If you’re a high school teacher who wants to examine your students scores on a recent test, for example, you could use dozens of different statistical analyses, depending on the specific questions you want to ask about the data. Broadly speaking, statistics fall into two categories: descriptive and inferential. Descriptive statistics can help you understand the data you've already collected from your students, while inferential statistics can help you determine how typical students in your sample compare to others in the population.
Measures of Central Tendency
Measures of central tendency are descriptive statistics that assess how spread out the data points in a sample are. The three primary measures of central tendency are mean, the basic average; median, the middle score; and mode, the most common score. These measures are useful for determining what a typical score in a sample looks like. For example, if the median test score in a class was 90, that means that half of the students scored higher than 90 and half scored below 90.
Measures of Dispersion
Measures of dispersion are another class of descriptive statistics that explain how spread out data points in a sample are. The most common measure of dispersion is standard deviation, a value calculated by measuring the distance of each point from the sample mean, squaring them, adding them together and then taking the square root. The larger the value of the standard deviation, the more spread out data points are. For example, if the mean score on the math test was 85 and the standard deviation was 5, that means about two-thirds of students scored between 80 and 90 points. If the standard deviation was 15, that means two-thirds of students scored between 70 and 100.
Tests of Difference
Tests of difference produce inferential statistics, allowing researchers to determine whether differences between groups in the sample occur at random or as the result of some variable. For example, a high school teacher might notice that girls scored an average of 95 points on the math test while boys scored an average of 85 points. The teacher might be tempted to conclude that girls are better than boys at the subject, but the difference in scores might be the result of random chance. The teacher could use a t-test or analysis of variance (ANOVA) to check how likely the different scores are to be the result of sampling error.
Tests of Relationship
If you wanted to determine whether students who studied more performed better on the exam, you could use a test of relationship, like a correlation or linear regression. To run either of these tests, you could ask students to report how many hours they studied for the exam. A correlation measures how closely related the "hours of study" variable is to the "exam score" variable. A perfect correlation would have a value of 1, while two completely unrelated variables would have a value of 0. A negative correlation would indicate that studying was actually related to decreased scores. You could incorporate other variables, like previous exam scores, into the calculation by using multiple regression.