Significant Differences

A-4 How do we know whether an observed difference can be generalized to other populations?

Data are “noisy.” The average score in one group could conceivably differ from the average score in another group not because of any real difference but merely because of chance fluctuations in the people sampled. How confidently, then, can we infer that an observed difference is not just a fluke—a chance result from the research sample? For guidance, we can ask whether the observed difference between the two groups is reliable and significant. These inferential statistics help us determine if results describe a larger population.

A-7

When Is an Observed Difference Reliable?

image
© The New Yorker Collection, 1988, Mirachi from cartoonbank.com. All Rights Reserved.

In deciding when it is safe to generalize from a sample, we should keep three principles in mind:

  1. Representative samples are better than biased (unrepresentative) samples. The best basis for generalizing is from a representative sample of cases, not from the exceptional and memorable cases one finds at the extremes. Research never randomly samples the whole human population. Thus, it pays to keep in mind what population a study has sampled. (To see how an unrepresentative sample can lead you astray, see Thinking Critically About: Cross-Sectional and Longitudinal Studies below.)

  2. Less-variable observations are more reliable than those that are more variable. As we noted earlier in the example of the basketball player whose game-to-game points were consistent, an average is more reliable when it comes from scores with low variability.

  3. More cases are better than fewer cases. An eager prospective student visits two university campuses, each for a day. At the first, the student randomly attends two classes and discovers both instructors to be witty and engaging. At the next campus, the two sampled instructors seem dull and uninspiring. Returning home, the student (discounting the small sample size of only two teachers at each institution) tells friends about the “great teachers” at the first school, and the “bores” at the second. Again, we know it but we ignore it: Averages based on many cases are more reliable (less variable) than averages based on only a few cases.

The greater variability of small samples explains why small schools often are top producers of high-achieving students—a finding that led several foundations to invest in the creation of smaller schools. Alas, underperforming schools also are disproportionately small, because small populations are more variable (Kahneman, 2011).

The point to remember: Smart thinkers are not overly impressed by a few anecdotes. Generalizations based on a few unrepresentative cases are unreliable.

THINKING CRITICALLY ABOUT

Cross-Sectional and Longitudinal Studies

A-5 What are cross-sectional studies and longitudinal studies, and why is it important to know which method was used?

When interpreting research results, smart thinkers consider how researchers arrived at their conclusions. One way studies vary is in the time period for gathering data.

cross-sectional study research in which people of different ages are compared with one another.

In cross-sectional studies, researchers compare different groups at the same time. When researchers compare intelligence test scores among people in differing age groups, older adults, on average, give fewer correct answers than do younger adults. This could suggest that mental ability declines with age, and indeed, that was the conclusion drawn from many early cross-sectional studies of intelligence.

longitudinal study research in which the same people are restudied and retested over a long period of time.

In longitudinal studies, researchers study and restudy the same people at different times in their life span. Around 1920, colleges began giving intelligence tests to entering students, and several psychologists saw their chance to study intelligence longitudinally. What they expected to find was a decrease in intelligence after about age 30 (Schaie & Geiwitz, 1982). What they actually found was a surprise: Until late in life, intelligence remained stable. On some tests, it even increased.

Why did these new results differ from the earlier cross-sectional findings? In retrospect, researchers realized that cross-sectional studies that compared 70-year-olds and 30-year-olds were comparing people not only of two different ages but also of two different eras.

They were comparing

  • generally less-educated people (born, say, in the early 1900s) with better-educated people (born after 1950).

  • people raised in large families with people raised in smaller families.

  • people from less-affluent families with people from more-affluent families.

Others have since pointed out that longitudinal studies have their own pitfalls. Participants who survive to the end of longitudinal studies may be the healthiest (and brightest) people. When researchers adjust for the loss of participants, as did one study following more than 2000 people over age 75 in Cambridge, England, they find a steeper intelligence decline, especially as people age after 85 (Brayne et al., 1999).

The point to remember: When interpreting research results, pay attention to the methodology used, such as whether it was a longitudinal or cross-sectional study.

When Is an Observed Difference “Significant”?

Perhaps you’ve compared men’s and women’s scores on a laboratory test of aggression, and you’ve found a gender difference. But individuals differ. How likely is it that the difference you observed was just a fluke? Statistical testing can estimate the probability of the result occurring by chance.

Here is the underlying logic: When averages from two samples are each reliable measures of their respective populations (as when each is based on many observations that have small variability), then their difference is probably reliable as well. (Example: The less the variability in women’s and in men’s aggression scores, the more confidence we would have that any observed gender difference is reliable.) And when the difference between the sample averages is large, we have even more confidence that the difference between them reflects a real difference in their populations.

statistical significance a statistical statement of how likely it is that an obtained result occurred by chance.

In short, when sample averages are reliable, and when the difference between them is relatively large, we say the difference has statistical significance. This means that the observed difference is probably not due to chance variation between the samples.

A-8

image See LaunchPad’s Video: Longitudinal and Cross-Sectional Studies, below, for a helpful tutorial animation.

In judging statistical significance, psychologists are conservative. They are like juries who must presume innocence until guilt is proven. For most psychologists, proof beyond a reasonable doubt means not making much of a finding unless the odds of its occurring by chance, if no real effect exists, are less than 5 percent.

When reading about research, you should remember that, given large enough or homogeneous enough samples, a difference between them may be “statistically significant” yet have little practical significance. For example, comparisons of intelligence test scores among hundreds of thousands of first-born and later-born individuals indicate a highly significant tendency for first-born individuals to have higher average scores than their later-born siblings (Rohrer et al., 2015; Zajonc & Markus, 1975). But because the scores differ by only one to three points, the difference has little practical importance.

image

A-9

The point to remember: Statistical significance indicates the likelihood that a result will happen by chance. But this does not say anything about the importance of the result.

image For a 9.5-minute video synopsis of psychology’s scientific research strategies, view LaunchPad’s Video: Research Methods below.

RETRIEVE IT

Can you solve this puzzle?

Question

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ANSWER: Averages based on fewer courses are more variable, which guarantees a greater number of extremely low and high marks at the end of the first term.

Question

RiL1kHjCiTKaHA78rhiHpw== statistics summarize data, while noCibxPsf/vg6MewSvWHng== statistics determine if data can be generalized to other populations.

REVIEW Statistical Reasoning in Everyday Life

Learning Objectives

Test Yourself by taking a moment to answer each of these Learning Objective Questions (repeated here from within Appendix A). Research suggests that trying to answer these questions on your own will improve your long-term memory of the concepts (McDaniel et al., 2009).

Question

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ANSWER: Researchers may use descriptive statistics to meaningfully organize data. A measure of central tendency is a single score that represents a whole set of scores. Three such measures that we use to describe data are the mode (the most frequently occurring score), the mean (the arithmetic average), and the median (the middle score in a group of data). Measures of variation tell us how diverse data are. Two measures of variation are the range (which describes the gap between the highest and lowest scores) and the standard deviation (which states how much scores vary around the mean, or average, score). Scores often form a normal (or bell-shaped) curve.

Question

0yx4lZ9AaqVNGOEjNtDdzI4Jof2CGJFxUePZrH/m8Q7pWvdTsdGD+wdSC+rS8E77cjqcqBs8LcNQ1exyYO+U2x2NkRZ3NWy8Lx2768WaSTza2e2BUMqmpHVCgL7WQRqQWEAFm+ylKLqEpIZDHwfHprHFJAxH5ZXLIpeKwzsNlEF9PEDU+HZrLOzFv02Vr+UcA83kDhe8scWaS+AL7swHpo1zR2onYtatsCv5IA==
ANSWER: When we say two things are correlated, we are saying that they accompany each other in their movements. The strength of their relationship is expressed as a correlation coefficient. Their relationship may be displayed in a scatterplot, in which each dot represents a value for the two variables. Correlations predict but cannot explain.

Question

HQRD59IR80jGWBwrK4WCf76qVb95QJgRGJ+ZGjc9Sd4TEJa7KeYgxsprmQ5sH7DEpAmiJMIkBAiFqNx5HwzfH9eUj72QY7kq0uq0hg0aLVd2VsQpJWnPpKfvRlIZ9JBGDyoZwHXu17Slj9s+YqsBe3BQjajHjPm8k2Q9vv07qi9xTxhGY4XrCYWdbEwaJM+y7laMum9ttUI=
ANSWER: Regression toward the mean is the tendency for extreme or unusual scores to fall back toward their average.

Question

ho63EkebXnfPAJy4iF3VYQtjtJE0thzXPTJIVfEvubMzhDBK4jshnhqU/lHnCha5OdPlVTYSbnWkw1Qln0E1dPwKtGxzhpdsXawi5HjqfhA7NW3CqhlkT8Xz39nIxVSfTw05b+1duCEhfh9vSBSD5zhUEchF5fDMvtqMzmNpTu05+mG4qhdZRvDcK+cJ29GNM5eq01HzDTIiWkD+OOMQl7bO4vx7hLFcaMcwKkXNELNIp+nVtm8w57Ce8kHnw/YvCGBmbX60TAo=
ANSWER: Researchers use inferential statistics to help determine the reliability and significance of a study finding. To feel confident about generalizing an observed difference to other populations, we would want to know that the sample studied was representative of the larger population being studied; that the observations, on average, had low variability; that the sample consisted of more than a few cases; and that the observed difference was statistically significant.

Question

AtZ4r7vlWoo5RrMvY6+da+Mdgk9hAke5+qiIcH+vocKGH4IjQl6BUCE5eg2K7f5d4NdERUzVMWTja866MQgbYR+DDkjXMsE2ii6WKmB1Q/xqhE+yuIKRFYaXJmmGLHfEWwUwxcX3QUX0IJCnPqMeLvDZU6B1lZdKZDA8ay1n3gxldyS4nmlXCVt9gKnKMIT+xu++EFb5b8RzmrLP1MN2SYI2LmpEj75IEy2xI3wds7yhS/MkKf1puym1HjRSPXzH+UUqSW6yqkMfccKFHpoP4t+HxW4c8ZmCK/9a/FzPNQPwif7q/2MSwEFIU6oq3hB72szmwMo4owbfiTa5ZEAidNpYNwQbMCYNwclsNnjzXlRNU3Wz
ANSWER: In a cross-sectional study, people of different ages are compared. In a longitudinal study, a group of people is studied periodically over a long period of time. To draw meaningful conclusions about a study's results, we need to know whether the study used a representative sample to draw its conclusions. Studies of intelligence and aging, for example, have drawn different conclusions depending on whether a cross-sectional or longitudinal study was used.

Terms and Concepts to Remember

Test yourself on these terms.

Question

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Experience the Testing Effect

Test yourself repeatedly throughout your studies. This will not only help you figure out what you know and don’t know; the testing itself will help you learn and remember the information more effectively thanks to the testing effect.

Question 16.1

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Question 16.2

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

Question 16.3

3. Another name for a bell-shaped distribution, in which most scores fall near the middle and fewer scores fall at each extreme, is a +jIVN7YiAOzQHsxz4cLRLj8W+oM= .

A-10

Question 16.4

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Question 16.5

5. If a study revealed that tall people were less intelligent than short people, this would suggest that the correlation between height and intelligence is PsZo0R5Y8yrofAbHg+qqcQ== (positive/negative).

Question 16.6

6. A uv9KSf3eN6x9nHyTN2Gqyg== provides a visual representation of the direction and the strength of a relationship between two variables.

Question 16.7

1+6dWmdWO99oJfaLpIIX9FzKiR1+2Ud+bFkDiRdkU64pM+qntFHS8wf7joJrNVvtW/K6tm0ra9VZ7bd9znBMqZS8gQpqj3a9oue92gFpY8Sa4neDuwUCo/i9aMKCXxs0UPU4upxr0JaGqvPP6LDpxP7OA/dQWqHpCKTNWp79JqLcLI2gdWLfnqur5VxLBhB554Jj0VaohSkzqiO6NQTIpaQz3x4+ow4ATPGTdRGP5DrRvnBxOryv0w==
ANSWER: Regression toward the mean is a statistical phenomenon describing the tendency of extreme scores or outcomes to return to normal after an unusual event. Without knowing this, we may inaccurately decide the return to normal was a result of our own behavior.

Question 16.8

8. In DJ1wVfeJ0Wj1Zvb7vSeTyKNZhA8WL7z6m5Ff0ODm1XGLUijB studies, a characteristic is assessed across different age groups at the same time.

Question 16.9

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Use image to create your personalized study plan, which will direct you to the resources that will help you most in image .