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How can data be represented using various methods?

How can a two way frequency table be used to
collect data?

How can a scatter plot estimate further
predictions for the data given?

Does the representation of all data sets look
exactly the same? Explain why or why not.

How can you visually represent the difference
between data sets? How can this representation
be used to find trends?

How are correlation and causation different?

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VOCABULARY

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Box-and-Whisker Plot EXAMPLE

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Box-and-whisker plot worksheet

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Box-and-Whisker Plot worksheet

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Summarize, represent, and interpret data on a single count or measurement variable.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Interpret linear models.

Students will interpret data given various data representations.

Look at a variety of social and political data.

Interpret differences in shape, center, and spread in the context of data sets, accounting for extreme data points.

Students will be able to categorize and compare data using a two-way frequency table.

Students will interpret relative frequencies in the context of the data, including joint, marginal, and
conditional relative frequencies.

Students will recognize possible associations and trends in the data.

Students will create scatterplots.

Students compute the correlation coefficient using technology.

Students will distinguish between correlation and causation and work with situations where they are
mistakenly interchanged.

Students will fit a linear or exponential function to a scatter plot.

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PRIORITY:
S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

S.ID.5​ Summarize categorical data for two categories in two-way frequency tables. Interpret relative
frequencies in the context of the data (including joint, marginal, and conditional relative frequencies).
Recognize possible associations and trends in the data.

SID.6​ Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the
data. Use given functions or choose a function suggested by the context. Emphasize linear and
exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for scatter plots that suggest a linear association

S.ID.9​ Distinguish between correlation and causation.

SUPPORTING:
S.ID.2​ Use statistics appropriate to the shape of the data distribution to compare center (median, mean)
and spread (interquartile range, standard deviation) of two or more different data sets.

S.ID.3​ Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

S.ID.7​ Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the
context of the data.

S.ID.8​ Compute (using technology) and interpret the correlation coefficient of a linear fit.

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data distribution
shape
center
spread
median
statistical
representation
outlier
data set
data trends
quantitative and categorical data
dot plot
Histogram
box plot
data distribution
statistical
representation
data trends
two-way frequency table
relative frequency table
shape
center
spread
outlier
joint
marginal
and
conditional
relative
frequencies
Bivariate data
Scatter plot
Slope
outlier
Causation
Correlation
statistical
representation
data set
data trends
quantitative and
categorical data,
standard deviation
correlation coefficient

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Two-way frequency tables and Venn diagrams (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Two way tables

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Given a Venn diagram or information about a relationship, create a two-way frequency table.

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Scatter plots: studying, shoe size, and test scores

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Aubrey wanted to see if there's a connection between the time a given exam takes place and the average score of this exam. She collected data about exams from the previous year. Plot the data in a scatter plot. And let's see, they give us a couple of rows here. This is the class. Then they give us the period of the day that the class happened. And then they give us the average score on an exam. And we have to be a little careful with the study-- maybe there's some correlation depending on what subject is taught during what period. But let's just use her data, at least, just based on her data, see if-- well, definitely do what they're asking us, plot a scatter plot, and then see if there is any connection. So let's see. On the horizontal axis, we have Period. And on this investigation, this exploration she's doing, she's trying to see, well, does the period of the day somehow drive average score? So that's why Period is on the horizontal axis. And the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. So let's plot each of these points. Period 1, average score 93-- right over there. Period 6, 87. Oh, that's not the right place, and then we can move it if we want-- 87, right over there. Period 2, 70. Period 4, 62-- right over there. Period 4 and 86, that's right over there. Period 1, 73. Period 3, average score of 73 as well. Period 1, 80, average score of 80. And then Period 3, average score of 96. So there we go. And it doesn't really seem like there's any obvious trend over here. So let's make sure that we got this right. And we did.

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Aubrey wanted to see if there's a connection between the time a given exam takes place and the average score of this exam. She collected data about exams from the previous year. Plot the data in a scatter plot. And let's see, they give us a couple of rows here.

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Constructing a scatter plot

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Practice plotting points to construct a scatter plot.

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Correlation coefficient intuition

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Match correlation coefficients to scatterplots to build a deeper intuition behind correlation coefficients.

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Sal explains the intuition behind correlation coefficients and does a problem where he matches correlation coefficients to scatter plots.

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I have this article right here from WebMD. And the point of this isn't to poke holes at WebMD. I think they have some great articles and they have some great information on their site. But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article, and the title of the article says "Eating breakfast may beat teen obesity." So they're already trying to create this cause-and-effect relationship. The title itself says if you eat breakfast then you're less likely-- or you won't be obese. You're not going to be obese. So the title right there already sets up this. That eating breakfast may beat teen obesity. And then they tell us about the study. "In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis-Saint Paul, Minnesota." And I won't talk too much about this. It looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over broad audience, that they were able to control for a lot of variables. But then they go on to say, "The researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates." And to some degree that first-- "than those who skipped breakfast." And to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains. You eat syrup, you eat waffles-- that all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Fruit Loops or whatever else. But we'll let that be right here. "In addition, regular breakfast eaters seemed more physically active then the breakfast skippers." So over here they're once again trying to create this other cause-and-effect relationship. Regular breakfast eaters seemed more physically active than the breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say "Over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers." So you could-- they're telling us that breakfast skipping-- this is the implication here-- is more likely, or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active. Breakfast skipping will make you obese. So you just say then, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause some to become more active? Does it really tell us that breakfast skipping can make you overweight or make it obese? Or, it is more likely, are they showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality, which is what it seems like they're implying. Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese. So it looks like they are kind of implying causality. They're implying cause and effect, but really what the study looked at is correlation. The whole point of this is to understand the difference between causality and correlation because they're saying very different things. Causality versus correlation. And, as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So the study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast and breakfast correlates with activity-- that all of these correlate. What they don't say-- and there's no data here that lets me know one way or the other-- what is causing what or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast-- and I'm not telling you that you should. I have no data one way or the other-- maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure. So they're trying to say, look, if you have breakfast it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows? Likewise they say breakfast skipping, or they're implying breakfast skipping, can cause obesity. But maybe it's the other way around. Maybe people who have high body fat-- maybe, for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. And these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude. That we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning, so you're more likely to eat breakfast. And obviously being physically active also makes it so that you burn calories. You have more muscle. So that you're not obese. So notice if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person-- let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast? Maybe that'll make things worse. And based on that, that the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. Maybe something other than physical activity. Maybe you have sleep, maybe people who sleep late and they're not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment-- or they have to run to school in the case of students-- and maybe that's why they skip breakfast. So once again, if you find someone that's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it will become even worse because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast. Maybe there's more stress in the-- who knows what it might be? And so when you have poverty maybe you're more likely to skip breakfast and maybe when there's poverty, and maybe you have two-- both your parents are working and the kids have to make their own dinner and whatever else-- maybe they also eat less healthy at all times of day and then that leads to obesity. So once again in this situation, if this is the reality of things, just telling someone to also eat breakfast regardless of what that breakfast is, even if it's Fruit Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner maybe by breakfast time you're not hungry still because you've binged so much on breakfast. So you skip breakfast. And this also leads to obesity. But once again, if this is the actual reality, doing the advice that that article's saying might actually be a bad thing. If you need an unhealthy dinner and then force yourself to eat a breakfast when you're not hungry, that might make the obesity even worse. So the whole point of this video isn't to say that the implications from that article are necessarily wrong. The important thing is to just realize that it might be wrong. And that just because you saw this correlation with the data, it doesn't mean that eating breakfast is going to somehow magically fight obesity.

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I have this article right here from WebMD. And the point of this isn't to poke holes at WebMD. I think they have some great articles and they have some great information on their site.

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Sal interprets a trend line that shows the relationship between study time and math test score for Shira.

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Estimating the line of best fit exercise

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Sal solves a problem where he has to estimate the line of best fit for a scatter plot.

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The slope of a line is rise over run. Learn how to calculate the slope of the line in a graph by finding the change in y and the change in x.

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Estimating equations of lines of best fit, and using them to make predictions

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Estimating equations of lines of best fit, and using them to make predictions

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Practice estimating the equation of a line of best fit through data points in a scatter plot. Then, use the equation to make a prediction.

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Eyeballing the line of best fit

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Given a random assortment of points, draw a line of best fit through them.

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Unit 4- Scatter plots and Two-way tables Review

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Introduction to Scatter Plots (12 Scenarios each for positive, negative, and no correlation)

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Estimating the line of best fit exercise

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Sal solves a problem where he has to estimate the line of best fit for a scatter plot.

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How Ice Cream Kills! Correlation vs. Causation

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Construct a box-and-whisker plot

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The owner of a restaurant wants to find out more about where his patrons are coming from. One day, he decided to gather data about the distance in miles that people commuted to get to his restaurant. People reported the following distances traveled. So here are all the distances traveled. He wants to create a graph that helps him understand the spread of the distances-- this is a key word-- the spread of distances and the median distance that people traveled or that people travel. What kind of graph should he create? So the answer of what kind of graph he should create, that might be a little bit more straightforward than the actual creation of the graph, which we will also do. But he's trying to visualize the spread of information. And at the same time, he wants the median. So what a graph captures both of that information? Well, a box and whisker plot. So let's actually try to draw a box and whisker plot. And to do that, we need to come up with the median. And we'll also see the median of the two halves of the data as well. And whenever we're trying to take the median of something, it's really helpful to order our data. So let's start off by attempting to order our data. So what is the smallest number here? Well, let's see. There's one 2. So let me mark it off. And then we have another two. So we've got all the 2's. And then we have this 3. Then we have this 3. I think we've got all the 3's. Then we have that 4. Then we have this 4. Do we have any 5's? No. Do we have any 6's? Yep. We have that 6. And that looks like the only 6. Any 7's? Yep. We have this 7 right over here. And I just realized that I missed this 1. So let me put the 1 at the beginning of our set. So I got that 1 right over there. Actually, there was two 1's. I missed both of them. So both of those 1's are right over there. So I have the 1's, 2's, 3's, 4's, no 5's. This is one 6. There was one 7. There's one 8 right over here. And then, let's see, any 9's? No 9's. Any 10s? Yep. There's a 10. Any 11s? We have an 11 right over there. Any 12s? Nope. 13, 14? Then we have a 15. And then we have a 20 and then a 22. So we've ordered all our data. Now it should be relatively straightforward to find the middle of our data, the median. So how many data points do we have? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. So the middle number is going to be a number that has 8 numbers larger than it and 8 numbers smaller than it. So let's think about it. 1, 2, 3, 4, 5, 6, 7, 8. So the number 6 here is larger than 8 of the values. And if I did the calculations right, it should be smaller than 8 of the values. 1, 2, 3, 4, 5, 6, 7, 8. So it is, indeed, the median. Now, when we're trying to construct a box and whisker plot, the convention is, OK, we have our median. And it's essentially dividing our data into two sets. Now, let's take the median of each of those sets. And the convention is to take our median out and have the sets that are left over. Sometimes people leave it in. But the standard convention, take this median out. And now, look separately at this set and look separately at this set. So if we look at this first bottom half of our numbers essentially, what's the median of these numbers? Well, we have 1, 2, 3, 4, 5, 6, 7, 8 data points. So we're actually going to have two middle numbers. So the two middle numbers are this 2 and this 3, three numbers less than these two, three numbers greater than it. And so when we're looking for a median, you have two middle numbers. We take the mean of these two numbers. So halfway in between two and three is 2.5. Or you can say 2 plus 3 is 5 divided by 2 is 2.5. So here we have a median of this bottom half of 2.5. And then the middle of the top half, once again, we have 8 data points. So our middle two numbers are going to be this 11 and this 14. And so if we want to take the mean of these two numbers, 11 plus 14 is 25. Halfway in between the two is 12.5. So 12.5 is exactly halfway between 11 and 14. And now, we've figured out all of the information we need to actually plot or actually create or actually draw our box and whisker plot. So let me draw a number line, so my best attempt at a number line. So that's my number line. And let's say that this right over here is a 0. I need to make sure I get all the way up to 22 or beyond 22. So let's say that's 0. Let's say this is 5. This is 10. That could be 15. And that could be 20. This could be 25. We could keep going-- 30, maybe 35. So the first thing we might want to think about-- there's several ways to draw it. We want to think about the box part of the box and whisker essentially represents the middle half of our data. So it's essentially trying to represent this data right over here, so the data between the medians of the two halves. So this is a part that we would attempt to represent with the box. So we would start right over here at this 2.5. This is essentially separating the first quartile from the second quartile, the first quarter of our numbers from the second quarter of our numbers. So let's put it right over here. So this is 2.5. 2.5 is halfway between 0 and 5. So that's 2.5. And then up here, we have 12.5. And 12.5 is right over-- let's see. This is 10. So this right over here would be halfway between, well, halfway between 10 and 15 is 12.5. So let me do this. So this is 12.5 right over here. So that separates the third quartile from the fourth quartile. And then our boxes, everything in between, so this is literally the middle half of our numbers. And we'd want to show where the actual median is. And that was actually one of the things we wanted to be able to think about when the owner of the restaurant wanted to think about how far people are traveling from. So the median is 6. So we can plot it right over here. So this right here is about six. Let me do that same pink color. So this right over here is 6. And then the whiskers of the box and whisker plot essentially show us the range of our data. And I can do this in a different color that I haven't used yet. I'll do this in orange. So essentially, if we want to see, look, the numbers go all the way up to 22. So they go all the way up to-- so let's say that this is 22 right over here. Our numbers go all the way up to 22. And they go as low as 1. So 1 is right about here. Let me label that. So that's 1. And they go as low as 1. So there you have it. We have our box and whisker plot. And you can see if you have a plot like this, just visually, you can immediately see, OK, what is the median? It's the middle of the box, essentially. It shows you the middle half. So it shows you how far they're spread or where the meat of the spread is. And then it shows, well, beyond that, we have the range that goes well beyond that or how far the total spread of our data is. So this gives a pretty good sense of both the median and the spread of our data.

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Box and whisker plot: how to construct (video) | Khan Academy

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The owner of a restaurant wants to find out more about where his patrons are coming from. One day, he decided to gather data about the distance in miles that people commuted to get to his restaurant. People reported the following distances traveled. So here are all the distances traveled.

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Constructing a box-and-whisker plot

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More practice making box plots to summarize data sets.

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using the graphing calculator in secondary mathematics

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Box-and-whisker plot

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This video shows all the steps to creating a Box and Whisker Plot: finding the median of a list of numbers, calculating the values for Quartiles I, II, and III, and identifying extreme values.

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Histograms

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Histograms are a neat way to organize massive amounts of related data for easy comparison and reference. If only history were this tidy.

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Correlation Coefficient

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Unit 4 Final Review