Monday, March 23, 2015
T-test overview
Often in social science situations, we want to see if there is a statistical difference between two groups.
That is, are any recorded differences due to "chance" (not significant), or are they NOT due to chance (significant)?
To determine if the differences are significant, we use a simple inferential statistical test called the t-test.
Here's how we solve for t:
t = X1 - X2/ Sm1 - Sm2
where X1 = mean (average) of the first group and X2 = mean (average) of the second group, and Sm = standard error of the mean.
So, for example, let's say we have two groups of students in an experiment where we're trying to test whether or not kids can learn their multiplication facts better via a TV show than in school.
Let's say we bring in 20 kids and we randomly assign them to two groups. The first group of 10 learns their multiplication facts via TV show, and the second group of 10 learns via the traditional classroom approach.
So we have something like this:
# of TV kids = 10
# of class kids = 10
Overall N = 20
We examine their scores and we see that the TV kids averaged a 2/10 on a post-test quiz measuring multiplication facts and the traditional class kids averaged a 6/10 on the same test.
Normally, you'd have to solve for the Sm, but-- since this is an 11-day online course-- I'm just going to provide it to you to simplify the process (plus, it's simple to solve and most statistical packages provide it as a matter of course, so almost never do you solve for it by hand). Let's say that the Sm = 1.05.
Ok, so here's what we do--
We solve for t like this:
t = 2 - 6/1.05
t = -4/1.05
t = 3.81 (we always take the absolute value)
This value, in and of itself, tells us nothing.
Just like chi-square, however, we have to be concerned about degrees of freedom (df).
The df for a t-test is simple-- you take the N for the group and subtract one. So, for the first group, the df is 9 (10-1), and for the second group it's also 9 (10-1).
9 + 9 = 18, so the df = 18.
So, now armed with this info, we can check out the t-value chart (either the one in the book, or one we find easily online-- like here for example-- t-test table), and we see that in order to be significant with 18 df (at the .05 level), the t-value needs to be greater than 2.10.
Since our t-value of 3.81 is higher than 2.10, we say that there is a significant difference between the groups.
We then look back at our original data and we see that the traditional kids scored, on average, much better than the TV kids, so we conclude that it's better to use the traditional method.
(Typically, in experimental settings, we never make such a claim without several iterations of the experiment).
Get it?
That is, are any recorded differences due to "chance" (not significant), or are they NOT due to chance (significant)?
To determine if the differences are significant, we use a simple inferential statistical test called the t-test.
Here's how we solve for t:
t = X1 - X2/ Sm1 - Sm2
where X1 = mean (average) of the first group and X2 = mean (average) of the second group, and Sm = standard error of the mean.
So, for example, let's say we have two groups of students in an experiment where we're trying to test whether or not kids can learn their multiplication facts better via a TV show than in school.
Let's say we bring in 20 kids and we randomly assign them to two groups. The first group of 10 learns their multiplication facts via TV show, and the second group of 10 learns via the traditional classroom approach.
So we have something like this:
# of TV kids = 10
# of class kids = 10
Overall N = 20
We examine their scores and we see that the TV kids averaged a 2/10 on a post-test quiz measuring multiplication facts and the traditional class kids averaged a 6/10 on the same test.
Normally, you'd have to solve for the Sm, but-- since this is an 11-day online course-- I'm just going to provide it to you to simplify the process (plus, it's simple to solve and most statistical packages provide it as a matter of course, so almost never do you solve for it by hand). Let's say that the Sm = 1.05.
Ok, so here's what we do--
We solve for t like this:
t = 2 - 6/1.05
t = -4/1.05
t = 3.81 (we always take the absolute value)
This value, in and of itself, tells us nothing.
Just like chi-square, however, we have to be concerned about degrees of freedom (df).
The df for a t-test is simple-- you take the N for the group and subtract one. So, for the first group, the df is 9 (10-1), and for the second group it's also 9 (10-1).
9 + 9 = 18, so the df = 18.
So, now armed with this info, we can check out the t-value chart (either the one in the book, or one we find easily online-- like here for example-- t-test table), and we see that in order to be significant with 18 df (at the .05 level), the t-value needs to be greater than 2.10.
Since our t-value of 3.81 is higher than 2.10, we say that there is a significant difference between the groups.
We then look back at our original data and we see that the traditional kids scored, on average, much better than the TV kids, so we conclude that it's better to use the traditional method.
(Typically, in experimental settings, we never make such a claim without several iterations of the experiment).
Get it?
Chi-square overview
When we talk about inferential statistics, we're simply determining whether or not the results we obtained were due to chance, or not due to chance.
(Inferential means that, if our sample is representative, we can INFER from our sample that the results are indicative of the entire population).
If they are not due to chance, we suggest a relationship between variables. If it is due to chance, we can not make such a claim.
Think of inferential stats as a light switch-- it's either on or off. In the social sciences, if the significance is .05 or lower (that is, we allow for 95% confidence), then we say the switch is "on" and the results are "significant"-- meaning that we are 95% sure that the results are NOT due to chance.
If p (the probability that the results would show up like this by chance) is HIGHER than .05, then we say there is NO significance-- which means we can't argue that the variables are related.
Well, we already know that if we have a nominal IV and a nominal DV, then we test using chi-square.
Chi-square is a simple statistical test.
Put simply, chi-square is the sum of the observed frequency minus the expected frequency, squared-- divided by the expected frequency.
The observed frequency is simply the number reported. The expected frequency is what you'd expect if it were completely by chance.
It's best explained with an example.
Suppose we had the following data:
We asked 97 people about their political affiliations (let's assume it's a random sample) and we got this:
Gender-------Republican------ Democrat----- Row Total
Male:----------- 23---------------- 17------------ 40
Female:--------- 20---------------- 37------------ 57
Column Total:---43----------------54------------ 97
Our hypothesis is:
H1: Women are more likely to be affiliated with the Democratic party than men.
By looking at the raw data, it's difficult to say, with certainty, that this is the case, so we test the hypothesis using chi square.
Our first order of business is to find the "expected" frequency.
The "expected" frequency is R x C / N (where R is the ROW total; C is the COLUMN total; and N is the overall number.
So the ROW total for men is 40.
The ROW total for women is 57.
The Column total for Republicans is 43.
The Column total for Democrats is 54.
The overall N is 97.
The expected frequency for males who should be Republicans based on chance is 40 x 43, which is 1,720 / 97 = 17.73.
Ok, so now we know that the observed frequency for men who are Republicans is 23, and the expected frequency is 17.73. This gives us a difference of (5.27). We square this value and get 27.77.
Once we have that value, we divide by the expected value and get this-- 27.77/17.73 = 1.57 (we always round to the nearest hundredth).
Remember, though, chi-square is the SUM OF, so we have compute it for each cell.
So, we repeat the process for each "cell" and then add up the totals.
Once we have the sum of the chi-squares, we check with a chi-square chart to see if it's significant at the .05 level. You can check here-- chi-square chart.
You'll note something called "degrees of freedom," or "df." A df helps us to determine what line to look at on the chart. The easiest way to remember df is this-- it's (R-1) x (C-1) where R is the number of rows and C is the number columns. In this case, we have 2 rows and 2 columns, which gives us a df of 1 because (R-1) = (2-1), and (C-1) = (2-1), and 1 x 1 = 1.
Also, remember that we use the .05 level of significance.
Go ahead and solve for the sum of chi-square and post what you get. Tell me if it's significant and whether the hypothesis is supported.
Jack
(Inferential means that, if our sample is representative, we can INFER from our sample that the results are indicative of the entire population).
If they are not due to chance, we suggest a relationship between variables. If it is due to chance, we can not make such a claim.
Think of inferential stats as a light switch-- it's either on or off. In the social sciences, if the significance is .05 or lower (that is, we allow for 95% confidence), then we say the switch is "on" and the results are "significant"-- meaning that we are 95% sure that the results are NOT due to chance.
If p (the probability that the results would show up like this by chance) is HIGHER than .05, then we say there is NO significance-- which means we can't argue that the variables are related.
Well, we already know that if we have a nominal IV and a nominal DV, then we test using chi-square.
Chi-square is a simple statistical test.
Put simply, chi-square is the sum of the observed frequency minus the expected frequency, squared-- divided by the expected frequency.
The observed frequency is simply the number reported. The expected frequency is what you'd expect if it were completely by chance.
It's best explained with an example.
Suppose we had the following data:
We asked 97 people about their political affiliations (let's assume it's a random sample) and we got this:
Gender-------Republican------ Democrat----- Row Total
Male:----------- 23---------------- 17------------ 40
Female:--------- 20---------------- 37------------ 57
Column Total:---43----------------54------------ 97
Our hypothesis is:
H1: Women are more likely to be affiliated with the Democratic party than men.
By looking at the raw data, it's difficult to say, with certainty, that this is the case, so we test the hypothesis using chi square.
Our first order of business is to find the "expected" frequency.
The "expected" frequency is R x C / N (where R is the ROW total; C is the COLUMN total; and N is the overall number.
So the ROW total for men is 40.
The ROW total for women is 57.
The Column total for Republicans is 43.
The Column total for Democrats is 54.
The overall N is 97.
The expected frequency for males who should be Republicans based on chance is 40 x 43, which is 1,720 / 97 = 17.73.
Ok, so now we know that the observed frequency for men who are Republicans is 23, and the expected frequency is 17.73. This gives us a difference of (5.27). We square this value and get 27.77.
Once we have that value, we divide by the expected value and get this-- 27.77/17.73 = 1.57 (we always round to the nearest hundredth).
Remember, though, chi-square is the SUM OF, so we have compute it for each cell.
So, we repeat the process for each "cell" and then add up the totals.
Once we have the sum of the chi-squares, we check with a chi-square chart to see if it's significant at the .05 level. You can check here-- chi-square chart.
You'll note something called "degrees of freedom," or "df." A df helps us to determine what line to look at on the chart. The easiest way to remember df is this-- it's (R-1) x (C-1) where R is the number of rows and C is the number columns. In this case, we have 2 rows and 2 columns, which gives us a df of 1 because (R-1) = (2-1), and (C-1) = (2-1), and 1 x 1 = 1.
Also, remember that we use the .05 level of significance.
Go ahead and solve for the sum of chi-square and post what you get. Tell me if it's significant and whether the hypothesis is supported.
Jack
Intro to Statistics (Overview)
Statistics are mathematical methods to collect, organize, summarize, and analyze data. Statistics provide valid and reliable results only when the data collection and research methods follow established scientific procedures. With the development of the computer, the science of statistics has changed dramatically.
Basic statistical procedures include:
descriptive statistics,
sample distribution, and
data transformation.
In descriptive statistics, the chapter presents the concept of data distribution, frequency distribution, cumulative frequency, histogram, bar chart, frequency polygon, normal curve, and skewness.
Summary statistics make data more manageable by measuring two basic tendencies of distributions: 1) central tendency; and 2) dispersion (variability). These statistics make it easier for researchers to understand data.
Central tendency statistics provide information about the grouping of numbers in a distribution by giving a single number that characterizes the entire distribution. Using the mode, median, and mean, researchers can figure out a typical score of a distribution.
In addition, dispersion measures describe the way scores are spread out about a central point. Using range, variance, and standard deviation, allows researchers to understand the characteristics of the data.
The term sample distribution—the distribution of some characteristic measured on an individual or other unit of analysis that were part of a sample. Additionally, it's important to understand the notion of a sampling distribution—a theoretical probability distribution of all values of a variable for a given sample size.
Most statistical procedures are based on the assumption that the data are normally distributed. When some anomalies arise, researchers can attempt to transform the data to achieve normality. Data transformation can be possible by multiplying or dividing each score by a certain number, or taking the square root or log of the scores.
Basic statistical procedures include:
descriptive statistics,
sample distribution, and
data transformation.
In descriptive statistics, the chapter presents the concept of data distribution, frequency distribution, cumulative frequency, histogram, bar chart, frequency polygon, normal curve, and skewness.
Summary statistics make data more manageable by measuring two basic tendencies of distributions: 1) central tendency; and 2) dispersion (variability). These statistics make it easier for researchers to understand data.
Central tendency statistics provide information about the grouping of numbers in a distribution by giving a single number that characterizes the entire distribution. Using the mode, median, and mean, researchers can figure out a typical score of a distribution.
In addition, dispersion measures describe the way scores are spread out about a central point. Using range, variance, and standard deviation, allows researchers to understand the characteristics of the data.
The term sample distribution—the distribution of some characteristic measured on an individual or other unit of analysis that were part of a sample. Additionally, it's important to understand the notion of a sampling distribution—a theoretical probability distribution of all values of a variable for a given sample size.
Most statistical procedures are based on the assumption that the data are normally distributed. When some anomalies arise, researchers can attempt to transform the data to achieve normality. Data transformation can be possible by multiplying or dividing each score by a certain number, or taking the square root or log of the scores.
Hypothesis Testing Overview
Hypothesis development in scientific research is important because the process refines and focuses research by excluding extraneous variables and permitting variables to be quantified.
Scientists rarely begin a research study without a problem or a question to test. Without research questions or hypotheses, research proves to be a waste of time.
Researchers develop studies based on existing theory and are thus able to make predictions about the outcome of their work. Therefore, hypothesis development is usually the culmination of a rigorous literature review (we don't have the time to conduct a full-scale literature review in this class, so our theories will be based on our own experiences).
Researchers should use hypotheses in scientific research to:
1) provide direction for a study;
2) eliminate trial-and-error research;
3) rule out intervening and confounding variables; and,
4) allow for quantification of variables.
In addition, hypotheses should be:
1) compatible with current knowledge in the area;
2) logically consistent;
3) stated concisely; and,
4) testable.
In hypothesis testing, a researcher either rejects or fails to reject the null hypothesis that the statistical differences being analyzed are due to chance or random error.
To determine the statistical significance of a research study, the research must set a probability level (significance level) against which the null hypothesis is tested. If the results of the study indicate a probability lower than this level, the researcher can reject the null hypothesis. If the research outcome has a high probability, the researcher fails to reject the null hypothesis. It is common practice in mass media research studies to set the probability level at .05, which means that either one or five times out of 100, significant results of the study occur because of random error or chance. Another way to think of this is to say that "we are 95% confident that our results are not due to chance."
All research contains error. Typically, two types of error (Type I error: the rejection of a null hypothesis that should not be rejected, and Type II error: the acceptance of a null hypothesis that should be rejected) are relevant to hypothesis testing.
There is always the possibility of making an error in rejecting or failing to reject a null hypothesis. It is not easy for researchers to balance these two error types, but one procedure, power analysis, helps researchers deal with the problem. Because power (the probability of rejecting the null hypothesis when it is true) indicates the probability that a statistical test of a null hypothesis will result in the conclusion that the phenomenon under study actually exists, if there is a difference, researchers are able to detect it.
Scientists rarely begin a research study without a problem or a question to test. Without research questions or hypotheses, research proves to be a waste of time.
Researchers develop studies based on existing theory and are thus able to make predictions about the outcome of their work. Therefore, hypothesis development is usually the culmination of a rigorous literature review (we don't have the time to conduct a full-scale literature review in this class, so our theories will be based on our own experiences).
Researchers should use hypotheses in scientific research to:
1) provide direction for a study;
2) eliminate trial-and-error research;
3) rule out intervening and confounding variables; and,
4) allow for quantification of variables.
In addition, hypotheses should be:
1) compatible with current knowledge in the area;
2) logically consistent;
3) stated concisely; and,
4) testable.
In hypothesis testing, a researcher either rejects or fails to reject the null hypothesis that the statistical differences being analyzed are due to chance or random error.
To determine the statistical significance of a research study, the research must set a probability level (significance level) against which the null hypothesis is tested. If the results of the study indicate a probability lower than this level, the researcher can reject the null hypothesis. If the research outcome has a high probability, the researcher fails to reject the null hypothesis. It is common practice in mass media research studies to set the probability level at .05, which means that either one or five times out of 100, significant results of the study occur because of random error or chance. Another way to think of this is to say that "we are 95% confident that our results are not due to chance."
All research contains error. Typically, two types of error (Type I error: the rejection of a null hypothesis that should not be rejected, and Type II error: the acceptance of a null hypothesis that should be rejected) are relevant to hypothesis testing.
There is always the possibility of making an error in rejecting or failing to reject a null hypothesis. It is not easy for researchers to balance these two error types, but one procedure, power analysis, helps researchers deal with the problem. Because power (the probability of rejecting the null hypothesis when it is true) indicates the probability that a statistical test of a null hypothesis will result in the conclusion that the phenomenon under study actually exists, if there is a difference, researchers are able to detect it.
Sunday, March 22, 2015
Exam #2 Date: April 6, 2015
The date of the second exam is April 6, 2015.
It will be an open-book, open-resource exam.
Jack
It will be an open-book, open-resource exam.
Jack
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