StatPac for Windows User's Guide
Basic Research Concepts
We understand the world by asking questions and searching for answers. Our construction of reality depends on the nature of our inquiry.
All research begins with a question. Intellectual curiosity is often the foundation for scholarly inquiry. Some questions are not testable. The classic philosophical example is to ask, "How many angels can dance on the head of a pin?" While the question might elicit profound and thoughtful revelations, it clearly cannot be tested with an empirical experiment. Prior to Descartes, this is precisely the kind of question that would engage the minds of learned men. Their answers came from within. The scientific method precludes asking questions that cannot be empirically tested. If the angels cannot be observed or detected, the question is considered inappropriate for scholarly research.
Defining the goals and objectives of a research project is one of the most important steps in the research process. Do not underestimate the importance of this step. Clearly stated goals keep a research project focused. The process of goal definition usually begins by writing down the broad and general goals of the study. As the process continues, the goals become more clearly defined and the research issues are narrowed.
Exploratory research (e.g., literature reviews, talking to people, and focus groups) goes hand-in-hand with the goal clarification process. The literature review is especially important because it obviates the need to reinvent the wheel for every new research question. More importantly, it gives researchers the opportunity to build on each other’s work.
The research question itself can be stated as a hypothesis. A hypothesis is simply the investigator's belief about a problem. Typically, a researcher formulates an opinion during the literature review process. The process of reviewing other scholar's work often clarifies the theoretical issues associated with the research question. It also can help to elucidate the significance of the issues to the research community.
The hypothesis is converted into a null hypothesis in order to make it testable because the only way to test a hypothesis is to eliminate alternatives of the hypothesis. Statistical techniques will enable us to reject or fail to reject a null hypothesis, but they do not provide us with a way to accept a hypothesis. Therefore, all hypothesis testing is indirect.
Defining a research problem provides a format for further investigation. A well-defined problem points to a method of investigation. There is no one best method of research for all situations. Rather, there are a wide variety of techniques for the researcher to choose from. Often, the selection of a technique involves a series of trade-offs. For example, there is often a trade-off between cost and the quality of information obtained. Time constraints sometimes force a trade-off with the overall research design. Budget and time constraints must always be considered as part of the design process.
There are three basic methods of research: 1) survey, 2) observation, and 3) experiment. Each method has its advantages and disadvantages.
The survey is the most common method of gathering information in the social sciences. It can be a face-to-face interview, telephone, mail, e-mail, or web survey. A personal interview is one of the best methods obtaining personal, detailed, or in-depth information. It usually involves a lengthy questionnaire that the interviewer fills out while asking questions. It allows for extensive probing by the interviewer and gives respondents the ability to elaborate their answers. Telephone interviews are similar to face-to-face interviews. They are more efficient in terms of time and cost, however, they are limited in the amount of in-depth probing that can be accomplished, and the amount of time that can be allocated to the interview. A mail survey is more cost effective than interview methods. The researcher can obtain opinions, but trying to meaningfully probe opinions is very difficult. Email and web surveys are the most cost effective and fastest methods.
Observation research monitors respondents' actions without directly interacting with them. It has been used for many years by A.C. Nielsen to monitor television viewing habits. Psychologists often use one-way mirrors to study behavior. Anthropologists and social scientists often study societal and group behaviors by simply observing them. The fastest growing form of observation research has been made possible by the bar code scanners at cash registers, where purchasing habits of consumers can now be automatically monitored and summarized.
In an experiment, the investigator changes one or more variables over the course of the research. When all other variables are held constant (except the one being manipulated), changes in the dependent variable can be explained by the change in the independent variable. It is usually very difficult to control all the variables in the environment. Therefore, experiments are generally restricted to laboratory models where the investigator has more control over all the variables.
It is incumbent on the researcher to clearly define the target population. There are no strict rules to follow, and the researcher must rely on logic and judgment. The population is defined in keeping with the objectives of the study.
Sometimes, the entire population will be sufficiently small, and the researcher can include the entire population in the study. This type of research is called a census study because data is gathered on every member of the population.
Usually, the population is too large for the researcher to attempt to survey all of its members. A small, but carefully chosen sample can be used to represent the population. The sample reflects the characteristics of the population from which it is drawn.
Sampling methods are classified as either probability or nonprobability. In probability samples, each member of the population has a known non-zero probability of being selected. Probability methods include random sampling, systematic sampling, and stratified sampling. In nonprobability sampling, members are selected from the population in some nonrandom manner. These include convenience sampling, judgment sampling, quota sampling, and snowball sampling. The advantage of probability sampling is that sampling error can be calculated. Sampling error is the degree to which a sample might differ from the population. When inferring to the population, results are reported plus or minus the sampling error. In nonprobability sampling, the degree to which the sample differs from the population remains unknown.
Random sampling is the purest form of probability sampling. Each member of the population has an equal and known chance of being selected. When there are very large populations, it is often difficult or impossible to identify every member of the population, so the pool of available subjects becomes biased.
Systematic sampling is often used instead of random sampling. It is also called an Nth name selection technique. After the required sample size has been calculated, every Nth record is selected from a list of population members. As long as the list does not contain any hidden order, this sampling method is as good as the random sampling method. Its only advantage over the random sampling technique is simplicity. Systematic sampling is frequently used to select a specified number of records from a computer file.
Stratified sampling is commonly used probability method that is superior to random sampling because it reduces sampling error. A stratum is a subset of the population that share at least one common characteristic. The researcher first identifies the relevant stratums and their actual representation in the population. Random sampling is then used to select subjects from each stratum until the number of subjects in that stratum is proportional to its frequency in the population. Stratified sampling is often used when one or more of the stratums in the population have a low incidence relative to the other stratums.
Convenience sampling is used in exploratory research where the researcher is interested in getting an inexpensive approximation of the truth. As the name implies, the sample is selected because they are convenient. This nonprobability method is often used during preliminary research efforts to get a gross estimate of the results, without incurring the cost or time required to select a random sample.
Judgment sampling is a common nonprobability method. The researcher selects the sample based on judgment. This is usually and extension of convenience sampling. For example, a researcher may decide to draw the entire sample from one "representative" city, even though the population includes all cities. When using this method, the researcher must be confident that the chosen sample is truly representative of the entire population.
Quota sampling is the nonprobability equivalent of stratified sampling. Like stratified sampling, the researcher first identifies the stratums and their proportions as they are represented in the population. Then convenience or judgment sampling is used to select the required number of subjects from each stratum. This differs from stratified sampling, where the stratums are filled by random sampling.
Snowball sampling is a special nonprobability method used when the desired sample characteristic is rare. It may be extremely difficult or cost prohibitive to locate respondents in these situations. Snowball sampling relies on referrals from initial subjects to generate additional subjects. While this technique can dramatically lower search costs, it comes at the expense of introducing bias because the technique itself reduces the likelihood that the sample will represent a good cross section from the population.
There are very few hard and fast rules to define the task of data collection. Each research project uses a data collection technique appropriate to the particular research methodology. The two primary goals for both quantitative and qualitative studies are to maximize response and maximize accuracy.
When using an outside data collection service, researchers often validate the data collection process by contacting a percentage of the respondents to verify that they were actually interviewed. Data editing and cleaning involves the process of checking for inadvertent errors in the data. This usually entails using a computer to check for out-of-bounds data.
Quantitative studies employ deductive logic, where the researcher starts with a hypothesis, and then collects data to confirm or refute the hypothesis. Qualitative studies use inductive logic, where the researcher first designs a study and then develops a hypothesis or theory to explain the results of the analysis.
Quantitative analysis is generally fast and inexpensive. A wide assortment of statistical techniques are available to the researcher. Computer software is readily available to provide both basic and advanced multivariate analysis. The researcher simply follows the preplanned analysis process, without making subjective decisions about the data. For this reason, quantitative studies are usually easier to execute than qualitative studies.
Qualitative studies nearly always involve in-person interviews, and are therefore very labor intensive and costly. They rely heavily on a researcher's ability to exclude personal biases. The interpretation of qualitative data is often highly subjective, and different researchers can reach different conclusions from the same data. However, the goal of qualitative research is to develop a hypothesis--not to test one. Qualitative studies have merit in that they provide broad, general theories that can be examined in future research.
The most important consideration in preparing any research report is the nature of the audience. The purpose is to communicate information, and therefore, the report should be prepared specifically for the readers of the report. Sometimes the format for the report will be defined for the researcher (e.g., a thesis or dissertation), while other times, the researcher will have complete latitude regarding the structure of the report. At a minimum, the report should contain an abstract, problem statement, methods section, results section, discussion of the results, and a list of references.
Validity refers to the accuracy or truthfulness of a measurement. Are we measuring what we think we are? This is a simple concept, but in reality, it is extremely difficult to determine if a measure is valid.
Face validity is based solely on the judgment of the researcher. Each question is scrutinized and modified until the researcher is satisfied that it is an accurate measure of the desired construct. The determination of face validity is based on the subjective opinion of the researcher.
Content validity is similar to face validity in that it relies on the judgment of the researcher. However, where face validity only evaluates the individual items on an instrument, content validity goes further in that it attempts to determine if an instrument provides adequate coverage of a topic. Expert opinions, literature searches, and open-ended pretest questions help to establish content validity.
Criterion-related validity can be either predictive or concurrent. When a dependent/independent relationship has been established between two or more variables, criterion-related validity can be assessed. A mathematical model is developed to be able to predict the dependent variable from the independent variable(s). Predictive validity refers to the ability of an independent variable (or group of variables) to predict a future value of the dependent variable. Concurrent validity is concerned with the relationship between two or more variables at the same point in time.
Construct validity refers to the theoretical foundations underlying a particular scale or measurement. It looks at the underlying theories or constructs that explain a phenomena. This is also quite subjective and depends heavily on the understanding, opinions, and biases of the researcher.
Reliability is synonymous with repeatability. A measurement that yields consistent results over time is said to be reliable. When a measurement is prone to random error, it lacks reliability. The reliability of an instrument places an upper limit on its validity. A measurement that lacks reliability will necessarily be invalid. There are three basic methods to test reliability: test-retest, equivalent form, and internal consistency.
A test-retest measure of reliability can be obtained by administering the same instrument to the same group of people at two different points in time. The degree to which both administrations are in agreement is a measure of the reliability of the instrument. This technique for assessing reliability suffers two possible drawbacks. First, a person may have changed between the first and second measurement. Second, the initial administration of an instrument might in itself induce a person to answer differently on the second administration.
The second method of determining reliability is called the equivalent-form technique. The researcher creates two different instruments designed to measure identical constructs. The degree of correlation between the instruments is a measure of equivalent-form reliability. The difficulty in using this method is that it may be very difficult (and/or prohibitively expensive) to create a totally equivalent instrument.
The most popular methods of estimating reliability use measures of internal consistency. When an instrument includes a series of questions designed to examine the same construct, the questions can be arbitrarily split into two groups. The correlation between the two subsets of questions is called the split-half reliability. The problem is that this measure of reliability changes depending on how the questions are split. A better statistic, known as Cronbach's alpha, is based on the mean (absolute value) interitem correlation for all possible variable pairs. It provides a conservative estimate of reliability, and generally represents the lower bound to the reliability of a scale of items. For dichotomous nominal data, the KR-20 (Kuder-Richardson) is used instead of Cronbach's alpha.
Most research is an attempt to understand and explain variability. When a measurement lacks variability, no statistical tests can be (or need be) performed. Variability refers to the dispersion of scores.
Ideally, when a researcher finds differences between respondents, they are due to true difference on the variable being measured. However, the combination of systematic and random errors can dilute the accuracy of a measurement. Systematic error is introduced through a constant bias in a measurement. It can usually be traced to a fault in the sampling procedure or in the design of a questionnaire. Random error does not occur in any consistent pattern, and it is not controllable by the researcher.
There are basically two kinds of research questions: testable and non-testable. Neither is better than the other, and both have a place in applied research.
Examples of non-testable questions are:
How do managers feel about the reorganization?
What do residents feel are the most important problems facing the community?
Respondents' answers to these questions could be summarized in descriptive tables and the results might be extremely valuable to administrators and planners. Business and social science researchers often ask non-testable research questions. The shortcoming with these types of questions is that they do not provide objective cut-off points for decision-makers.
In order to overcome this problem, researchers often seek to answer one or more testable research questions. Nearly all testable research questions begin with one of the following two phrases:
Is there a significant difference between ...?
Is there a significant relationship between ...?
Is there a significant relationship between the age of managers and their attitudes towards the reorganization?
Is there a significant difference between white and minority residents with respect to what they feel are the most important problems facing the community?
A research hypothesis is a testable statement of opinion. It is created from the research question by replacing the words "Is there" with the words "There is", and also replacing the question mark with a period. The hypotheses for the two sample research questions would be:
There is a significant relationship between the age of managers and their attitudes towards the reorganization.
There is a significant difference between white and minority residents with respect to what they feel are the most important problems facing the community.
It is not possible to test a hypothesis directly. Instead, you must turn the hypothesis into a null hypothesis. The null hypothesis is created from the hypothesis by adding the words "no" or "not" to the statement. For example, the null hypotheses for the two examples would be:
There is no significant relationship between the age of managers and their attitudes towards the reorganization.
There is no significant difference between white and minority residents with respect to what they feel are the most important problems facing the community.
All statistical testing is done on the null hypothesis...never the hypothesis. The result of a statistical test will enable you to either 1) reject the null hypothesis, or 2) fail to reject the null hypothesis. Never use the words "accept the null hypothesis".
There are two types of hypothesis testing errors. The first one is called a Type I error. This is a very serious error where you wrongly reject the null hypothesis. Suppose that the null hypothesis is: Daily administrations of drug ABC will not help patients. Also suppose that drug ABC is really a very bad drug, and it causes permanent brain damage to people over 60. In your research, you ask for volunteers, and all of the sample is under 60 years of age. The sample seems to improve and you reject the null hypothesis. There could be very serious consequences if you were to market this drug (based on your sample). Type I errors are often caused by sampling problems.
A Type II error is less serious, where you wrongly fail to reject the null hypothesis. Suppose that drug ABC really isn't harmful and does actually help many patients, but several of your volunteers develop severe and persistent psychosomatic symptoms. You would probably not market the drug because of the potential for long-lasting side effects. Usually, the consequences of a Type II error will be less serious than a Type I error.
One of the most important concepts in statistical testing is to understand the four basic types of data: nominal, ordinal, interval, and ratio. The kinds of statistical tests that can be performed depend upon the type of data you have. Different statistical tests are used for different types of data.
Nominal and ordinal data are nonparametric (non-continuous or categorical). Interval and ratio scales are called parametric (continuous). Some statistical tests are called parametric tests because they use parametric data. Others are called nonparametric tests because they use nonparametric data. All statistical tests are designed to be used with a specific kind of data, and may only be performed when you have that kind of data.
Nominal data is characterized by non-ordered response categories.
Examples of nominal data
What is your sex?
____ Male ____ Female
What program are you in?
___ Health Services
___ Human Services
Do you have health insurance?
___ Yes ___ No ___ Don't know
What school did you attend?
___ Park Elementary
___ West Side
What should be done with the program?
___ Close it down
___ Seek government funding
___ Hold a private fund raiser
What state do you live in? _________________________
Note: This question is called an open-ended question because it calls for a verbatim response. Even though the categories (i.e., the states) are not listed, the question is still considered nominal because the data can be categorized after it is collected.
Which of the following meats have you eaten in the last week? (Check all that apply)
___ Hamburger ___ Pot roast ___ Liver
___ Hotdogs ___ Bacon ___ Steak
___ Pork chops ___ Sausage ___ Other
Note: This question is called a multiple response item because respondents can check more than one category. Multiple response simply means that a respondent can make more than one response to the same question. The data is still nominal because the responses are non-ordered categories.
What are the two most important issues facing our country today?
________________________ and ________________________
Note: This question is an open-ended multiple response item because it calls for two verbatim responses. It is still considered nominal data because the issues could be categorized after the data is collected.
Ordinal data is characterized by ordered response categories.
Examples of ordinal data
What is your highest level of education?
___ Grade school
___ Some high school
___ High school graduate
___ Some college
___ College graduate
___ Advanced degree
How many beers have you drunk in the last week?
___ None ___ One to five ___ Six to ten ___ Over ten
How would you rate your progress?
What has the trend been in your business over the past year?
___ Decreasing ___ Stable ___ Increasing
Please rate the quality of this lecture?
___ Low ___ Medium ___ High
Use a circle to indicate your level of agreement or disagreement with the following statement: Abortion should be a decision between a woman and her doctor.
Agree Agree Neutral Disagree Disagree
1 2 3 4 5
What is your annual family income?
___ Under $12,000
___ $12,000 to $23,999
___ $24,000 to $49,999
___ $50,000 to $74,999
___ $75,000 or more
Interval and ratio data
Interval and ratio data are such that each numeric interval represents one unit of measurement. Ratio scales also have the property of an absolute "zero-point". Interval and ratio-scaled questions are preferable in research design because they offer the most versatility in the kinds of analyses that may be performed.
Examples of interval and ratio data
What is your age? _______
How many children do you have? ________
What was your SAT score? ________
How many years of school have you completed? _______
What percent of your work time do you spend .... ? _______
How many collective bargaining sessions have you been involved in? ______
What is the average class size in your school? ________
What was your family income last year? ___________
How many units have you completed? (Circle) 0 1 2 3
What was your GPA as an undergraduate student? _____
How many times have you been arrested? _____
What does significance really mean?
Many researchers get very excited when they have discovered a "significant" finding, without really understanding what it means. When a statistic is significant, it simply means that you are very sure that the statistic is reliable. It doesn't mean the finding is important.
For example, suppose we give 1,000 people an IQ test, and we ask if there is a significant difference between male and female scores. The mean score for males is 98 and the mean score for females is 100. We use an independent groups t-test and find that the difference is significant at the .001 level. The big question is, "So what?". The difference between 98 and 100 on an IQ test is a very small difference...so small, in fact, that its not even important.
Then why did the t-statistic come out significant? Because there was a large sample size. When you have a large sample size, very small differences will be detected as significant. This means that you are very sure that the difference is real (i.e., it didn't happen by fluke). It doesn't mean that the difference is large or important. If we had only given the IQ test to 25 people instead of 1,000, the two-point difference between males and females would not have been significant.
Significance is a statistical term that tells how sure you are that a difference or relationship exists. To say that a significant difference or relationship exists only tells half the story. We might be very sure that a relationship exists, but is it a strong, moderate, or weak relationship? After finding a significant relationship, it is important to evaluate its strength. Significant relationships can be strong or weak. Significant differences can be large or small. It just depends on your sample size.
Many researchers use the word "significant" to describe a finding that may have decision-making utility to a client. From a statistician's viewpoint, this is an incorrect use of the word. However, the word "significant" has virtually universal meaning to the public. Thus, many researchers use the word "significant" to describe a difference or relationship that may be strategically important to a client (regardless of any statistical tests). In these situations, the word "significant" is used to advise a client to take note of a particular difference or relationship because it may be relevant to the company's strategic plan. The word "significant" is not the exclusive domain of statisticians and either use is correct in the business world. Thus, for the statistician, it may be wise to adopt a policy of always referring to "statistical significance" rather than simply "significance" when communicating with the public.
One important concept in significance testing is whether to use a one-tailed or two-tailed test of significance. The answer is that it depends on your hypothesis. When your research hypothesis states (or implies) the direction of the difference or relationship, then you use a one-tailed probability. For example, a one-tailed test would be used to test these null hypotheses: Females will not score significantly higher than males on an IQ test. Blue collar workers will not have significantly lower education than white collar workers. Superman is not significantly stronger than the average person. In each case, the null hypothesis (indirectly) predicts the direction of the expected difference. A two-tailed test would be used to test these null hypotheses: There will be no significant difference in IQ scores between males and females. There will be no significant difference between blue collar and white collar workers. There is no significant difference in strength between Superman and the average person. A one-tailed probability is exactly half the value of a two-tailed probability.
There is a raging controversy (for about the last hundred years) on whether or not it is ever appropriate to use a one-tailed test. The rationale is that if you already know the direction of the difference, why bother doing any statistical tests. The safest bet is to always state your hypotheses so that two-tailed tests are appropriate.
Whenever we perform a significance test, it involves comparing a test value that we have calculated to some critical value for the statistic. It doesn't matter what type of statistic we are calculating (e.g., a t-statistic, a chi-square statistic, an F-statistic, etc.), the procedure to test for significance is the same.
1. Decide on the critical alpha level you will use (i.e., the error rate you are willing to accept).
2. Conduct the research.
3. Calculate the statistic.
4. Compare the statistic to a critical value obtained from a table or compare the probability of the statistic to the critical alpha level.
If your statistic is higher than the critical value from the table or the probability of the statistic is less than the critical alpha level:
Your finding is significant.
You reject the null hypothesis.
The probability is small that the difference or relationship happened
by chance, and p is less than the critical alpha level (p < R ).
If your statistic is lower than the critical value from the table or the probability of the statistic is higher than the critical alpha level:
Your finding is not significant.
You fail to reject the null hypothesis.
The probability is high that the difference or relationship happened
by chance, and p is greater than the critical alpha level (p > R ).
Modern computer software can calculate exact probabilities for most test statistics. When StatPac (or other software) gives you an exact probability, simply compare it to your critical alpha level. If the exact probability is less than the critical alpha level, your finding is significant, and if the exact probability is greater than your critical alpha level, your finding is not significant. Using a table is not necessary when you have the exact probability for a statistic.
Bonferroni's theorem states that as one performs an increasing number of statistical tests, the likelihood of getting an erroneous significant finding (Type I error) also increases. Thus, as we perform more and more statistical tests, it becomes increasingly likely that we will falsely reject a null hypothesis (very bad).
For example, suppose our critical alpha level is .05. If we performed one statistical test, our chance of making a false statement is .05. If we were to perform 100 statistical tests, and we made a statement about the result of each test, we would expect five of them to be wrong (just by fluke). This is a rather undesirable situation for social scientist.
Bonferroni's theorem states that we need to adjust the critical alpha level in order to compensate for the fact that we're doing more than one test. To make the adjustment, take the desired critical alpha level (e.g., .05) and divide by the number of tests being performed, and use the result as the critical alpha level. For example, suppose we had a test with eight scales, and we plan to compare males and females on each of the scales using an independent groups t-test. We would use .00625 (.05/8) as the critical alpha level for all eight tests.
Bonferroni's theorem should be applied whenever you are conducting two or more tests that are of the same "type" and the same "family". The same "type" means the same kind of statistical test. For example, if you were going to do one t-test, one ANOVA, and one regression, you would not make the adjustment because the tests are all different. The same "family" is a more elusive concept, and there are no hard and fast rules. "Family" refers to a series of statistical tests all designed to test the same (or very closely related) theoretical constructs. The bottom line is that it's up to the individual researcher to decide what constitutes a "family".
Some things are more obvious than others, for example, if you were doing t-tests comparing males and females on a series of questionnaire items that are all part of the same scale, you would probably apply the adjustment, by dividing your critical alpha level by the number of items in the scale (i.e., the number of t-tests you performed on that scale). The probabilities of the tests would be called the family error rates. However, suppose you have a series of independent questions, each focusing on a different construct, and you want to compare males and females on how they answered each question. Here is where the whole idea of Bonferroni's adjustment becomes philosophical. If you claim that each t-test that you perform is a test of a unique "mini"-hypothesis, then you would not use the adjustment, because you have defined each question as a different "family". In this case, the probability would be called a statement error rate. Another researcher might call the entire questionnaire a "family", and she would divide the critical alpha by the total number of items on the questionnaire.
Why stop there? From a statistician's perspective, the situation becomes even more complex. Since they are personally in the "statistics business", what should they call a "family"? When a statistician does a t-test for a client, maybe she should be dividing the critical alpha by the total number of t-tests that she has done in her life, since that is a way of looking at her "family". Of course, this would result in a different adjustment for each statistician--an interesting dilemma.
In the real world, most researchers do not use Bonferroni's adjustment because they would rarely be able to reject a null hypothesis. They would be so concerned about the possibility of making a false statement, that they would overlook many differences and relationships that actually exist. The "prime directive" for social science research is to discover relationships. One could argue that it is better to risk making a few wrong statements, than to overlook relationships or differences that are clear or prominent, but don't meet critical alpha significance level after applying Bonferroni's adjustment.
The best known measures of central tendency are the mean and median. The mean average is found by adding the values for all the cases and dividing by the number of cases. For example, to find the mean age of all your friends, add all their ages together and divide by the number of friends. The mean average can present a distorted picture of central tendency if the sample is skewed in any way.
For example, let's say five people take a test. Their scores are 10, 12, 14, 18, and 94. (The last person is a genius.) The mean would be the sums of the scores 10+12+14+18+94 divided by 5. In this example, a mean of 29.6 is not a good measure of how well people did on the test in general. When analyzing data, be careful of using only the mean average when the sample has a few very high or very low scores. These scores tend to skew the shape of the distribution and will distort the mean.
When you have sampled from the population, the mean of the sample is also your best estimate of the mean of the population. The actual mean of the population is unknown, but the mean of the sample is as good an estimate as we can get.
The median provides a measure of central tendency such that half the sample will be above it and half the sample will be below it. For skewed distributions this is a better measure of central tendency. In the previous example, 14 would be the median for the sample of five people. If there is no middle value (i.e., there are an even number of data points), the median is the value midway between the two middle values.
The distribution of many variables follows that of a bell-shaped curve. This is called a "normal distribution". One must assume that data is approximately normally distributed for many statistical analyses to be valid. When a distribution is normal, the mean and median will be equal to each other. If they are not equal, the distribution is distorted in some way.
Variability is synonymous with diversity. The more diversity there is in a set of data, the greater the variability. One simple measure of diversity is the range (maximum value minus the minimum value). The range is generally not a good measure of variability because it can be severely affected by a single very low or high value in the data. A better method of describing the amount of variability is to talk about the dispersion of scores away from the mean.
The variance and standard deviation are useful statistics that measure the dispersion of scores around the mean. The standard deviation is simply the square root of the variance. Both statistics measure the amount of diversity in the data. The higher the statistics, the greater the diversity. On the average, 68 percent of all the scores in a sample will be within plus or minus one standard deviation of the mean and 95 percent of all scores will be within two standard deviations of the mean.
There are two formulas for the variance and standard deviation of a sample. One set of formulas calculates the exact variance and standard deviation of the sample. The statistics are called biased, because they are biased to the sample. They are the exact variance and standard deviation of the sample, but they tend to underestimate the variance and standard deviation of the population.
Generally, we are more concerned with describing the population rather than the sample. Our intent is to use the sample to describe the population. The unbiased estimates should be used when sampling from the population and inferring back to the population. They provide the best estimate of the variance and standard deviation of the population.
The standard error of the mean is used to estimate the range within which we would expect the mean to fall in repeated samples taken from the population (i.e., confidence intervals). The standard error of the mean is an estimate of the standard deviation of those repeated samples.
The formula for the standard error of the mean provides an accurate estimate when the sample is very small compared to the size of the population. In marketing research, this is usually the case since the populations are quite large. However, when the sample size represents a substantial portion of the population, the formula becomes inaccurate and must be corrected. The finite population correction factor is used to correct the estimate of the standard error when the sample is more than ten percent of the population.
When the sample size is small (less than 30), the z value for the area under the normal curve is not accurate. Instead of a z value, we can use a t value to derive the area under the curve. In fact, many researchers always use the t value instead of the z value. The reason is that the t values are more accurate for small sample sizes, and they are nearly identical to the z values for large sample sizes. Unlike the z value, the values for t depend upon the number of cases in the sample. Depending on the sample size, the t value will change.
Degrees of freedom literally refers to the number of data values that are free to vary.
For example, suppose I tell you that the mean of a sample is 10, and there are a total of three values in the sample. It turns out that if I tell you any two of the values, you will always be able to figure out the third value. If two of the values are 8 and 12, you can calculate that the third value is 10 using simple algebra.
(x + 8 + 12) / 3 = 10 x = 10
In other words, if you know the mean, and all but one value, you can figure out the missing value. All the values except one are free to vary. One value is set once the others are known. Thus, degrees of freedom is equal to n-1.