The science of statistics deals with the collection, analysis, interpretation, and presentation of data.
We see and use data in our everyday lives. In the public health fields of kinesiology and nutrition, statistics allows us to answer critical questions, such as: Does a new exercise program improve bone density? Do dietary changes lower the risk of heart disease?
In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing (e.g., showing the distribution of BMI scores) and by using numbers (for example, finding an average daily caloric intake or mean reaction time). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions about a population are correct (e.g., concluding a vaccine works based on a trial sample).
Effective interpretation of data (inference) is based on good procedures for producing data (such as running a controlled trial or conducting surveillance) and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data and its implications for human health. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in your future public health career.
Probability
Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is > or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996/2,000 is equal to 0.498 which is very close to 0.5, the expected probability.
The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. In public health, we use probabilities to predict the likelihood of an infant developing obesity, of an athlete experiencing an ACL tear, or whether a community health program will achieve a 10% increase in physical activity. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data and translate that uncertainty into actionable health recommendations.
Key Terms
In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.
Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In a national health survey, a sample of 5,000 adults is randomly selected across various states. This sample is supposed to represent the average daily physical activity and nutritional status of the entire country's adult population.
From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter. A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.
One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.
A variable, usually notated by capital letters such as X and Y, is a characteristic or measurement that can be determined for each member of a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. If we let X equal the number of steps a person takes in a day, then X is a numerical variable. We can do math with values of X to calculate someone’s weekly average of steps per day.
Categorical variables place the person or thing into a category or group. If we let Y be a person's primary dietary pattern, then some examples of Y include Vegetarian, Omnivore, Vegan, and Mediterranean. Y, is a categorical variable. You do not perform math with values of Y, but categorical variables are important for examining differences among groups.
Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.
Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is 22/40 and the proportion of women students is 18/40. Mean and proportion are discussed in more detail in later chapters.
NOTE
The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean," and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean."
Determine what the key terms refer to in the following study. We want to know the average (mean) daily consumption of protein (in grams) by all varsity collegiate athletes at State University. We randomly surveyed 120 varsity athletes from the university's various sports teams. Three of those athletes reported consuming 85 grams, 110 grams, and 92 grams of protein per day, respectively.
- Answer
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The Population is all varsity collegiate athletes currently attending State University.
The Sample is the 120 randomly surveyed varsity athletes from State University's various sports teams.
The Parameter is the average (mean) daily protein consumption (in grams) for all varsity collegiate athletes at State University.
The Statistic is the average (mean) daily protein consumption (in grams) calculated from the 120 athletes in the sample.
The Variable (X) is the daily consumption of protein (in grams) by one individual varsity athlete at State University.
The Data are the measured protein amounts reported by the sampled athletes. Examples of the data are 85 g, 110 g, and 92 g.
Solution
The population is all first year students attending ABC College this term.
The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population).
The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term.
The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample.
The variable could be the amount of money spent (excluding books) by one first year student. Let X = the amount of money spent (excluding books) by one first year student attending ABC College.
The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225.
Determine what the key terms refer to in the following study.
An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.
- Answer
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The population is all medical doctors listed in the professional directory.
The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population.
The sample is the 500 doctors selected at random from the professional directory.
The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample.
The variable X = whether an individual doctor has been involved in a malpractice suit.
The data are either: yes, was involved in one or more malpractice lawsuits, or no, was not.