Welcome in this section of the course. What we're going to do is wrap it all up  and bring it all together. In this first lecture, we're going to talk about all the  concepts that we've covered throughout this course, so we can see how they all  fit together. Then, over the course of the next two lectures, we'll look at an actual real-world example work from beginning to end. So, let's jump in and remind  ourselves of pretty much everything we've been learning throughout this course. Remember all the way back to the beginning, we talked about the idea of really  what is data, and data, if you remember, is just the idea that we have factual  information used as a basis for reasoning, discussion, or calculation, or that  information was just the idea of a measurement or a value that describes  something, whether it's an object, a person, a place, or a thing. From that  information, we wanted to be able to gain inference, where we use that  information to come to some kind of conclusion. And the goal of drawing those  conclusions is really to make better decisions in the context of our problem.  Right, a lot of times we were asking questions about who, what, where, when,  why, how, and so we're trying to use data to inform us about better conclusions  and decisions we can make about whatever problem we're running into. From  there, we learned about the idea of a data table, we looked at how data was  structured when looking at individual observations of data all the way to different variables inside of data. With those variables, we looked at two different types of variables. We looked at categorical or qualitative variables that summarize  things as categories like month of year or season or day of week or type of  weather, then we also looked at quantitative or numerical variables that really  used a numerical value to be able to summarize things like number of casual  users, number of registered users, or temperature or humidity. Remember, you  know something is numerical in nature if you can do some basic arithmetic on it. For example, what is the average temperature? Well, that makes sense.  However, when we have a categorical variable, what's the average day of  week? That doesn't make any sense. So, we have all this data, and we learned  that data is pretty much everywhere, it's being gathered all the time. In fact,  you're producing a lot of data yourself, and so a lot of times what we do is we  need to think about how to be able to gather this data, so we can make correct  inference. And so we talked about this idea of gathering data, and it involved  four main pieces: the population, the sample, the statistic, and the parameter.  The population, remember, was a set of all objects or all individuals that you  were interested in finding something out about. Of course, you couldn't talk to  the entire population, so instead you took a sample, which is really just a subset  of the population that information was actually obtained about. Now, what is that  information you obtain about a sample? Some measure you compute from it.  Well, that's what we refer to as a statistic. A statistic is just a calculation from our sample. That statistic, if you remember, is what we use to estimate a parameter.  The parameter is some measure calculated from a population, but just like the 

population is unknown, the parameter is unknown. So we're using the sample to  give us an idea about the population, and from that sample we get a statistic to  estimate the parameter from that population, and so we talked about this idea of taking samples, because the better the sampling that we had, the more  believable it's going to be when it comes to that statistic estimating that  parameter, and we talked about the idea of randomness and having  randomness helping us make the sample representative of the population. This  again protects us from having certain pieces of information overly influence our  sample, with again the goal of sampling being that we can make good inference  from the statistic to the parameter. That's the main idea of what we're trying to  do, right. We're trying to estimate something about a group of people we cannot  see in totality, so we can see a small subset of them, and if we take that subset  really, really well, then what we can do is we can rely on what that subset is  telling us about the population. Of course, there are incorrect ways of doing this, as well. Bad sampling methods lead to bias, sort of that garbage in, garbage out kind of scenario. Good sampling methods wouldn't lead to this kind of situation.  Now, remember, bias means that we're basically favoring certain outcomes over  other outcomes in a sample. There are very common types of bias that are out  there. They're typically broken into two groups: selection bias and sampling bias. And we talked about a couple examples from each. For selection bias, we talked about under coverage and nonresponse. For sampling bias, we moved into the  world of convenience sampling, or voluntary sampling. Either way, none of these techniques were good ways of being able to ensure that we had a reasonable  and reliable sample. So, of course, we talked about good sampling methods as  well. We talked about four common ones: the simple random sampling  technique, where every single observation, and really set of observations, has  an equal chance of being selected, a stratified random sampling technique, or  we learned about breaking our sample into groups ahead of time, and then  taking a piece of each one of those groups to make our sample. We talk also  about cluster sampling, where again we take the population and break them into groups ahead of time, but instead of looking at all of the groups, like we did in  stratified sampling, we look at only some of the groups, sort of a sample of the  groups of the population, and from each one of those sampled groups we take  another sample, and then we have the idea of systematic sampling, sort of a  quick and dirty way of being able to get some notion of sampling from there, we  moved into ethical considerations around data. The gathering of data, of course, leads to questions around ethical collection and use of that data. As Christians,  we are held an even higher standard around ethical considerations in  observational studies and experiments, we must keep the interest of the subject  we are collecting data from at the forefront, and there are a lot of things in  process to be able to help us do this. Institutional review boards, making sure  we get informed consent, and then also making sure we respect confidentiality, 

but again making sure we keep the subject at the forefront is one of the best  ways of being able to use data ethically. From there, once we learned how to be  able to collect our data, we moved into exploring different types of variables. For categorical variables, we explored both within a category as well as across  categories for continuous variables or quantitative variables, numeric variables.  We explored center and spread, and really sort of the overall look of the variable themselves. And accordingly, we talked about this idea visually, always, always,  always try and visualize your data. It helps you see things a lot better. For  qualitative categorical variables, we looked at a couple of different charts. We  looked at a pie chart, which was a graph in which a circle is divided into sections that represent a proportion of the whole, whereas a bar chart has numerical  values of variables represented by the height or the length of lines or rectangles. Now a lot of times a pie chart is when we're looking at again the population or  the groups as a whole, and really what their proportional breakdown is. A bar  chart can be used to compare just subsets of groups. You don't have to look at  all of them for numeric or quantitative variables, we use things like line graphs  and scatterplots. A line graph uses lines to connect individual data points over  time, where a scatterplot uses the two variables and plots individual  combinations of points for them. So, again, you have one variable on an x axis,  one variable on a y axis, and we're seeing what values take place together, and  we really are looking for the pattern of the resulting points to see if there's any  kind of relationship that's been revealed when exploring variables numerically.  On the other hand, we looked at sort of measures of spread or center, and so  when looking at measures of center, we looked at things like the mode that  would be great for categorical variables or qualitative variables. Basically, it's the value of a variable that happens the most often. For numeric or quantitative  variables, when coming to center, we looked for things like mean and median,  where the mean is just really summing up all the values and dividing by the  number of values, it's just the average, where the median is the number right in  the middle. Now, when it came to spread, on the other hand, or variability, we  looked at things like range, which was the difference between the largest and  the smallest values, as well as measures of variability called variance and  standard deviation. Remember, these two things are tied together. Variance is a  measure of dispersion around the mean, however, variance is also measured in  squared units. Standard deviation is the square root of variance to be able to  help. With the units great, so now we started looking at this data, and we started exploring our data, and starting to understand what our data looked like. Then  we had to take a little bit of a detour, and we started talking about the notion of  probability. The reason was when we were looking at probabilities, we needed to define these, so we could look at distributions of data, so when looking at the  probability, remember the probability that an event happens is just a numerical  measure of the likelihood of that event's occurrence, and it takes a value from 

zero to one. The closer to zero the event is unlikely, the closer to one the event  is very likely, with a probability of zero being it will never occur, and a probability  of one being it will always occur, and if we were going to talk about probabilities,  we also had to talk about a couple of things, like the law of large numbers. The  law of large numbers states that as the number of independent trials increases,  in the long run, remember we talked about this in the long run, in the long run,  the proportion for a certain event gets closer and closer and closer to a single  value. We call that single value the probability of the event. Remember, although that the law of large numbers is real, the law of averages is really just a myth.  From there we were able to build a probability distribution, so the probability  distribution for a random variable describes how probabilities are distributed  over the values of these random variables, and so now we can start piecing  everything together. We have probabilities, so we know that certain values  inside of a variable may occur more often or less often, and then we can even  look inside of a variable to get again those notions of center or spread, so all  these concepts are rolling up to this idea of a distribution, and relative  frequencies were what we used to estimate the probability of an event occurring, especially when it came to discrete random variables. In fact, probability  distributions for discrete random variables are best described with tables or  graphs or equations, like the example we worked through here. Remember, we  worked a lot through this example, where x was the number of TVs sold at a  small department store in a single day, and X can only take the value 0, 1, 2, 3,  4, 5, Once we looked at this, we could look at the actual distribution below. You  know, how many days did we sell zero TVs? Well, 90 days. How many days did  we sell one TV? 85 days, and so on and so forth. From there, we were able to  calculate the probability of any one of those occurrences, so the probability of us selling zero TVs was 0.25, because it was 90 out of 365 So now we have this  probability, but we also looked at the idea of being able to measure an average,  or again a notion of center for this distribution, so we calculated what we call the expected value, and the expected value of the number of TVs we were selling  on any given day was 1.88 TVs with a variance of 2.522 So again, notice how  we wrapped up all the concepts together. Once we sampled our data, we started looking at our data. We looked at it in terms of measures of center and spread.  Then we looked at probabilities, and then we wrapped all those concepts up  together to get an idea of a distribution of a variable, as well as an idea of  summarizing that variable in terms of center, spread, or probability of something  happening. From there, we moved into a specific example of this, specifically  the binomial distribution. Remember, the binomial distribution looks at the  probabilities of the number of successes occurring in n independent trials. Think  about the flip of a coin and how many times you were able to get heads in 10  flips of a coin, for example. Now, the binomial probability function was  comprised of two intuitive pieces. It was the number of outcomes providing 

exactly that many successes in those n trials, as well as the probability of a  particular sequence of trial outcomes with that many successes in n trials. So  we combined these together and looked at a couple of different examples, then  we moved on to the idea of continuous random variables, so again we're  generalizing even more, so with these continuous random variables, we said  they can assume any value in an interval on the real line, or really in a collection of intervals on the real line, and so the probability of a random variable  assuming a value inside of an interval was really the area underneath the curve, so we looked at this distribution, and we were looking at area underneath. It's  very similar to what we did when it came to the discrete example for zero TVs.  We just looked at essentially all the observations with zero. The problem is  there's infinite. Many observations on a continuous random variable, so here we  have to look at ranges. We looked at a couple examples of these as well. If you  remember, we looked at the uniform probability distribution. The uniform is a  common distribution where every single possible value has an equal chance of  happening, but more importantly, we also looked at the normal probability  distribution. The normal probability distribution is one of the most common and  important distributions for describing a continuous random variable. In fact, it is  the foundation of statistical inference. It helped us develop things like hypothesis testing and confidence intervals, and we also saw that it appeared in nature and  in real world data. If you remember certain characteristics of the normal  distribution, the normal distribution had some very useful characteristics. First of  all, it was perfectly symmetric. It was unimodal, basically meaning it had one big  hump of data in the middle. It is mean and median and mode were all equal to  each other. It was asymptotic to the x axis, which means that it could take any  value from negative infinity to positive infinity, even though the closer you were  to the middle of the distribution, the more where that data actually is, and the  further from the middle, the less data values we typically actually see. And  again, it's completely defined by that middle, that average, that mean, and the  standard deviation, the spread of the distribution. From there, we learned about  some great things we could do with the normal distribution, like we learned  about the empirical rule that basically told us within one standard deviation of  the mean lies about 68% of the data you have in a normal distribution. Two  standard deviations from the mean you have about 95% Three standard  deviations about 99.7% So we can see a variety of different data points here,  and what the real look of a normal distribution does. And so we could do some  very quick mathematics, if you knew what the mean and standard deviation of  your normal distribution was, you could figure out some easy values where you  think your data is going to be located. However, we then sort of saw that we  can't do this for all values inside of a normal distribution, so we converted our  normal distributions from any normal distribution that we had to a standardized  normal distribution. The value of converting these things to a standardized 

normal distribution was basically for the point that we could answer this  probability question about any single point inside of the distribution of our normal from there, what we did is we learned about the idea that we could have some  sampling error, and we started talking about sampling distributions. In fact, we  looked at the idea, if we were to take many, many, many, many, many, many,  many samples and calculate the same statistic from each one of those samples, would that statistic have a predictable pattern? And it did that with the central  limit theorem. If we were to use large samples, basically a sample size of 50 or  more, the central limit theorem said that the sampling distribution of the sample  mean is approximately normally distributed, regardless of what the population  distribution looks like, and now everything has now been wrapped up together,  right. So we talked about probabilities, we talked about center, we talked about  spread, all for setting up the foundation of a distribution. Why? Because of the  central limit theorem. The central limit theorem says that sample means follow a  very specific distribution, that normal distribution that we talked about. Now, if  we have a small sample, the sampling distribution of x bar is approximately  normal only if the population actually is normal. So, take large enough samples,  and sample means will basically approximate a normal distribution, regardless  of what the population looks like. From there we actually got the sampling  distribution of x bar, so again the sampling distribution of x bar is basically a  collection of all possible x bars from all possible samples of the same size from  your population, and we learned that the sampling distribution of x bar is  normally distributed with a mean being the overall population mean, and the  standard deviation being the population standard deviation divided by the  square root of our sample size, the bigger the sample, the smaller the standard  deviation of the mean, because of the fact that we were more exact and more  reliable in terms of our estimate, we also learned about the idea of p hat. P hat is a sample proportion estimating a population proportion. We learned how  proportions actually also follow the central limit theorem, because they're  basically like averages, so therefore they followed a normal distribution. We  learned about their expected value and their standard. Deviation, as well, once  we had the idea of what the distributions look like, then we could measure some notion of error. If we know that statistics follow a predictable pattern, wonderful.  From that predictable pattern, we know how wrong we're going to be. So that's  how we built our interval estimates, right? So we had this idea of our original  point estimate, plus or minus some margin of error that we created. Now, again,  the purpose of an interval estimate is to provide some idea of how close we  actually are, right? So, as a notion of wiggle room, really, to make people rely on our results a little bit better, we're admitting the fact that we know that there's  going to be some variability in our guess, but that variability is accounted for by  the fact that we know what patterns exist inside of data in general, and so we  talked about the idea of a confidence interval, and again, I know confidence 

intervals were a little bit weird, because confidence meant something we  probably weren't really used to. Remember, confidence intervals are just interval estimates, where we have this confidence, where confidence was the  confidence in the procedure itself. If we were to take many, many, many, many  samples, all of the same size, that each produced a different confidence interval, 95% of them would contain the truth. And we looked at that visually. Here, for  example, we have all of these sample means from the same population, and we  build confidence intervals from all these sample means. Not all of them are  actually going to capture the truth. Now, unfortunately, you don't know which one your sample is, but you're relying on the fact that more often than not, this  procedure gives you a good answer. So that's what we do. From there we were  able to actually calculate the confidence interval specifically for proportions as  well as for means. They are basically laid out in these equations that we saw.  Remember, though, when looking at the means, we did not use the normal  distribution anymore. We needed to switch over to t distribution, since we are  now estimating both mu and sigma. I know if you're going to make two  estimates, you have to add a little extra wiggle room. The t distribution does that  for us. Then, lastly, we moved into the world of hypothesis testing, and we talked about this example, or we talked about this example of a coin. You have a coin  you believe is fair, so we tested it, and we kept flipping heads over and over and over and over again, and then you started saying, hmm, well, the probability of  you getting heads this many times in a row in a fair coin is probably really small.  I no longer believe the coin is fair. Congratulations, you just did a hypothesis  test. It's just we use more formal language, like null hypothesis, test statistic, p  value, and decisions. So that's really what we do. A hypothesis test uses the  data to help us evaluate an initial claim about the parameter from a population.  And there are four main steps to hypothesis testing, like we just talked about.  We had to state our hypotheses, where the null hypothesis was our original  idea, and the alternative is what we were trying to test. The test statistics were,  we gathered all of our data to be able to do this. The p value told us the  probability we saw the data that we did, and then based on that p value, we  made a decision on our null hypothesis. We were basically trying to figure out  how likely was it that we saw a sample, for example, a sample mean as far  away from the original guess, mu naught that we actually had. Now, remember,  hypothesis tests, much like confidence intervals, are not always right, and so  there were different types of errors that we could do, right? So, sometimes we  get it right, sometimes we get it wrong. When we get it wrong, we have either a  type one or a type two error. If you think that you should reject the null  hypothesis, but you actually shouldn't have, the null hypothesis was true, you've  made a type one error. If you think that you should not reject the null hypothesis, but you actually should have, the null hypothesis was false. Then you have a  type two error. Whew, can you believe it? That's everything we covered in this 

course. Data is everywhere, but the nice part is we can do amazing things with  data. We can explore it, we can understand it, we can make inferences from it.  It's really sort of God giving us a glimpse of this world through data, however, we must use the information that we've seen here to find out things wisely and  ethically. So that lecture summarizes everything that we've talked about. That is  the end of this lecture, and I look forward to seeing you in the next one.



Last modified: Monday, June 29, 2026, 8:34 AM