Section 1 - Descriptive Statistics
MAIN IDEA QUESTION
What measures can we use to summarize sets of data?
VOCABULARY
statistics - The branch of mathematics concerned with collecting, organizing, analyzing, and drawing conclusions from numerical data
descriptive statistics - The branch of statistics that provides a means of summarizing data
frequency distribution - An arrangement of scores from a sample that indicates how often a particular score is present
histogram - Bar graph
central tendency - An index of the central location within a distribution of scores; the most representative score in a distribution of scores (the mean, median, and mode are measures of central tendency)
mean - The average of all scores, arrived at by adding scores together and dividing by the number of scores
median - The point in a distribution of scores that divides the distribution exactly in half when the scores are listed in numerical order
mode - The most frequently occurring score in a set of scores
normal distribution - A distribution of scores that produces a symmetrical, bell-shaped curve in which the right half mirrors the left half and in which the mean, median, and mode all have the same value
Statistics, the branch of mathematics concerned with collecting, organizing, analyzing, and drawing conclusions from numerical data, is a part of all of our lives. For example, we are all familiar with claims and counterclaims regarding the effects of smoking. The U.S. government requires cigarette manufacturers to include a warning that smoking is dangerous to people's health on every package of cigarettes and in their advertisements; the government's data show clear statistical links between smoking and disease. At the same time, the tobacco industry has long minimized the negative effects of smoking.
Statistics is at the heart of many debates. How do we determine the nature and strength of the effects of heredity on behavior? What is the relationship between learning and schedules of reinforcement? How do we know if the "double standard" regarding male and female sexual practices has shifted over time? These questions cannot be answered without using statistics.
Suppose as an instructor of college psychology you wanted to evaluate your class's performance on its initial exam. Where might you begin? You would probably start by using descriptive statistics, the branch that provides a means of summarizing data and presenting it in a usable and convenient form. For instance, you might first simply list the scores the pupils had received on the test. There are many ways you could sort them in a meaningful way; you might sort them in order from highest to lowest score. By indicating the number of people who obtained each score, you would have produced what is called frequency distribution - an arrangement of scores from a sample that indicates how often a particular score is present.
Another way of summarizing scores is to consider them visually. You could construct the histogram, or bar graph. Arranging the scores from highest to lowest allows us to visually inspect the data. Most often, visual inspection is insufficient. There may be so many scores in a sample that it is difficult to construct a meaningful visual representation. Also, our interpretation of patterns on a graph or table are often biased or inaccurate. In cases in which a precise means of summarizing the data is desirable, psychologists use measures of central tendency. Central tendency is an index of the central location within a distribution of scores. There are three major measures: the mean, the median, and the mode.
THE MEAN: FINDING THE AVERAGE
The most familiar measure of central tendency is the mean; the technical term for an average, which is simply the sum of all scores in a set, divided by the number of scores making up the set. The mean is an accurate reflection of the central score in a set of scores. Still, the mean does not always provide the best measure of central tendency. The mean is very sensitive to extreme scores which can present a deceptive picture of a set of scores, especially in cases where the mean is based on a relatively small number of scores.
THE MEDIAN: FINDING THE MIDDLE
A measure of central tendency which is less sensitive to extreme scores than the mean is the median. The median is the point in a distribution of scores that divides the distribution exactly in half. It lies in the middle of a distribution of scores. EX: 10, 8, 7, 4, 3... the media is 7. Two scores lay before and after the score of 7.
One feature of the median is that it is insensitive to extreme scores. The median divides a set of scores in half, and the magnitude of the scores is of no consequence in this process. The median is often used instead of the mean when extreme scores might be misleading. For example, government statistics on income are typically presented using the median as the measure of central tendency because the median corrects for the small number of extreme cases of very wealthy individuals, whose high incomes might otherwise inflate the mean income.
THE MODE: FINDING WHAT IS MOST FREQUENT
The final measure of central tendency is the mode. The mode is the most frequently occurring score in a set of scores. Some distributions, of course, may have more than one score occurring most frequently. In instances where two scores are of equal frequency, we would say there are two modes - a case known as bimodal distribution.
The mode is often used as a measure of preference or popularity. For instance, if teachers wanted to know who was the most popular child in their elementary school classrooms, they might develop a questionnaire that asked the students to choose someone with whom they would like to participate in some activity. After the tally, the mode would probably provide the best indication of which child was the most popular.
COMPARING THE THREE M's: MEAN VERSUS MEDIAN VERSUS MODE
If a sample is sufficiently large, there is generally little difference between the mean, the median, and the mode. With large samples, scores typically form what is called a normal distribution. A normal distribution is a distribution of scores that produces a symmetrical, bell-shaped curve. EX: Distribution of IQ scores, hours students study in a week - in both examples, there would be a large center and few extremes.
The mean, median, and mode differ, however, when distributions are not normal. In cases where the distributions are skewed, or not symmetrical, there is a "hump" at one end. EX: If we gave a calculus exam to a group of students enrolled in an elementary algebra class, we would expect that most of the students would fail the test, leading to low scores being overrepresented in the distribution. On the other hand, if we gave the same students a test of elementary addition problems, the scores would probably form a distribution in which high scores predominated. Both distributions would be skewed (in opposite directions) and the mean, median, and mode are different from one another.
Section 2 - Measures of Variability
MAIN IDEA QUESTION
How can we assess the variability of a set of data?
VOCABULARY
variability - The spread, or dispersion, of scores in a distribution
range - The difference between the highest score and the lowest score in a distribution
standard deviation - An index of the average deviation of a set of scores from the center of the distribution
Although measures of central tendency provide information about where the center of a distribution lies, often this information is insufficient. For example, suppose a psychologist was interested in determining the nature of people's eye movements while they were reading in order to perfect a new method to teach reading. It would not be enough to know how most people moved their eyes; it would also be important to know how much individual people's eye movements differed or varied from one another.
A second important characteristic of a set of scores provides this information: variability. Variability is a term that refers to the spread, or dispersion, of scores in a distribution.
THE RANGE: HIGHEST MINUS LOWEST
The simplest measure of variability is the range. The range is the difference between the highest score in a distribution and the lowest score. A range is simple to calculate, which is about its only virtue. The problem with this measure of variability is it is based entirely on extreme scores, and a single score that is very different from the others can distort the picture of the distribution as a whole.
THE STANDARD DEVIATION: DIFFERENCES FROM THE MEAN
The most frequently used method of characterizing the variability of a distribution of scores is the standard deviation. The standard deviation bears a conceptual relationship to a mean. A standard deviation is an index of the average deviation of a set of scores from the center of the distribution. EX: In the general population, IQ scores of intelligence fall into a normal distribution, and they have a mean of 100 and a standard deviation of 15. Consequently an IQ score of 100 does not deviate from the mean, whereas an IQ score that is three standard deviations above the mean (or 145) is very unusual (higher than 99% of all IQ scores).
The calculation of the standard deviation follows the logic of calculating the difference of individual scores from the mean of the distribution. Not only does it provide an excellent indicator of the variability of a set of scores, it provides a means for converting initial scores on standardized tests such as the SAT (the college admissions exam) into the scales used to report results. In this way, it is possible to make a score of 585 on the verbal section of the SAT exam, for example, equivalent from one year to the next even though the specific test items differ from year to year.
Section 3 - Using Statistics to Answer Questions: Inferential Statistics and Correlation
MAIN IDEA QUESTIONS
How do we generalize from data?
How can we determine the nature of a relationship, and the significance of differences, between two sets of scores?
VOCABULARY
population - All the members of a group of interest
sample - A representative subgroup of a population of interest
inferential statistics - The branch of statistics that uses data from samples to make predictions about the larger population from which the sample is drawn
significant outcome - An outcome in which the observed outcome would be expected to have occurred by chance with a probability of .05 or less
positive relationship - A relationship established by data that shows high values of one variable corresponding with high values of another, and low values of the first variable corresponding with low values of the other
negative relationship - A relationship established by data that shows high values of one variable corresponding with low values of the other
correlation coefficient - A numerical measure that indicates the extent of the relationship between two variables
Suppose you were interested in whether there is a relationship between smoking and anxiety. Would it be reasonable to simply look at a group of smokers and measure their anxiety using some rating scale? Probably not. It clearly would be more informative if you compared their anxiety with the anxiety exhibited by a group of nonsmokers.
Once you decided to observe anxiety in two groups of people, you would have to determine just who would be your subjects. A population consists of all the members of a group of interest. Obviously, however, this would be impossible because of the size of the two groups; instead, you would limit your subjects to a sample of smokers and nonsmokers. A sample, in formal statistical terms, is a subgroup of a population of interest that is intended to be representative of the larger population.
The obvious question is whether the two samples (smokers and nonsmokers) differ in the degree of anxiety their members display. The two samples can be examined in terms of central tendency and variability. The more important question, however, is whether the magnitude of difference between the two distributions is sufficient to conclude that the distributions truly differ from one another, or if, instead, the differences are attributable merely to chance.
To answer the question of whether the samples are truly different from one another, psychologists use inferential statistics. Inferential statistics is the branch of statistics that uses data from samples to make predictions about a larger population, permitting generalizations to be drawn.
THE CORRELATION COEFFICIENT: MEASURING RELATIONSHIPS