates that the data points are close to the mean, whereas a high SD indicates that the data points are spread out over a wider range of values.
• For instance, the heights of professional basketball players are likely to have a lower SD compared to a random sample of the general population, as basketball players' heights are less spread out (they're typically taller than average).

Statistical Variability

Statistical variability, or statistical variation, refers to the measure of how much the data values in a sample differ among themselves. It's not just about how they spread out from the mean but how they relate to one another.

• Even in a sample where the mean is very stable, you might have high variability if the individual data points vary significantly from each other.
• Situations with high variability tend to require more complex analysis and careful interpretation since the data may suggest a wider range of underlying causes or factors.